# Public-input continuation: calculations completed and physical questions retained

**132 additional executable components passed 65221 scoped checks.** The [coverage ledger](public-input-ledger.json) records what each completes, and the [figure gallery](/research/metastable/full/public-inputs/index.html) exposes its data, assumptions and provenance. This continuation closes several concrete omissions in the first E24–E35 batch: released posterior propagation, a source Bonn baseline, thermal inverse weak rates, solved formation and charged few-body kernels, a measured-data abundance likelihood, full-EOS stellar contraction and real observational/isomer inputs.

The central physical milestone is still unmet: **one viable persistent field, consistently matched nuclear interactions and currents, an accessible threshold crossing, formation and survival, and a physical spacetime history**. Some remaining tasks need a new physical response or empirical covariance; others are substantial solver extensions with public equations. The ledger distinguishes those categories rather than calling every unfinished calculation “missing data.” Earlier numerical results and failed runs remain preserved.

## Executed components

| Completed component | Executed scope | Frozen evidence |
|---|---|---|
| W04-screened-continuation-C2 | All141 finite screened-contact cells and certificate, with unchanged source-width gate failed | [7 scoped checks](runs/20261008T083447928357Z-W04-screened-continuation-C2/manifest.json) |
| W10-Gaussian-upper-neutral-block-cost | One complete5778-column upper neutral-even represented block; strict2e-6 promotion excluded by a represented witness | [27 scoped checks](runs/20261008T084340391725Z-W10-Gaussian-upper-neutral-block-cost/manifest.json) |
| W21-nonidentifiability | Bounded Re-only rank and chronology/inventory non-identifiability alternatives | [11 scoped checks](runs/20261008T075722716663Z-W21-nonidentifiability/manifest.json) |
| W29-six-case-tau120-production | Six fixed planar cases,57000 steps and726 frames through dimensionless tau120 with return-controlled collision | [22 scoped checks](runs/20261008T082632719274Z-W29-six-case-tau120-production/manifest.json) |
| W20-molecular-forcing-tube-method | A conditional 10^7-second population tube is certified for the original case03 first domain using the fixed96-node atomic source graph, all41 populations and unchanged nonunit mass. All three bound families and eight preregistered radii were evaluated; radius1 fails the inward-temperature condition. | [6 scoped checks](runs/20261005T074140278056Z-W20-molecular-forcing-tube-method/manifest.json) |
| W08-helium-boundfree-channels-method | All25 labeled Coulomb states and66 spin-resolved bound-free channels have independently reviewed angular, threshold, radial, thermal and absolute-tail controls. | [2506 scoped checks](runs/20261005T103445506986Z-W08-helium-boundfree-channels-method/manifest.json) |
| W09-pending27-independent-synthesis | All27 pending projected-pair candidates and216 fresh full512 Hamiltonian products have an independently checked saved-evidence synthesis;23 signed policies pass and four fail. | [231 scoped checks](runs/20261005T110634262292Z-W09-pending27-independent-synthesis/manifest.json) |
| W13-quarkmass-sensitivities-method | Complete source-specific local responses for88 poles,10 thresholds and28 regular current supports | [1336 scoped checks](runs/20261005T054523775716Z-W13-quarkmass-sensitivities-method/manifest.json) |
| W20-molecular-mixing-lower-method | Eight conditional lower transition kernels and mass-zero contraction bounds for the full41-level operator | [16 scoped checks](runs/20261005T063142719956Z-W20-molecular-mixing-lower-method/manifest.json) |
| W20-molecular-stationary-enclosure-method | Full41-level stationary population and parameter-response interval enclosures at a point and throughout one rectangle | [31019 scoped checks](runs/20261005T053638481437Z-W20-molecular-stationary-enclosure-method/manifest.json) |
| W27-triaxial-production-combine | Six independently reviewed full-m triaxial nonlinear cases through T200 with two amplitudes and finite angular/radial/time/box refinements | [2354 scoped checks](runs/20261005T033150137042Z-W27-triaxial-production-combine/manifest.json) |
| W02-quartet-full-phase-synthesis | Four independently reviewed neutral LO quartet full-phase integrals at3/5 MeV and600/1200 MeV cutoffs, retaining l=0..4 | [35 scoped checks](runs/20261005T014721094724Z-W02-quartet-full-phase-synthesis/manifest.json) |
| W08-helium-radiative-census-synthesis | Complete independently reviewed closed 25-level radiative/collision 30-case transient census | [427 scoped checks](runs/20261004T211201832899Z-W08-helium-radiative-census-synthesis/manifest.json) |
| W05-fourbody-conversion-benchmark | Conversion-derived full three-body regulator limit and quantified rejection of the separated-dimer completion | [456 scoped checks](runs/20261004T202054925963Z-W05-fourbody-conversion-benchmark/manifest.json) |
| W14-axion-static-census-production | Complete LO/NLO aligned three-sector vacuum root census and18 full ten-coordinate static supports | [165 scoped checks](runs/20261004T191641970503Z-W14-axion-static-census-production/manifest.json) |
| W02-quartet-bin-synthesis | Neutral LO quartet coherent l0..4 finite bins and same-H spin-weighted mixture at four declared cases | [103 scoped checks](runs/20261004T041239868634Z-W02-quartet-bin-synthesis/manifest.json) |
| E10-spectral-dense-continuation-production | Numerically stationary free45-component fit in the original bounded coadd likelihood | [1152 scoped checks](runs/20261004T035729512575Z-E10-spectral-dense-continuation-production/manifest.json) |
| P4-axion-pion-NLO | Strict NLO vacuum pion pole gaps and neutral alignment at the frozen discrete-sector input | [1153 scoped checks](runs/20261004T032304378062Z-P4-axion-pion-NLO/manifest.json) |
| W10-Gaussian-phase-polished-continuation-production | Bounded polished same-H Gaussian schedule capture | [144 scoped checks](runs/20261004T025604096938Z-W10-Gaussian-phase-polished-continuation-production/manifest.json) |
| W28-fermi-fission-production | Three same-functional asymptotically isolated binary-fission endpoint comparisons | [267 scoped checks](runs/20261004T021005808326Z-W28-fermi-fission-production/manifest.json) |
| W08-helium-collision-precision-production | Closed25-level HeII collision-only fixed-bath dynamics at20 declared temperature/energy cases | [2537 scoped checks](runs/20261003T222353411802Z-W08-helium-collision-precision-production/manifest.json) |
| W05-fourbody-projection-benchmark | Finite-source-mode physical descent rejection and exact static/Feshbach identities | [107 scoped checks](runs/20261003T223509360913Z-W05-fourbody-projection-benchmark/manifest.json) |
| W04-charged-odd-audit | Finite-screen quartet total-L0 pp odd waves 1 and 3 with independently normalized coherent breakup | [115 scoped checks](runs/20261003T210732675805Z-W04-charged-odd-audit/manifest.json) |
| W13-quarkmass-bracketed-survey | 110 declared source-specific inverse-NLO pole cases and ten continuous numerical thresholds | [410 scoped checks](runs/20261003T204902133504Z-W13-quarkmass-bracketed-survey/manifest.json) |
| W05-STM-attached-contact | Static-action spin/normal-order coefficients and quantified global spectator-support obstruction | [93 scoped checks](runs/20261003T192457580506Z-W05-STM-attached-contact/manifest.json) |
| W13-quarkmass-matching | Source-specific NLO isosymmetric quark-mass ERP and leading current matching with distinct barred intercept | [690 scoped checks](runs/20261003T192357984026Z-W13-quarkmass-matching/manifest.json) |
| W03-paired-curvature-budget-production | Curvature-aware actual-charge stopping and independently diagnosed cold reference with bounded B1 seed census | [375 scoped checks](runs/20261003T184423636738Z-W03-paired-curvature-budget-production/manifest.json) |
| W05-STM-contact-derivation | Exact isolated-DD spectral obstruction and conditional finite-block identities | [56 scoped checks](runs/20261003T182902516499Z-W05-STM-contact-derivation/manifest.json) |
| W02-coherent-breakup-synthesis | Four-case same-H neutral LO coherent ell0..4 CM differential points and finite bins | [90 scoped checks](runs/20261003T070802616726Z-W02-coherent-breakup-synthesis/manifest.json) |
| W07-atomic-VP-production | First-order static electronic vacuum-polarization energy and atomic threshold components | [57 scoped checks](runs/20261003T064522393857Z-W07-atomic-VP-production/manifest.json) |
| W03-paired-volume-continuation-A1 | Bounded same-Hamiltonian separate-mean-charge finite L4 Gaussian seed census | [394 scoped checks](runs/20261003T062634112832Z-W03-paired-volume-continuation-A1/manifest.json) |
| W03-paired-volume-continuation-B2 | Bounded same-Hamiltonian separate-mean-charge finite L4 Gaussian seed census | [394 scoped checks](runs/20261003T062901596918Z-W03-paired-volume-continuation-B2/manifest.json) |
| W27-axisymmetric-production | Conditional finite-amplitude axisymmetric phi6 backreaction and finite-time convergence controls | [85 scoped checks](runs/20261003T023731930584Z-W27-axisymmetric-production/manifest.json) |
| W20-molecular-kinetics-production | Conditional time-dependent CO level kinetics, H2 proton-mediated spin conversion and gas energy with a prescribed Planck reservoir | [212 scoped checks](runs/20261003T020428615811Z-W20-molecular-kinetics-production/manifest.json) |
| W23-GNSS-fresh-production | Preregistered seven-day frozen-model processed-clock validation | [1120 scoped checks](runs/20261003T020128144049Z-W23-GNSS-fresh-production/manifest.json) |
| W23-GNSS-temporal | Training-only temporal model comparison and conditional processed-product uncertainty; original third day is reused, not a new blind validation. No receiver response, Gaussian significance, event probability or physical exclusion. | [17 scoped checks](runs/20261003T014345831882Z-W23-GNSS-temporal/manifest.json) |
| E28-STM-fourbody-operator | Actual fermionic FY pair kernels and embedded STM 3+1 contact controls | [300 scoped checks](runs/20261003T012657182307Z-E28-STM-fourbody-operator/manifest.json) |
| E21-domain-walls-production | Gradient-resolved conditional finite-box wall and bulk-excitation dynamics with canonical inertia | [65 scoped checks](runs/20261003T005017200095Z-E21-domain-walls-production/manifest.json) |
| W03-cold-charge-increment-production | Precision-controlled finite cold-charge density and Helmholtz increments at the frozen L4A reference | [58 scoped checks](runs/20261003T004614613211Z-W03-cold-charge-increment-production/manifest.json) |
| W08-hydrogenic-two-photon-production | Source-normalized approximate2s↔1s channel coupled to finite ordinary E1/continuum populations and pair-energy ledger | [436 scoped checks](runs/20261003T002752613991Z-W08-hydrogenic-two-photon-production/manifest.json) |
| W03-paired-charge-response-B | Analytic coupled Gaussian charge response and bounded finite-volume curvature repair | [27 scoped checks](runs/20261002T235807925724Z-W03-paired-charge-response-B/manifest.json) |
| E31-stellar-microphysics-interface | Ordinary source-authenticated microphysics adapter and independent native thermodynamic/nuclear accounting controls | [294 scoped checks](runs/20261003T000352021101Z-E31-stellar-microphysics-interface/manifest.json) |
| E29-matched-transfer-subset | Same-STM transfer thermal coverage and inverse-amplitude detailed balance with closed-subsystem Saha evolution | [158 scoped checks](runs/20261002T233210243361Z-E29-matched-transfer-subset/manifest.json) |
| E28-charged-asymptotic-quartet | LO charged elastic point-Coulomb-reference screening candidates at one declared k/cutoff/partial wave | [67 scoped checks](runs/20261002T180944135969Z-E28-charged-asymptotic-quartet/manifest.json) |
| E28-charged-asymptotic-doublet | LO charged elastic point-Coulomb-reference screening candidates at one declared k/cutoff/partial wave | [37 scoped checks](runs/20261002T181401505440Z-E28-charged-asymptotic-doublet/manifest.json) |
| W06-np-M1-production | Same-source-draw ordinary LWL np M1 scalar contact matching and finite-energy transfer with both sign branches | [491 scoped checks](runs/20261002T173014347683Z-W06-np-M1-production/manifest.json) |
| W08-hydrogenic-cascade-production | Finite-level charge-neutral ordinary Coulomb E1 populations and energy ledger | [287 scoped checks](runs/20261002T173536290414Z-W08-hydrogenic-cascade-production/manifest.json) |
| W03-paired-volume-L3 | Colder and broader np paired starts with full coupled sufficient stability tests and same-ensemble normal competitors | [111 scoped checks](runs/20261002T163445174412Z-W03-paired-volume-L3/manifest.json) |
| E28-fourbody-triplet | Microscopic same-force frozen-triton neutron J1-positive-parity elastic continuum and authenticated unpolarized L0 spin average | [50 scoped checks](runs/20261002T161505505732Z-E28-fourbody-triplet/manifest.json) |
| E06-scalar-hydrodynamics | Conservative 1D compressible ideal-gas/scalar shock dynamics with reciprocal same-functional work, analytic controls and grid/time refinement | [37 scoped checks](runs/20261002T151355513739Z-E06-scalar-hydrodynamics/manifest.json) |
| W19-material-transport-production | Conservative spatial two-phase ten-species transport of authenticated saved single-cohort wind and helium-exhaustion release inputs | [25 scoped checks](runs/20261002T152911955527Z-W19-material-transport-production/manifest.json) |
| E28-doublet-higher-synthesis | Reviewed neutral LO doublet spectator l1..4 outgoing breakup flux at both3/5MeV and600/1200MeV cutoffs | [19 scoped checks](runs/20261002T160255747887Z-E28-doublet-higher-synthesis/manifest.json) |
| E17-fermi-angular | Same-functional fixed-charge nonspherical Hessian with translation zero-mode control and conditional all-angular-momentum positivity argument | [78 scoped checks](runs/20261002T154910351865Z-E17-fermi-angular/manifest.json) |
| E08-history-contract | Explicit spacetime/ensemble provenance contract, frozen component mismatch audit, and correctly paired conditional detection from actual saved E08 histories | [132 scoped checks](runs/20261002T150703457054Z-E08-history-contract/manifest.json) |
| W21-isotope-exchange | Conservative finite-time parent/daughter exchange, conditional time-ordered history kernels and metadata-gated identifiability | [43 scoped checks](runs/20261002T150051218776Z-W21-isotope-exchange/manifest.json) |
| E17-fermi-profile-synthesis | Self-consistent spherical finite-bias fixed-charge scalar/relativistic Thomas-Fermi profiles with same-functional attraction and controlled constrained radial Hessian | [17 scoped checks](runs/20261002T143916605596Z-E17-fermi-profile-synthesis/manifest.json) |
| W23-GNSS-clocks | Actual ESA processed GPS-clock sampling, reference-invariant covariance and heldout product-template controls | [187 scoped checks](runs/20261002T062354709266Z-W23-GNSS-clocks/manifest.json) |
| W08-hydrogenic-production | Matched ordinary hydrogenic 1s photoionization/radiative recombination with stimulated emission, Saha/Planck balance, charge-neutral kinetics and energy ledger | [508 scoped checks](runs/20261002T144143151210Z-W08-hydrogenic-production/manifest.json) |
| E23-snapshot-likelihood | Conditional binomial snapshot-probability fit, finite-sample bootstrap and classification/calibration sensitivity for68 released atomic observations | [29 scoped checks](runs/20261002T062007216298Z-E23-snapshot-likelihood/manifest.json) |
| E28-fourbody-scattering | Microscopic frozen-target finite-rank neutron-triton elastic continuum | [73 scoped checks](runs/20261002T051954553720Z-E28-fourbody-scattering/manifest.json) |
| W20-molecular-cooling | CO-H2 statistical equilibrium with CMB, gas/photon energy balance and fixed-composition optically thin controls | [1643 scoped checks](runs/20261002T061100483981Z-W20-molecular-cooling/manifest.json) |
| E21-isotropic-domains | Isotropic expanding Fourier-shell Gaussian noise and finite-time spatial spectator Langevin controls on declared nonperiodic sample points | [38 scoped checks](runs/20261002T051103348444Z-E21-isotropic-domains/manifest.json) |
| E29-ordinary-weak-capture | Ordinary continuum electron-proton capture with exact tree-level point-current recoil, external-state and spin normalization, plus a separately labelled published muonic-hydrogen spin/radiative benchmark. | [56 scoped checks](runs/20261002T033607509184Z-E29-ordinary-weak-capture/manifest.json) |
| W27-W30-nonspherical | Linear l=1,2,3 perturbations of one ordinary canonical phi6 radial benchmark and analytic TT/source-energy controls | [93 scoped checks](runs/20261002T033859036026Z-W27-W30-nonspherical/manifest.json) |
| E28-charged-threebody | Finite-screen truncated charged quartet three-body outgoing flux pilot | [11 scoped checks](runs/20261002T034622487037Z-E28-charged-threebody/manifest.json) |
| E28-charged-cut | Exact charged-pair spectral cut and driven outgoing response with screening-removal controls | [141 scoped checks](runs/20261002T030901547406Z-E28-charged-cut/manifest.json) |
| E35-projected-canonical-comparison | Same-Hamiltonian L2 exact-charge canonical variational bounds with projected pure-Gaussian and projected kinetic Gibbs trials, plus exact internal-ground and local-quartet projectors | [20 scoped checks](runs/20261002T031242741570Z-E35-projected-canonical-comparison/manifest.json) |
| W26-oxygen-multireference | Ordinary O2 spin-pure valence CASSCF ordering and diatomic bond curves with independent minimal-basis FCI | [143 scoped checks](runs/20261002T032239764501Z-W26-oxygen-multireference/manifest.json) |
| E10-native-exposures | All 17 public ESPRESSO exposures acquired and conditional native-pixel registration for six Fe II lines | [259 scoped checks](runs/20261002T032837087378Z-E10-native-exposures/manifest.json) |
| P4-discrete-axion | Exact discrete-sector LO/NLO vacuum, planar walls, LO off-valley and ideal-matter consistency | [132 scoped checks](runs/20261002T030017444607Z-P4-discrete-axion/manifest.json) |
| E28-doublet-exclusive | Independent neutral LO doublet l0 spin/isospin summed outgoing breakup flux | [83 scoped checks](runs/20261002T004220697119Z-E28-doublet-exclusive/manifest.json) |
| E35-paired-minima | Stationary A/B L3 T2 np-paired restricted Gaussian local minima, certified by a positive all-channel enlarged Hessian | [7 scoped checks](runs/20261002T005429768233Z-E35-paired-minima/manifest.json) |
| E29-nn-magnetic-capture | Conditional finite-photon one-body nn capture benchmark with spin and independent multipole controls | [180 scoped checks](runs/20261001T201455286282Z-E29-nn-magnetic-capture/manifest.json) |
| E28-pd-quartet | Charged quartetS/P/D/F below-breakup elasticLO/strictNLO screening controls | [1328 scoped checks](runs/20261001T195822977198Z-E28-pd-quartet/manifest.json) |
| E35-canonical-correlations | Exact canonical L2 four-nucleon thermal density correlations and local quartet occupancy with independent free controls | [107 scoped checks](runs/20261001T200953928144Z-E35-canonical-correlations/manifest.json) |
| E29-atomic-populations | Ordinary helium charge kinetics from public CI/RR/DR fits with conditional cooling histories | [99 scoped checks](runs/20261001T200517723231Z-E29-atomic-populations/manifest.json) |
| E29-atomic-inclusive | Inclusive finite-mass boosted spectator spectral and kinematic response | [63 scoped checks](runs/20261001T195600924014Z-E29-atomic-inclusive/manifest.json) |
| E35-cold-pairing | Independently audited colder normal-root pp/nn/np stability at frozen mean charges | [2 scoped checks](runs/20261001T200436164207Z-E35-cold-pairing/manifest.json) |
| E24-basis-audit | Independent spin and distributional audit of released central-basis transformation | [36 scoped checks](runs/20261001T193750915211Z-E24-basis-audit/manifest.json) |
| E29-network-contract | Mass compatibility gate and consistent conditional four-species transfer evolution | [42 scoped checks](runs/20261001T194240334391Z-E29-network-contract/manifest.json) |
| E28-exclusive-partial | LO quartet l0..4 independently integrated exclusive flux at3,5MeV | [181 scoped checks](runs/20261001T193548336872Z-E28-exclusive-partial/manifest.json) |
| E24-posterior | Correlated public ordinary-phase posterior propagated across three regulators | [63 scoped checks](runs/20260929T214425497398Z-E24-posterior/manifest.json) |
| E25-Bonn | Public Bonn A/B/C baseline and tensor-threshold reconstruction | [33 scoped checks](runs/20260929T203530228825Z-E25-Bonn/manifest.json) |
| E26-thermal | All thermal crossings and reverse allowed one-body weak rates | [710 scoped checks](runs/20260929T193546300964Z-E26-thermal/manifest.json) |
| E28-scattering | Neutral LO/NLO nd scattering with a triton-fitted force; LO rearrangement amplitudes | [71 scoped checks](runs/20260929T193546599340Z-E28-scattering/manifest.json) |
| E29-current | Ordinary np radiative capture with calibrated two-body EM currents, source table and thermal controls | [19 scoped checks](runs/20260929T201030373477Z-E29-current/manifest.json) |
| E29-source-rates | Retrieved historical He3 input and executed Eq.15 using evaluated reference reactions | [105 scoped checks](runs/20260929T194839837161Z-E29-source-rates/manifest.json) |
| E29-transfer | Thermal rearrangement from solved LO three-body amplitudes | [46 scoped checks](runs/20260929T201313664523Z-E29-transfer/manifest.json) |
| E30-source-BBN | Evaluated-analogue BBN with full thermal weak conversion and rate controls | [42 scoped checks](runs/20260929T201633081739Z-E30-source-BBN/manifest.json) |
| E30-compensation | Published correlated binding and photodestruction compensation mechanism | [5 scoped checks](runs/20260929T201029195741Z-E30-compensation/manifest.json) |
| E30-likelihood | Conditional abundance likelihood with recomputed correlated errors | [64 scoped checks](runs/20260929T203401050040Z-E30-likelihood/manifest.json) |
| E30-STM-BBN | BBN driven by one solved three-body reaction, with a controlled S-wave tail bracket | [21 scoped checks](runs/20260929T213258904696Z-E30-STM-BBN/manifest.json) |
| E28-response-consistency | Bound-state consistency audit of the solved-transfer BBN substitution | [19 scoped checks](runs/20260929T223004324566Z-E28-response-consistency/manifest.json) |
| E28-charged | Charged trinucleon bound states with all LO Coulomb kernels and a calibrated perturbative NLO electromagnetic force | [48 scoped checks](runs/20260929T224825117361Z-E28-charged/manifest.json) |
| E28-pd-scattering | Charged proton-deuteron continuum phases with strict NLO insertions, neutral-limit and screening controls | [78 scoped checks](runs/20260929T225832801441Z-E28-pd-scattering/manifest.json) |
| E28-fourbody | Antisymmetrized LO three/four-nucleon bound states, held-out helium predictions and correlated nn response | [69 scoped checks](runs/20260929T231113457755Z-E28-fourbody/manifest.json) |
| E28-fourbody-refinement | Shorter-range three/four-body regulator study with larger independent bases | [30 scoped checks](runs/20260929T232225018002Z-E28-fourbody-refinement/manifest.json) |
| E28-spin-complete | Three/four-body predictions with a complete internal-spin variational space | [96 scoped checks](runs/20260929T233754706450Z-E28-spin-complete/manifest.json) |
| E28-spin-validation | Reserved independent full-spin variational energies, radii and weak currents | [10 scoped checks](runs/20260930T004330382653Z-E28-spin-validation/manifest.json) |
| E28-threshold-refinement | Infinite-volume dineutron calibration with recomputed correlated few-body responses | [18 scoped checks](runs/20260930T005420065273Z-E28-threshold-refinement/manifest.json) |
| E29-trinucleon-weak | Microscopic one-body trinucleon weak strengths with independent binding and current controls | [28 scoped checks](runs/20260930T005525250915Z-E29-trinucleon-weak/manifest.json) |
| E29-trinucleon-thermal | Thermal trinucleon conversion with frozen masses/currents and finite electron populations | [88 scoped checks](runs/20260930T040332257433Z-E29-trinucleon-thermal/manifest.json) |
| E29-axial-contact | Ordinary axial-contact calibration transferred to reserved pp fusion and conditional nn weak overlaps | [28 scoped checks](runs/20260930T005709594476Z-E29-axial-contact/manifest.json) |
| E30-PRIMAT | Independent primary-author BBN implementation and two-backend baseline controls | [21 scoped checks](runs/20260929T214035686025Z-E30-PRIMAT/manifest.json) |
| E31-contraction | Self-consistent pure-H contraction and degeneracy/cooling endpoints | [27 scoped checks](runs/20260929T193751335229Z-E31-contraction/manifest.json) |
| E32-coupled | Source coupled-channel deuteron threshold excursions with continuum and work controls | [38 scoped checks](runs/20260929T220308671706Z-E32-coupled/manifest.json) |
| E32-continuum-check | Reserved continuum and time refinement of the fastest coupled passage | [7 scoped checks](runs/20260929T220837446601Z-E32-continuum-check/manifest.json) |
| E33-data | Public measured production limits reconstructed with extraction controls | [10 scoped checks](runs/20260929T200528580709Z-E33-data/manifest.json) |
| E34-measured | Measured-isomer energy response and conditional M1 radiative lifetime sensitivity | [16 scoped checks](runs/20260929T200930246324Z-E34-measured/manifest.json) |
| E34-environment | Released thorium-isomer pumping/decay fits and thermal environmental-quenching population control | [17 scoped checks](runs/20260930T181150182339Z-E34-environment/manifest.json) |
| E35-virial | Finite-temperature dilute EOS across a shallow binding threshold | [47 scoped checks](runs/20260929T201314827757Z-E35-virial/manifest.json) |
| E35-tightbinding | Published alpha-cluster locality diagnostic independently reconstructed from spin traces and lattice kernels | [24 scoped checks](runs/20260929T234708649318Z-E35-tightbinding/manifest.json) |
| E35-lattice-manybody | Source many-nucleon CCSD benchmarks with independent exact few-body energies and finite-volume alpha-breakup diagnostics | [37 scoped checks](runs/20260930T001827978384Z-E35-lattice-manybody/manifest.json) |
| E35-NO2B-audit | Exact separation of the public NO2B Hamiltonian and coupled-cluster approximation errors | [32 scoped checks](runs/20260930T022824997049Z-E35-NO2B-audit/manifest.json) |
| E35-lattice-HF | Finite-range public A/B mean-field calculations through oxygen-16 in two boxes and two spatial starts | [116 scoped checks](runs/20260930T010050322377Z-E35-lattice-HF/manifest.json) |
| E35-HF-stability | Independent orbital-curvature classification of every computed A/B mean-field branch | [64 scoped checks](runs/20260930T011129639631Z-E35-HF-stability/manifest.json) |
| E35-lattice-matching | Independent projected two-nucleon operator and energy audit of the public finite-range A/B Hamiltonians | [48 scoped checks](runs/20260930T015504482144Z-E35-lattice-matching/manifest.json) |
| E35-HF-species | Independent spin/isospin-species real-orbital stability of all 32 source A/B Hartree-Fock branches | [196 scoped checks](runs/20260930T015717334889Z-E35-HF-species/manifest.json) |
| E35-HF-unrestricted | Charge-conserving complex spinor stability of all 32 finite-range A/B mean-field branches | [204 scoped checks](runs/20260930T020233243143Z-E35-HF-unrestricted/manifest.json) |
| E35-HF-paths | Explicit normalized paths and escape-barrier upper bounds between all pairs of finite-nucleus mean-field minima | [128 scoped checks](runs/20260930T020842489784Z-E35-HF-paths/manifest.json) |
| E35-finite-CCSD | Finite-range A/B correlated alpha nuclei with independently exact complex two-body controls | [68 scoped checks](runs/20260930T030807570905Z-E35-finite-CCSD/manifest.json) |
| E35-finite-clusters | Correlated finite-range A/B nuclei through oxygen-16 with reserved alpha and reference-overlap controls | [141 scoped checks](runs/20260930T164749835215Z-E35-finite-clusters/manifest.json) |
| E35-finite-exact | Exact finite-range A/B nuclei in all small-box momentum sectors and controlled CCSD method comparisons | [204 scoped checks](runs/20260930T183857233297Z-E35-finite-exact/manifest.json) |
| E28-neutral-breakup | Neutral LO l=0..4 continuum across breakup with inclusive loss, contour and independent PV controls | [2029 scoped checks](runs/20260930T195637530757Z-E28-neutral-breakup/manifest.json) |
| E29-atomic-capture | Correlated atomic capture with analytic Coulomb final states, resolved atomic thresholds and charge-neutral initial ion populations | [287 scoped checks](runs/20260930T200800704743Z-E29-atomic-capture/manifest.json) |
| E35-thermal-ensembles | Finite-temperature A/B Gaussian mean-field ensembles in two volumes with competing starts, and complete canonical two/three-nucleon traces | [1440 scoped checks](runs/20260930T194538217202Z-E35-thermal-ensembles/manifest.json) |
| E35-canonical-alpha | Complete canonical finite-range A/B four-nucleon thermal traces over all 14,400 fixed-charge states | [126 scoped checks](runs/20260930T194712301119Z-E35-canonical-alpha/manifest.json) |
| E28-pd-partial-waves | Charged doublet P/D/F spectator waves below breakup with LO and strict NLO Coulomb-subtracted phase shifts | [267 scoped checks](runs/20260930T205654839089Z-E28-pd-partial-waves/manifest.json) |
| E35-thermal-density | Matched-density thermal A/B finite boxes with complete charge-conserving Gaussian stability and followed negative modes | [946 scoped checks](runs/20260930T210235602482Z-E35-thermal-density/manifest.json) |
| E29-atomic-mass | Finite-nuclear-mass correlated atomic thresholds, contact response and exclusive spectator recoil diagnostics | [105 scoped checks](runs/20260930T210506521193Z-E29-atomic-mass/manifest.json) |
| E28-exclusive-breakup | Independently integrated, absolutely normalized neutral quartet l=0 exclusive breakup flux | [85 scoped checks](runs/20260930T213850623642Z-E28-exclusive-breakup/manifest.json) |
| E35-thermal-pairing | All pp/nn/np anomalous thermal curvature of 32 matched-density normal minima | [418 scoped checks](runs/20260930T213334223420Z-E35-thermal-pairing/manifest.json) |

## Correlated ordinary nuclear inputs: E24

The released [Bub et al. posterior](https://github.com/bubjason/beft) is frozen at commit `0a8f4d8090f46d46d3a3b1807c5fb76cf1acdd21`. Nine NLO-full, NLO-reduced and N3LO-reduced model/regulator combinations provide **225,000 singlet predictions and 1,728 coupled deuteron solves**. The singlet scattering solver is independently refined; representative bound states are recalculated with half the radial spacing. Parameter columns remain paired across posterior samples. Shuffling those columns is an explicit counterexample showing that marginal parameter errors cannot replace covariance.

The source included deuteron binding and nn scattering in its calibration, so their recovery is **not independent held-out validation**. At R=2 fm, N3LO-reduced binding has a parameter-only median of 2.3678 MeV and 95% interval 2.2376–2.5379 MeV. Adding the stated source Gaussian EFT truncation model gives 2.0807–2.6955 MeV. Each selected joint draw supplies its own breakdown scale and error coefficient, retaining their correlation with the couplings. The NLO-full R=2 fm prediction is much broader after truncation, -1.5598 to 5.5887 MeV. Negative Gaussian error-model draws are not evidence that our phase contains unbound deuterons. These intervals illustrate the distinction between fitted-parameter uncertainty and omitted-order uncertainty, not a precision prediction for another field background.

A separate audit finds a mismatch between the released standard-basis column labels/marginals and the paper's stated basis transformation. The channel-basis coefficients are used directly. No silent column relabeling or inferred row pairing repairs that source discrepancy. Source-current consistency, precise pp electromagnetic corrections and finite-background response remain separate requirements. [Source analysis](https://arxiv.org/abs/2408.02480).

## The Bonn A/B/C benchmark: E25

Machleidt's public Fortran potential core is compiled after changes limited to obsolete text declarations. The [published A/B/C parameter table](https://arxiv.org/abs/nucl-th/9307028) is used without refitting. Relativistic momentum-space bound and zero-energy scattering equations are solved independently. The ordinary bindings are **2.224589, 2.224676 and 2.224495 MeV**, within several electronvolts of the source values. The reported D-state probabilities are representation-dependent outputs, not observables.

The tensor subtraction is independently projected from the [published operators](https://arxiv.org/abs/1806.11270), including the relative S/D Fourier phase and upper-spinor factors. The zero-energy Birman–Schwinger crossing avoids inversion of a singular scattering matrix. Infinite-volume unitarity occurs at multipliers **0.42986, 0.57957 and 0.64197**. The [2024 paper](https://arxiv.org/abs/2403.14474) instead quotes approximate a=−100 fm points; our exact −100-fm targets are **0.39854, 0.55972 and 0.62562**, versus its 0.4029, 0.5624 and 0.6278. At the paper's quoted multipliers, our calculated scattering lengths are **−115.94, −115.64 and −115.46 fm**. Momentum and angular refinement do not erase this difference, but the paper describes its lengths only as approximately −100 fm and gives no precision budget. The comparison therefore does **not** establish an inconsistency with a precision source datum. Its exact lengths and subtraction implementation would permit a stronger reproduction test. This qualification is distinct from the older Lee OBE discrepancy. An adjustable tensor multiplier remains a Hamiltonian sensitivity, not a fundamental field prediction.

## Few-body consistency, currents and formation: E26, E28, E29

The neutral three-channel STM equations now have an explicit principal-value subtraction and outgoing-wave residue, fixing the absolute scattering normalization. The ordinary triton fixes the LO force and its perturbative NLO range correction once per cutoff. Reserved neutron–deuteron scattering gives doublet/quartet lengths **0.4025/5.0891 fm at LO** and **0.9000/6.0661 fm at NLO** for the finest high-cutoff calculation. These do not reproduce the usual measured references near 0.65/6.35 fm at precision accuracy. The retained discrepancy is a physical truncation/model issue; mesh and unitarity checks are not substitutes for it. [Three-body source](https://arxiv.org/abs/1406.1359), [quartet reference](https://arxiv.org/abs/nucl-th/9710073).

The charged bound-state calculation now includes all LO bubble, box and triangle Coulomb kernels, together with the resummed pp propagator. An analytic subtraction handles the zero-photon-mass logarithm, checked against independent adaptive quadrature. The neutral limit agrees with the previous independent solver, reciprocity holds, and momentum, angular and energy-derivative refinements pass. The [published LO helium-3 curve](https://arxiv.org/abs/1405.7961) is reconstructed in its own isospin-symmetric input convention; the retained 6–13-keV discrepancy is comparable to its graphical line width and is not described as exact source agreement.

With physical nn inputs, tritium is fitted once. Omitting the **NLO electromagnetic three-body term** gives helium-3 bindings of 8.1754, 8.6087 and 9.0946 MeV at cutoffs 600, 1200 and 1800 MeV. Fitting that required term restores 7.718043 MeV by construction, reproducing the [published renormalization mechanism](https://arxiv.org/abs/1402.5441). This fitted datum supplies no independent prediction or field derivative. The stored NLO operator is a single perturbative insertion, including the gauged effective-range vertex; it is never resummed into a spurious deep pole.

An nn-only perturbation leaves the npp sector unchanged in this diagnostic, but the neutral NLO response is not fixed by our ordinary calibration. At inverse nn length 0.1 fm⁻¹, keeping the ordinary NLO force fixed gives tritium binding of about 11.36–12.10 MeV over those cutoffs. This is an **unresolved short-distance response**, not a converged alternative-phase prediction. The need for additional input when varying scattering length at NLO has an established [short-range EFT precedent](https://arxiv.org/abs/1106.3837). The following additional calculations supply continuum, four-body bound-state and one-body-current controls while retaining that missing physical response.

The charged extension also now solves **proton–deuteron S-wave scattering below breakup**. No scattering datum is fitted: the strong and electromagnetic three-body terms come from the ordinary binding calibration. A separately solved composite-deuteron Coulomb phase is subtracted using the same screening and propagator conventions. At relative momenta 20, 35 and 50 MeV, the NLO nuclear phases at a 1200-MeV cutoff are approximately **−4.458°, −16.005° and −28.612°**. Momentum and screening-window changes are below 0.040° and 0.016° respectively, and the independent neutral solver agrees, including its NLO insertion. The 600-to-1200-MeV NLO cutoff differences are larger, up to 0.714°; they remain visible. These predictions complete the charged S-wave component; the subsequent doublet P/D/F extension is reported below. Charged breakup, four-body reactions and a changed-field current model remain unfinished.

**A separate coordinate-space calculation now solves fully antisymmetrized three- and four-nucleon bound states.** Correlated Gaussian basis functions carry explicit np-triplet and np/nn/pp-singlet projectors, point pp Coulomb and one scalar three-body interaction. Two-body observables and tritium binding fix the ordinary parameters independently at each range. Neither helium binding is fitted. This is a local Gaussian LO Hamiltonian, not the momentum-cutoff STM convention; outputs from these families cannot be substituted for one another. [Variational method and Volkov benchmark](https://arxiv.org/abs/nucl-th/9508023), [Gaussian few-body precedent](https://arxiv.org/abs/1605.03898).

The first variational implementation used one internal spin seed. Particle antisymmetrization alone did **not** make that internal space complete: a subsequent current calculation exposed an artificially enforced relation between Fermi and Gamow–Teller strengths. The corrected calculation spans all six fixed-charge S=1/2 internal vectors for three nucleons, and all twelve S=0 vectors for four. Off-diagonal spin matrix elements, Hermiticity, total-spin eigenvalues and exact nesting of the earlier restricted space are checked. The earlier runs remain labeled **restricted variational controls**. The table below and current predictions use the expanded space, including a fresh ordinary-tritium calibration at each regulator.

The source Volkov Hamiltonian gives calculated energies −0.545921, −8.464885 and −30.416990 MeV for A=2,3,4, consistent with the source's quoted energy precision. An independent radial solver also checks the two-body energy and radius. Our converged two-body rms radius is 3.59427 fm, differing from the source table's 3.44 fm; that discrepancy is retained rather than claiming every source observable was recovered. Independent random seeds, analytic matrix-element checks and basis growth distinguish numerical error from the following physical spread:

| Gaussian range (fm) | Predicted He3 binding (MeV) | Predicted He4 binding (MeV) | Last He4 basis change (MeV) |
|---:|---:|---:|---:|
| 0.8 | 7.887625 | 24.429610 | 0.000742 |
| 1 | 7.863045 | 24.472588 | 0.000280 |
| 1.25 | 7.819234 | 25.108009 | 0.000083 |
| 1.5 | 7.771132 | 26.480468 | 0.000035 |
| 2 | 7.684844 | 31.299452 | 0.000005 |
| 2.5 | 7.625257 | 36.816819 | 0.000003 |

Measured He3/He4 bindings are 7.718043/28.295674 MeV. The range scan does not by itself establish a controlled regulator limit or reproduce these held-out data. The last basis changes are much smaller than the model differences. Variational energies are upper bounds **at fixed interactions**; refitting tritium in a larger spin space changes the interaction too, so the two separately calibrated series are not ordered by that variational theorem. Higher-order, tensor, electromagnetic and three-body terms require a consistent calculation, not a hypothetical-phase refit chosen to recover our nuclei.

A reserved R=2-fm calculation starts all three spatial/spin spaces again with independent random seeds and holds the already fitted Hamiltonian fixed. Its maximum binding difference is 1.42984e-05 MeV, maximum radius difference 3.73927e-07 fm, and maximum Fermi/GT difference 2.61497e-08. This tests the new complete-spin optimizer independently of its inherited restricted-space starting functions.

The weak-overlap calculation exposed another numerical-input problem: calibrating the shallow dineutron in a 60-fm radial box did not fix its infinite-volume binding. The nominal 0.1-MeV case actually had an exterior pole at 0.10125582 MeV. A separate refinement now fits the decaying exterior boundary condition, checks two matching radii, and compares independent finite-difference boxes through 240 fm with a mesh extrapolation. The original finite-box results remain archived; the response and weak-current calculations below use the corrected couplings. Ordinary states are unchanged.

For the R=2-fm family, changing only nn binding gives the following shifts, with the ordinary three-body term fixed:

| Imposed nn binding (MeV) | Tritium binding shift (MeV) | He4 binding shift (MeV) |
|---:|---:|---:|
| 0.1 | 1.824508 | 2.573279 |
| 2.5 | 6.924111 | 9.101755 |

The npp nucleus remains unchanged under this particular perturbation. These are **correlated finite-regulator Hamiltonian responses**, distinct from the STM response above; neither is a field prediction.

The archived trinucleon wavefunctions also supply **actual Fermi and Gamow–Teller matrix elements**. The one-body operators act on the full antisymmetric states; no weak datum is fitted. The exact SU(4) result, basis changes and an independent phase-space quadrature are checked. Positron Coulomb repulsion is included with the opposite sign to electron attraction. [One-body and higher-order current conventions](https://arxiv.org/abs/2405.07889), [additional axial-current input](https://arxiv.org/abs/1611.10004).

| R=2-fm nn input | Fermi magnitude | GT / √3 | Tritium β− Q (MeV) | He3 β+ Q (MeV) |
|---|---:|---:|---:|---:|
| ordinary | 0.999572 | 0.980803 | -0.014623 | -1.007375 |
| Bnn=0.1 MeV | 0.994524 | 0.982207 | -1.839131 | 0.817133 |
| Bnn=2.5 MeV | 0.968640 | 0.968561 | -6.938734 | 5.916736 |

This exposes a useful ordinary-phase failure: the LO binding error can exceed the roughly 19-keV tritium beta endpoint, and some regulator choices incorrectly close that decay. A result that passes variational convergence therefore need not predict the correct weak stability. Rates calculated with empirical nuclear Q values are stored separately as **current checks**, never substituted into the Hamiltonian's own threshold predictions. Under the R=2-fm nn-only perturbations, helium-3 beta-plus decay can open while tritium beta decay closes. Allowed one-body rates are archived; radiative/recoil and atomic corrections and competing reactions remain necessary for a precision or complete survival calculation.

**Thermal trinucleon conversion is now calculated from these same frozen masses and currents.** Four crossing pairs include electron capture, positron capture, both neutrino absorptions, both allowed beta directions and their inverse three-body processes. The electron channels see the helium-3 charge; the positron channels see the tritium charge. Each process and its inverse share their Coulomb factor. Equal temperatures recover reverse/forward=exp[(μe−μν−Δ)/T], with no spin-degeneracy factor because both nuclei have J=1/2. Both vacuum limits independently reproduce the previous adaptive Coulomb phase integrals. A failed first evaluation exposed cancellation in `1−f` for a nearly filled electron state; direct Fermi hole populations restore detailed balance without relaxing the criterion.

| R=2-fm interaction | T (MeV) | μe (MeV) | T→He3 rate (s⁻¹) | He3→T rate (s⁻¹) | Relaxation time (s) |
|---|---:|---:|---:|---:|---:|
| Ordinary | 0.05 | 0 | 1.22493e-06 | 5.97918e-11 | 816335 |
| Ordinary | 0.05 | 0.75 | 3.85797e-07 | 6.15611e-05 | 16142.9 |
| Ordinary | 0.5 | 0 | 0.0406677 | 0.0150696 | 17.9413 |
| Bnn=0.1 MeV | 0.05 | 0 | 3.57904e-15 | 0.00122963 | 813.252 |
| Bnn=0.1 MeV | 0.05 | 0.75 | 5.65434e-21 | 0.00635049 | 157.468 |
| Bnn=0.1 MeV | 0.5 | 0 | 0.00526937 | 0.0750515 | 12.4501 |
| Bnn=2.5 MeV | 0.05 | 0 | 9.26663e-56 | 6.2736 | 0.159398 |
| Bnn=2.5 MeV | 0.05 | 0.75 | 2.86234e-62 | 6.33481 | 0.157858 |
| Bnn=2.5 MeV | 0.5 | 0 | 1.72412e-05 | 6.60128 | 0.151485 |

The relaxation time is 1/(forward+reverse) for an isolated two-state population. The corresponding stationary helium fraction is forward/(forward+reverse), conditional on a fixed bath. The electron chemical potential includes rest energy. The μe=0.75-MeV cases are explicit environmental controls, not a derived plasma density or a charge-neutral evolutionary model. Separate colder-neutrino tables leave thermal equilibrium. These results retain the ordinary LO threshold error and the one-body current approximation; the bound atomic capture extension below supplies another channel, while energy/charge backreaction and competing strong reactions are still needed for an actual survival history. [Plotted temperatures and populations](/research/metastable/full/public-inputs/E23-trinucleon-plasma.json).

**A further current calculation fits a normalized Gaussian two-body axial contact to ordinary tritium and reserves pp fusion as a prediction.** The source empirical product gA GT/√3=1.213±0.002 is adjusted for the Fermi strength calculated here. The coefficient multiplies the explicitly archived spin/isospin operator and normalized coordinate regulator; it is not equated to a coefficient in a different subtraction scheme. With gA fixed, the quoted error propagates only the empirical axial-product uncertainty. It omits Hamiltonian and current truncation error.

| Range (fm) | Fitted axial coefficient (fm³) | One-body pp Λ² | With fitted contact pp Λ² |
|---:|---:|---:|---:|
| 0.8 | -0.386442 ± 0.021610 | 6.804167 | 6.729758 |
| 1 | -0.574989 ± 0.032107 | 6.942765 | 6.871571 |
| 1.25 | -0.867809 ± 0.048051 | 7.121220 | 7.051909 |
| 1.5 | -1.235537 ± 0.067415 | 7.304779 | 7.235604 |
| 2 | -2.251097 ± 0.117906 | 7.689532 | 7.616993 |
| 2.5 | -3.740068 ± 0.186829 | 8.095682 | 8.016478 |

The pp continuum includes Coulomb repulsion and uses the same pair potential as the bound-state calibration. Four low energies determine the zero-energy overlap, with independent radial, outer-boundary and matching-radius refinements. The [Solar Fusion II](https://arxiv.org/abs/1004.2318) reference Λ²=7.035 is a theoretical evaluation, not a directly measured pp-fusion datum. None of the pp results is fitted to it. The remaining range dependence is evidence against treating this partial correction as a precision prediction. It is an LO Hamiltonian with one current improvement, **not** a complete NLO calculation. The same ordinary fitted coefficient is also applied to two conditional dineutron overlaps, with its change in a different field left undetermined. The calculation therefore tests transfer between reactions while keeping calibration, reserved prediction and missing field response distinct.

The p+(nn)→n+d transfer rate is calculated from the same actual LO amplitude. At Bnn=0.1 MeV and T=0.005 MeV it is approximately **6.0×10⁻¹⁶ cm³/s**; the independently reconstructed strong-rate analogy is substantially larger. The omitted above-breakup S-wave contribution is bounded by unitarity for each temperature. This calculation improves on a dimensional rate estimate but retains a fixed ordinary three-body force, neutral LO kinematics and no higher partial waves.

Ordinary n+p radiative capture now includes [Rupak's M1 and E1 currents](https://arxiv.org/abs/nucl-th/9911018), calibrated to declared thermal and photodisintegration inputs. Reserved Table I energies agree within the rounding/model comparison tolerance. The ordinary np amplitude is never copied into a forbidden nn S-wave 0→0 one-photon transition. This supplies a current benchmark; it does not produce a new allowed nn capture operator.

All four crossed allowed nn/deuteron weak processes are integrated with exact electron mass, Fermi factors, thermal lepton distributions, final-state blocking and the calculated radial overlap. Equal-temperature rates satisfy the spin-weighted detailed-balance relation; the zero-temperature limit recovers the prior vacuum lifetime. A 3.2-MeV dineutron can have its displayed vacuum beta decay closed and still convert through a thermal environment. The actual BBN integration uses PRyMordial's electron-neutrino temperature history. Recoil, microscopic two-body weak-current response and energy backreaction are outside this heavy-nucleus allowed approximation.

## Evaluated strong rates and source corrections: E29–E30

The historical ENDF/B-IV He3(n,p)T reference is recovered, including **5327 barn at 0.0253 eV**. DD and DT reference reactions use the explicitly identified [ENDF/B-VIII.1 release](https://www.nndc.bnl.gov/endf-releases/). ENDF interpolation laws are implemented, including charged-particle INT=6. Coulomb and phase-space factors are removed at tabulated nodes before interpolation; doing that after a coarse raw interpolation creates spurious amplitudes. The reduced-amplitude low-energy continuation is explicit.

The Kneller analogy includes four reversible strong transfer channels and derives all Q values from one mass table. In particular, He3+(nn)→alpha+n has Q=20.58−Bnn MeV; the displayed historical equation omits the initial He3 binding. The modern DT reference retains its resonance, whereas the exact historical nonresonant coefficients have not been recovered. A reserved DD-channel analogy predicts only **0.216–0.284 times** the evaluated cross section. This motivates broad rate scenarios and prevents a precision interpretation of the novel rates. [Historical dineutron paper](https://arxiv.org/abs/astro-ph/0312388).

The [MacDonald–Mullan compensation model](https://arxiv.org/abs/0904.1807) is also executed at its declared square-well and rate-formula scope. Its printed wells bind nn and pp at G≈1.04347 and 1.06317, while stronger deuterium binding moves its formation epoch ahead of those pairs. Exact Planck rather than Boltzmann photodestruction changes the chosen B=0.15 MeV, T9=4 example by about 14.9%. The printed ordinary deuteron well gives **2.30861 MeV**, an 84-keV baseline mismatch, and the source beta-endpoint sign differs from mass conservation by **1.56467 MeV**. These differences are retained. This reproduces the compensation mechanism and source formulas, not the paper's complete final hydrogen abundance; its phenomenological new capture normalization is not promoted to a spin-allowed microscopic amplitude.

## Actual primordial populations and observations: E30

PRyMordial integrates nine species with four reversible strong transfers and all crossed thermal weak conversions. An analytic augmented Jacobian prevents numerical failure from a disconnected zero-abundance ninth species. Zero-extension recovery, baryon conservation, positivity, integrated flow checks, an independent Radau solve and tighter tolerances are archived. Thermal quadrature is checked against adaptive integration; a separate abundance run tests a 256-point generalized quadrature. Numerical solver error is much smaller than the novel-amplitude uncertainty.

The independent [primary-author PRIMAT implementation](https://github.com/CyrilPitrou/primat) is pinned at commit `4bf97d5082eee54b9df50d88f43182bb651fea80`. Both C and Python backends reproduce its published default abundances. With the same declared baryon-density and neutron-lifetime inputs as our PRyMordial baseline, PRIMAT gives Y=0.24694730 and 10⁵ D/H=2.457430; the relative code differences are 0.0249% and -0.0320%. Independent backend, small/large-network and refinement checks pass. Remaining rate-library and thermal conventions are preserved, so this is not a claim of identical complete inputs or an independent dineutron-network implementation.

| Case | Helium nucleon fraction | D/H × 10⁵ | Peak nn per baryon |
|---|---:|---:|---:|
| standard-recovery | 0.24688574 | 2.458217 | 0 |
| B0.1 | 0.24688580 | 2.458217 | 1.1384e-14 |
| B2.0 | 0.24707744 | 2.457781 | 4.3536e-06 |
| B2.5 | 0.25685316 | 2.274468 | 0.00039753 |
| B3.0 | 0.27113773 | 0.730732 | 0.0050158 |
| B3.2 | 0.27601198 | 0.009709 | 0.010844 |
| B2.5-scale0.2 | 0.25299150 | 2.401573 | 0.00090779 |
| B2.5-scale5.0 | 0.26093026 | 1.980778 | 0.0001667 |
| B3.2-vacuum-weak | 0.27601679 | 0.009704 | 0.010845 |
| B2.5-refined | 0.25685332 | 2.274469 | 0.00039753 |

Replacing p+(nn)↔n+d by the independently solved three-body amplitude gives these additional integrations:

- L600-N144-lower: Y=0.25685467, 10⁵ D/H=2.255628.
- L1200-N144-lower: Y=0.25685470, 10⁵ D/H=2.255440.
- L1200-N144-upper: Y=0.25685463, 10⁵ D/H=2.255440.
- L1200-N192-lower: Y=0.25685468, 10⁵ D/H=2.255440.

Those integrations retain the other three explicit analogue transfers. A zero-to-unitarity upper bracket on the unresolved above-breakup S-wave contribution, two cutoffs and momentum/thermal refinement expose the remaining numerical and tail dependence. They **connect a computed formation amplitude to an actual abundance network**; they do not complete the other four-body or radiative amplitudes.

An additional consistency calculation exposes a stronger limitation. **The same fixed ordinary three-body force used for that transfer amplitude predicts tritium binding of 11.767154 MeV when Bnn=2.5 MeV**, a shift of 3.285354 MeV. The one-rate substitution retains 8.4818 MeV in its network. It is therefore a **hybrid sensitivity experiment, not a matched realization of that Hamiltonian**. Momentum refinement resolves the mismatch; changing cutoff from 600 to 1200 MeV changes this binding by only about 0.0033 MeV. No hypothetical-phase refit is used to hide it.

Holding He3 binding fixed solely to expose the mass-balance consequence, tritium's displayed beta Q value becomes -3.266777 MeV and He3 electron capture has the opposite sign. Neither that fixed-He3 illustration nor a Q sign supplies the missing charged-wavefunction matrix element or decay rate. At Bnn=0.1 MeV the same model already shifts tritium by about 0.917 MeV: the earlier negligible-abundance-change result for weak nn binding belongs to the fixed-other-bindings model, not every correlated nuclear change. A consistent new network needs the remaining binding and current responses together.

The observation experiment uses [LBT 2026](https://arxiv.org/abs/2601.22238) He4/H=0.08146±0.00058 and [Cooke et al.](https://arxiv.org/abs/1710.11129) log10(D/H)=−4.5974±0.0052. Actual number ratios from the final network are compared; the helium CMB rest-mass convention is not substituted for the observational definition. Independent Gaussian summary likelihoods are declared. Twelve ordinary nuclear-rate nuisances, the frozen neutron-lifetime prior and the BBN-independent baryon-density prior are propagated by **separate network response calculations in ordinary BBN and the 2.5-MeV analogue model**, with held-out joint perturbations testing local linearity. These priors and rate-correlation conventions follow the [PRyMordial publication](https://pmc.ncbi.nlm.nih.gov/articles/PMC11266446/).

With that ordinary-input covariance, conditional χ² for the two abundance summaries is **2.242** for ordinary BBN and **94.875** for the 2.5-MeV analogue model. These are specific model/data comparisons, not a significance for “a dineutron exists.” Unknown hypothetical-amplitude covariance, correlated larger-nucleus changes and a physical field history are not supplied by an ordinary-rate prior. No universal 2.5-MeV bound or interpolated exclusion limit is claimed; a phase absent from our observed primordial region is not constrained by this likelihood.

## Self-consistent stellar contraction: E31

The earlier polytropic ignition roots now have an actual stellar counterpart: the public [Adams–Grohs contraction-only limiting experiment](https://arxiv.org/abs/1612.04741), run with MESA's full ordinary EOS, opacity, ionization and electron degeneracy. Initial material is pure hydrogen, with no unexplained CNO seed and all nuclear abundance changes and heating disabled. Each track reaches a recorded cooling endpoint after central temperature falls below half its attained peak in a degenerate core.

| Initial mass (M☉) | Peak central temperature (MK) | Age at peak (Myr) | Cooling endpoint (Myr) |
|---:|---:|---:|---:|
| 0.25 | 9.3849 | 716.882 | 3483.387 |
| 1 | 45.3934 | 254.474 | 948.639 |
| 5 | 329.9024 | 5.837 | 6.109 |
| 1 (refined) | 45.4209 | 251.070 | 934.747 |

Doubling temporal/spatial resolution changes the one-solar-mass peak temperature by about **0.061%**, endpoint age by **1.5%**, and integrated radiated energy by **0.20%**. Contraction supplies light even with nuclear heating disabled. These are not predictions for every universe with unbound deuterium, nor biological lifetimes. A complete alternative stellar network still needs the common field-dependent weak/strong reactions and appropriate EOS/opacity response; later explosions and ejecta remain separate.

## Coupled-channel passage through the threshold: E32

The transition study now propagates the **full source Bonn B S/D Hamiltonian**, including the discretized continuum, along a prescribed 1→0.4→1 tensor excursion. The minimum lies below the independently verified infinite-volume unitarity crossing, 0.57957. Each smooth ramp lasts the stated interval, with a fixed ℏ/MeV hold between ramps; ℏ/MeV is about 6.58×10⁻²² seconds. Both endpoint Hamiltonians are identical, but a wavefunction that spread into the continuum need not return to the bound state when the coupling returns.

| One-way ramp (ℏ/MeV) | Bound fraction after return | Supplied work (MeV) |
|---:|---:|---:|
| 0.02 | 0.230530 | 5.654540 |
| 0.2 | 0.200913 | 4.270318 |
| 2 | 0.112847 | 2.822178 |

The final bound-state projector is separated from intermediate overlap with the original deuteron. Energy change is reconciled against work supplied by the time-dependent interaction. The complete instantaneous Hamiltonian is exponentiated at each midpoint; an earlier split propagator failed its work check and remains archived. The 80-point continuum was also insufficient for the intermediate ramp. Finer momentum/time calculations, an independently solved adaptive ODE control, a commuting cycle and the exact sudden limit test the replacement. These are finite-duration unitary calculations, not evidence of irreversible decay, thermal capture or hysteresis.

The reserved fastest-passage calculation gives a 256-point returned fraction of **0.229580**. Its momentum-grid difference is 0.000949618 in absolute probability and its subsequent time-step difference is 2.43e-08. These are numerical refinement differences, not a physical uncertainty distribution for an alternative field.

The passage time and background energy source are **prescribed**, not derived from a viable field configuration. A stationary bound endpoint alone therefore cannot prove survival through a history; conversely, the computed disruption does not establish that such a history occurs in nature. Weak/radiative channels and self-consistent field backreaction remain outside this controlled Hamiltonian experiment.

## Measurements, genuine isomers and collective thermodynamics: E33–E35

The [JLab tritium electrodisintegration search](https://arxiv.org/abs/2109.14524) supplies real experimental limits beyond the earlier synthetic-identifiability exercise. Native PDF vectors reproduce the published efficiency and five exclusion curves, with an explicit coordinate calibration and digitization bound. At about 2.2 MeV, the two-sigma cross-section-ratio limit is **0.008699**, consistent with the paper's rounded 0.009. The limit weakens sharply near threshold as efficiency decreases. A production-ratio limit requires a production amplitude before it becomes a binding-model constraint; missing event data and response covariance prevent an independent raw-event likelihood.

The isomer extension uses [measured Th229 radius and quadrupole changes](https://www.nature.com/articles/s41467-025-64191-7). An independently integrated spheroidal Coulomb model yields coefficients −485.095 and 11.339 MeV and **Kα=6104.7±2305.0** from the declared input errors. This is consistent in scope with the source's linearized rounded result; unknown octupole differences are tested separately. For a homogeneous photon-coupled scalar and fixed reduced M1 current, explicit coupling and photon-energy dependence give dlnτ/dlnα=−1−3K≈**-18315.1**. This is a conditional radiative response, not an absolute lifetime or an arbitrary scalar matrix element between different angular momenta. Internal conversion and microscopic-current response remain environment/model inputs.

**The isomer branch now also tests formation and survival with released measurements.** The [2024 primary study](https://www.nature.com/articles/s41467-024-49631-0) releases its plotted data through [Figshare](https://doi.org/10.6084/m9.figshare.25919569.v1). The original download route failed; the public API supplied files whose upstream MD5 hashes were verified. The ROOT macros are parsed as data, never executed. All 89 in-range background-subtracted decay bins, including negative values, enter the Gaussian fit; the overflow entry is saved separately. Linear amplitudes are profiled analytically, lifetimes numerically, and an independent optimizer and direct population ODE check the result.

The fitted **mean lifetimes** are 56.489330 s under irradiation and 622.486751 s in the dark, reproducing the source macro fits. Their half-lives are 39.155 and 431.475 s. The ratio is 11.020; under constant pumping and loss, the inferred additional loss rate is 0.016096 per second. This population timescale includes the irradiation environment and is not a vacuum nuclear lifetime. Exact finite-bin averaging leaves the fitted dark timescale unchanged when its amplitude is free. Allowing a residual constant background changes the dark mean lifetime to 739.832 s, while excluding the first 200 s gives 599.343 s. These are sensitivity checks, not reasons to select a preferred lifetime after observing it. Source-wide systematic errors and unreleased covariance are not reconstructed from plot points.

The [2025 temperature study](https://arxiv.org/abs/2505.03852v2) supplies a second control. Fitting all nine printed lifetime summaries to τ(T)=641/[1+β exp(−Ea/kT)] gives **β=28.210±10.887 and Ea=0.033279±0.006460 eV**, consistent with the source's rounded values; χ²=5.638 for 7 degrees of freedom under the declared independent-error model. Celsius is converted to kelvin. The unquenched input 641±4 s is a mean lifetime, not a half-life; separate refits at both error limits and leave-one-temperature-out predictions are retained. No anomalous temperature is discarded.

Using that lifetime curve in N(t)=Rτ[1−exp(−t/τ)] and fitting only one overall production normalization to the twelve 900-s populations leaves a residual χ²=44.482 for a nominal 11 degrees of freedom. This conditional residual diagnostic excludes uncertainty in the transferred curve and cross-dataset covariance; it is not a joint exclusion significance. The visible temperature structure motivates a fuller carrier/trap and detection model. The separate room-temperature pumping table gives 47.704 s rather than the printed 45.9±21.8 s. Its 0.083-source-sigma central difference remains unresolved; that table is not claimed as an exact fit reproduction. The first run's stricter central-value comparison and the subsequent table-schema failure are preserved, with independent numerical controls replacing neither the data nor the source values.

These ordinary environmental changes provide a necessary comparison for any proposed background-induced isomer response. They complete measured population/decay and thermal-rate controls, while a microscopic quenching current, a specified field coupling and material history remain separate requirements. [Data, fits and sensitivity controls](/research/metastable/full/public-inputs/E26-isomer-environment.json).


The collective extension evaluates the **Beth–Uhlenbeck second virial coefficient**, combining the shallow bound pole and continuum rather than counting only the newly bound pair. The coefficient is continuous across the threshold and recovers the analytic unitary limit b₂=**0.53033009**. The retained dilute domain has positive compressibility; 232 rows are excluded by the declared truncation criterion. A two-body binding crossing therefore does not by itself imply a dilute thermodynamic jump in this model. This calculation complements the earlier independently solved lattice two-body sector. It does not replace new alpha-containing dense many-body ensembles, finite-size scaling, interfaces or nucleation. [Virial source](https://arxiv.org/abs/nucl-th/0507064).

The [published alpha-cluster locality test](https://arxiv.org/abs/1602.04539) is also independently executed. Explicit direct and exchange spin/isospin traces, a lattice smearing convolution and an independent Fourier calculation reconstruct the compact-alpha A/B potentials without refitting any source parameter. At the nearest separation, interaction A gives about −7.08 MeV; interaction B gives −30.33 MeV, including about −26.26 MeV from its additional local term. All twenty A/B markers are extracted from the source SVG, with a maximum potential discrepancy of 0.0259 MeV, below the declared figure-resolution tolerance. This is a source-kernel reproduction. The compact-cluster formula excludes the origin and does not include alpha deformation; inserting an arbitrary core would not recover the original many-body phase diagram. New many-alpha ensembles and interface dynamics remain distinct work.

**Public many-body code also permits actual A=8,12,16 calculations.** [NuLattice](https://github.com/NuLattice/NuLattice), pinned at `90c60deafdfd5d6821d06d49aea057e5d4145427`, supplies the coupled-cluster singles/doubles equations for its on-site SU(4) two- and three-body Hamiltonian. An independent exact relative-coordinate solver removes total translation for A≤4. Together they reproduce all six exact and six CCSD entries of [source Table 5](https://arxiv.org/abs/2509.08771) to its printed precision. CCSD is additionally checked against the exact two-particle solution and by amplitude residuals; a deliberately perturbed amplitude must fail that residual control. The first residual implementation accidentally compared aliased arrays returned by the source's in-place update routines; its record is superseded, with the reason retained.

Small numerical residuals do not remove either approximation in that implementation. The source drops the residual normal-ordered three-body operator, retaining only its zero-, one- and two-body pieces (NO2B), before solving CCSD. A new independent **exact NO2B calculation separates this Hamiltonian error from the subsequent CCSD state approximation**. Its occupation-polynomial potential agrees with the actual source normal-ordered tensors on 102 number states in each of six cases. The reference is localized, so NO2B can break translation symmetry; the exact calculation retains all absolute coordinates, including 531,441 states for A=4 in the L=3 box. A separate full-Hamiltonian calculation checks the translation-reduced solver where that symmetry does hold.

At a=1.7 fm, the three-body exact energy is -16.261431 MeV; exact NO2B gives -7.607348 MeV and the source CCSD/NO2B result is -6.682753 MeV. Of the 9.578678-MeV discrepancy, **8.654083 MeV comes from the Hamiltonian truncation and 0.924595 MeV from CCSD for that truncated Hamiltonian**. NO2B dominates all six source comparisons. The [plotted decomposition](/research/metastable/full/public-inputs/E18-lattice-manybody.json) contains every value and residual. The earlier runs are preserved, with this more precise interpretation replacing attribution of the entire discrepancy to omitted correlations.

Changing only the normal-ordering reference to a normalized zero-momentum Slater state produces another exactly solvable truncated Hamiltonian. For the same A=3, a=1.7-fm case its exact energy is -31.860497 MeV. This is a change of approximation, not of the underlying physical force. Both references reproduce the full three-body expectation in their own reference state; they do not reproduce its operator on every state. No reference is selected to fit the full exact energy. The observed reference dependence is retained as a warning against treating this truncation as controlled for clustered nuclei.

The larger-nucleus calculation therefore reports both approximations and the reference dependence explicitly. At source spacing a=2.5 fm, the energies relative to a separately calculated alpha threshold are:

| Nucleons A | E − (A/4) E(alpha), L=3 (MeV) | Same diagnostic, L=4 (MeV) |
|---:|---:|---:|
| 4 | 0.000000 | 0.000000 |
| 8 | 4.216772 | 3.183909 |
| 12 | 8.049268 | 6.145378 |
| 16 | 11.421485 | 8.858457 |

The two box sizes change both the nuclear state and its reference threshold. Nucleus-dependent references also produce different truncated effective Hamiltonians from the same underlying force. These numbers cannot be read as a controlled alpha-decay energy without controlling those effects and the omitted operators and correlations. The **on-site source Hamiltonian differs from the finite-range Elhatisari A/B interaction** used in the preceding locality test; results are never combined as one force family. This completes a public many-body source benchmark and approximation/volume diagnostic. It does not establish a thermodynamic gas/liquid transition, interface tension, nucleation rate or matched field response.

**The finite-range A/B interactions themselves are also calculated in a declared mean-field space.** The same pinned repository supplies nonlocal contacts, local spin/isospin terms and one-pion exchange. This calculation uses a continuous Hamiltonian, while the original lattice study used a finite-time transfer matrix. The repository's explicit NLEFT constants are applied to both families; its B example would otherwise inherit different defaults. Its gA=1.287 also differs from the original rounded 1.29. These convention differences are recorded rather than claiming an exact reproduction of the original many-body phase diagram.

Four internal species occupy each real spatial orbital. Tracing the full source interaction gives a reduced Hartree–Fock contraction, independently checked against the source's full contraction and an energy derivative. The particle count, density projector, fixed-point residual and Hamiltonian–density commutator are checked for each of 32 cases: A=4,8,12,16, two box sizes, two families and two spatial starts. The following table selects the lower of the two converged energies in each L=4 case:

| Family | Nucleons A | E − (A/4) E(alpha), L=4 (MeV) | Two-start energy difference (MeV) |
|---|---:|---:|---:|
| A | 4 | 0.000000 | 20.550102 |
| A | 8 | 13.771338 | 10.584856 |
| A | 12 | 14.020694 | 8.893160 |
| A | 16 | -0.092760 | 62.738118 |
| B | 4 | 0.000000 | 18.353755 |
| B | 8 | -12.055343 | 29.257559 |
| B | 12 | -44.390517 | 12.114274 |
| B | 16 | -80.826510 | 54.491765 |

Interaction B favors collective binding more strongly in this approximation. However, the energy spread between starting configurations is sometimes tens of MeV; convergence does not establish the global minimum. The first stability calculation evaluates the full occupied/unoccupied real-orbital Hessian with the four species sharing spatial orbitals. Finite orbital rotations independently check its curvature. All 32 states have positive curvature in that restricted space.

**The operator convention and the omitted orbital directions are now tested explicitly.** A normalized antisymmetric two-particle projection of the public operators is compared with independent relative-coordinate Fourier equations in both spin channels, two volumes and both interaction families. Every kinetic, contact, local and pion-exchange matrix agrees within 1.4e-12 MeV. This requires the source's down-first spin/isospin convention and both transposes in its density-product construction; the first unaligned and failed comparisons remain preserved. The complete finite-box spectra agree, and the independent earlier solver is also checked at its original rounded axial coupling. No coupling or energy is fitted in this audit.

The stability space is then enlarged twice: first to independent real orbitals for all four species, and then to **all complex proton and neutron spinors at fixed charge numbers**, including spin mixing and imaginary particle-hole amplitudes. The complex density contraction is verified against independent antisymmetric Wick energy, and the lowest Hessian modes pass residual and finite-rotation controls. The final classification is {'charge_conserving_HF_local_minimum': 32}; its smallest tested curvature is 1.005222 MeV per squared total particle-hole amplitude. The earlier shared-orbital curvature must be divided by four to use this same normalization. Degenerate Fock orbital bases are compared by an explicit change of basis, rather than by comparing entries in arbitrary eigensolver coordinates. These checks support charge-conserving **Hartree–Fock local minima**, while leaving pairing and other many-body correlations outside the calculation.

Finally, explicit principal-angle paths connect each pair of minima. Their projectors, particle numbers, endpoint energies and stationary endpoint derivatives are checked throughout. At L=4, the chosen-path escape-barrier upper bounds range from 7.780 to 208.358 MeV. These maxima bound the least mean-field barrier **from above**; no optimized minimum-action path, collective inertia or tunneling lifetime is supplied. They are finite-nucleus paths, not interfaces between thermodynamic phases or a passing fundamental field. Correlations, volume/density scaling and physical dynamics remain necessary. The A/B calculations remain separate from the on-site CCSD Hamiltonian.

**The same finite-range A/B force now also has correlated alpha calculations**, extending the mean-field energies above. This is a two-body Hamiltonian; the separate on-site benchmark's NO2B three-body approximation is absent. Complex CCSD amplitudes retain the pion-exchange spin couplings. Their equations are checked directly with connected amplitudes extracted from an independently diagonalized two-nucleon wavefunction. Perturbing those exact singles/doubles by ±1% tests **local recovery of a known root**, not independent global ground-state discovery; less informed starts in the larger box did not recover the physical ground root and remain archived. The recovered energies differ from the exact values by at most 7.54e-10 MeV. Alpha predictions use only the existing mean-field singles and zero doubles as their initial guess. Every selected two- and four-body solution passes an independent amplitude fixed-point check.

This extension found an additional public-code issue. The source Fock builder skips the interaction contraction when the corresponding sparse kinetic entry is absent. Explicit zero kinetic entries restore those terms without changing the physical Hamiltonian. The largest missing matrix element among these cases is 0.442256 MeV. The corrected full field and reference energy agree with independent Wick contractions. Particle-only sums are accelerated with sparse matrix products, checked against original routines on random complex inputs. Nonconverged original iterations remain preserved; the selected solutions use real-coefficient, SVD-stabilized DIIS for the unchanged complex equations. This is a numerical repair and solver control, not a fitted nuclear parameter.

| Interaction | Box length L | Alpha HF energy (MeV) | Alpha CCSD energy (MeV) | CCSD − HF (MeV) |
|---|---:|---:|---:|---:|
| A | 3 | -35.037887 | -35.748605 | -0.710718 |
| B | 3 | -39.832070 | -40.319362 | -0.487292 |
| A | 4 | -31.199665 | -31.843351 | -0.643686 |
| B | 4 | -33.182409 | -33.635581 | -0.453172 |

These are finite-volume CCSD roots reached from the existing best Hartree–Fock states, with singles and doubles included. **CCSD is not a variational bound**, and exact two-body agreement does not bound omitted three- and four-particle correlations. Controlled volume extrapolation, correlated multi-alpha thermodynamic ensembles and transition dynamics remain unfinished. The later finite-box Gaussian and exact few-particle thermal traces supply separate scoped controls. [Plotted comparison and frozen controls](/research/metastable/full/public-inputs/E22-finite-range-correlations.json).

**The correlated calculation now extends through oxygen-16**, with 16 solutions at A=4,8,12,16 in the same two boxes. The best existing mean-field state initializes each calculation; no nuclear parameter is refitted. Some fixed localized references have almost zero overlap with that state. Pivoted QR selects occupied sites with controlled overlap before solving, and the reserved alpha energy checks that change of reference. This does not establish reference independence of the many-body truncation or identify a global ground state.

An external execution interruption left ten solved cases and one unfinished iteration history. The original records remain unchanged. The recovery rebuilds every Hamiltonian and contraction control, checks frozen amplitude hashes, and recomputes every energy and residual; it does not inherit unrecorded validation verdicts. The remaining predictions begin from their declared mean-field singles and zero doubles. Pairwise complex residual inner products reduce solver memory without changing the Pulay metric.

| Interaction | L | A | CCSD energy (MeV) | CCSD − HF (MeV) | Same-box alpha-breakup difference (MeV) |
|---|---:|---:|---:|---:|---:|
| A | 3 | 4 | -35.748605 | -0.710718 | 0.000000 |
| A | 3 | 8 | -70.875223 | -1.377719 | 0.621988 |
| A | 3 | 12 | -102.682128 | -2.071607 | 4.563689 |
| A | 3 | 16 | -136.672763 | -2.609765 | 6.321658 |
| B | 3 | 4 | -40.319362 | -0.487292 | 0.000000 |
| B | 3 | 8 | -94.037343 | -0.976196 | -13.398619 |
| B | 3 | 12 | -174.904839 | -1.129128 | -53.946753 |
| B | 3 | 16 | -276.745930 | -1.152964 | -115.468482 |
| A | 4 | 4 | -31.843351 | -0.643686 | 0.000000 |
| A | 4 | 8 | -50.021196 | -1.393204 | 13.665506 |
| A | 4 | 12 | -81.588284 | -2.009984 | 13.941769 |
| A | 4 | 16 | -127.370873 | -2.479454 | 0.002531 |
| B | 4 | 4 | -33.635581 | -0.453172 | 0.000000 |
| B | 4 | 8 | -79.307349 | -0.887189 | -12.036187 |
| B | 4 | 12 | -145.021945 | -1.084201 | -44.115202 |
| B | 4 | 16 | -214.611351 | -1.055206 | -80.069027 |

The last column is E(A) − A E(4)/4 using alphas in the **same finite box and approximation**. It diagnoses the response of these solutions, not a rigorous breakup bound or dense-matter phase boundary. Two box sizes do not control the infinite-volume limit, and CCSD truncation remains unbounded by the two-body checks. [Plotted data and controls](/research/metastable/full/public-inputs/E24-correlated-clusters.json).

**An exact small-box calculation now measures the correlated approximation directly.** For each interaction, all eight total-momentum sectors cover the complete fixed-charge Hilbert space at A=2,3,4 in an L=2 box. Translation orbits include fermionic signs. Hermiticity, normalization and independent eigenvector residuals are checked, and the two-particle energy matches the separate Fourier calculation. The lowest state is found in the zero-momentum sector in every tested case.

This particularly small box exposes a periodic-image convention in the source implementation. Positive and negative neighbors coincide at L=2. Its nonlocal smearing adds both weights, while its distance-based local smearing counts the site only once. The calculation restores the six directed local-neighbor contributions required by the Fourier form factor, then compares each kinetic, pion-exchange and contact operator independently. No coupling is fitted. The original failed comparison is preserved; this correction changes none of the earlier L=3 or L=4 operators.

| Interaction | A | Full Fock dimension | Exact energy (MeV) | CCSD − exact (MeV) |
|---|---:|---:|---:|---:|
| A | 2 | 256 | -22.614812489 | 1.77635684e-14 |
| A | 3 | 1920 | -56.519135804 | not converged |
| A | 4 | 14400 | -115.214664955 | 27.7082934 |
| B | 2 | 256 | -27.673041869 | 1.81188398e-13 |
| B | 3 | 1920 | -70.365983908 | not converged |
| B | 4 | 14400 | -144.223972226 | 23.4593219 |

The CCSD comparison uses exact C1/C2 coefficients to initialize connected singles/doubles at a determinant with large exact-state overlap. This is deliberately a **method-error diagnostic initialized with known information**, not a blind predictive ground-state search. The omitted higher connected amplitudes are not fitted. The Hamiltonian is identical in the exact and CCSD calculations, contains only two-body terms, and uses the corrected complete Fock field. An independent full-space construction of exp(-T) H exp(T) checks its reference energy and every single/double projected equation, bypassing the source diagram routines; residuals in omitted higher excitations are recorded separately. A converged energy difference characterizes these local roots and reference choices, rather than the separate on-site benchmark's NO2B Hamiltonian error. Numerical convergence alone does not remove it, and its size cannot be assigned as an error bar to a different box or heavier nucleus.

Both exact three-particle ground energies pass their eigenvector controls, but neither corresponding CCSD root meets the declared iteration tolerance. A separate dense eigensolve checks the degenerate ground eigenspace and selects a reference-aligned vector using its spectral projector. Expanded DIIS history and a tighter linear-algebra cutoff are checked against perturbed exact pairs before retrying those roots. Separate trust-region attempts fail their reserved pair convergence test and are retained as unsuccessful alternatives. No solver tolerance is enlarged, no spin component is removed, and no energy difference is assigned to an unaccepted root. These failures do not prove that no CCSD solution exists. [Exact and correlated data](/research/metastable/full/public-inputs/E25-exact-finite-nuclei.json), [method attempts and logs](public-method-controls.json).

Checkpoint21 preserves all128 prior scientific components and adds four independently reviewed scoped results. Review/controller checks remain separate from author component counts.

The full141-cell finite screened-contact certificate is independently reconstructed, but its source width0.0096595478245381 fails the unchanged2e-8 target. Original source/phase gates, zero-range certification and physical flux remain unadmitted. [Certificate review](w04-full-independent-screened-certificate-review-20261008.md).

One upper neutral-even Gaussian block has all5778 columns/181 panels independently reviewed. Its represented Rayleigh interval[3.4714277938751195e-7,3.4715099061157804e-7] is entirely below the unchanged2e-6 strictminimum threshold; this is not a physical negative mode or rigorous root/operator enclosure. Three blocks remain uncomputed. [Upper-block review](w10-upper-physical-independent-review-20261008.md).

The Re-only historical calculation demonstrates quantified non-identifiability:225 chronology-strength ridge points,183 admitted under the declared4.6Gyr ceiling, and zero marginalized rank with free chronology or heterogeneous inventories. Its inventory ceiling is conditional, not an empirical confidence interval or heavy-nucleus theta constraint. [Historical inference](isotope-nonidentifiability-findings-20261008.md).

Six fixed planar collision cases reach dimensionless tau120,57000 steps and726 frames with an independent full-history review. The saved finite-box histories show rebound and two surviving walls at tau120. This is a return-controlled collision in a separate field branch, with no complete annihilation to measured pure radiation, ordinary-matter matching, formation/lifetime or infinite-volume claim.

A qualified diagnostic inventory now joins13 cached and27 fresh candidates into40 comparisons while retaining all failures and cohort provenance; it supplies no new physical state or stationary phase. The six-case tiny hydrodynamic method has separate numerical/controller reviews; its31-case production remains unreleased. The oxygen anchor and physical-flux method reviews remain finite algebra/control results, with no real oxygen eightsector correction or physical encounter production. Original20261005 methods and failed attempts remain individually accessible in the frozen catalog. Historical input gaps are attached only to named failed attempts and do not admit scientific evidence. [Tiny method review](w15-complete-tiny-method-review-20261008.md), [bounded method reviews](independent-bounded-reviews-20261008.md), [flux controller review](w27-full-method-review-findings-20261008.md).

The original fixed optical method now has a complete independent review of 103 records: 65 tables and 6,305 coefficients, 24 scalar supports and 72 coordinates, two crossing boxes and ten coordinates, and twelve physical keys with 36 jets and 1,116 components. Five bounded numerical allocations and their metadata JOIN retain the original precision and comparison gates. This adds review evidence for an existing method, with no new scientific component or independent physical confirmation. The failed reviews remain preserved. Weighted fusion, ordinary alternatives, field matching, the full physical Hessian and a minimum certificate remain open. [Complete optical method review](w22-complete-partition-review-20261008.md).

For the selected LO canonical QCD axion vacuum potential, an analytic stationary census proves only the ordinary static minimum within|theta|<=0.5 for every finite positive quark-mass ratio; the equal-mass cusp is a maximum. This scoped negative result constrains no f_a and excludes neither finite-density nor oscillatory backgrounds. Full field viability remains open. [Static census](canonical-qcd-axion-static-vacuum-census-20261008.md).

## Shared histories and observer conditioning

Explicit action, parameter, initial-ensemble, spacetime and source identities now accompany paired realization IDs, weights and lineage. Six actual component audits retain incompatible or missing physical connections. A legitimate conditional demonstration reuses the same saved geometric histories for observer survival and opportunity: independent RNG replay reconstructs all 240 histories and 1,200 aggregate rows exactly. Conditioning changes the no-detection probability, and replacing an expectation of exponentials by the exponential of a mean changes it again. A malformed requested-domain bug found in review was repaired without changing the numerical histories. This contract verifies declared compatibility and source bytes; it supplies no physical cosmological population or calibrated solitude likelihood. [Audit, reconstruction and corrected review](history-contract-findings.md).

## Finite-time isotope exchange and an inference failure

Chronological parent/daughter transport now replaces instantaneous mixing in a separate two-compartment control. An analytic compartment-mode solution independently reconstructs every inventory and the time-dependent sensitivity kernel. Unequal elemental mobility adds temporal information, but the weakest formal mode lies outside the linear model's validity range; a 10-ppm nonlinear injection produces zero nominal coverage in the best-measured linear coordinate. This approximation failure is retained. All inventories, exchange histories and covariance are synthetic. The 88 actual source rows remain ineligible for a new empirical joint fit because analysis/batch associations, covariance and closure/exchange records are missing. [Conservation, identifiability, failed approximations and independent review](isotope-exchange-findings.md).

## Self-consistent conserved-charge pockets

One common scalar/relativistic-Fermi energy functional now determines spherical finite-bias profiles at charges 3,000, 10,000 and 30,000. Enforcing fixed total charge removes the negative radial direction of the unconstrained grand-potential Hessian; an independent finite-element operator and fixed-charge finite differences confirm positive lowest radial eigenvalues. Pressure cancellation near a formula switch was found in review and repaired in new runs, with the failed review preserved. These are prescribed dark-sector configurations with ordinary-matter coupling exactly zero. The subsequent angular Hessian controls local nonspherical stability within this functional; global stability, formation, evaporation and physical matter matching remain open. [Profiles, fixed-charge derivation and corrected review](fermi-profile-findings.md).

## Matched atomic photoionization and radiative recombination

The same ordinary hydrogenic Coulomb kernel now supplies 1s photoionization and its Milne inverse, including stimulated recombination. Independent analytic radial integrals and energy-space quadrature check normalization; equal-temperature Planck/Saha balance holds for number and energy moments. Seventy-five prescribed He+/He++ reservoir cases obey charge-neutral kinetics and the binding-energy ledger. Omitting stimulation gives a 26% balance error in the hottest benchmark. This ground-continuum subsystem has no excited-state cascades, neutral helium, complete plasma evolution or weak-destruction lifetime. [Equations, source comparison and independent review](hydrogenic-balance-findings.md).

## A genuine four-body elastic continuum in a frozen-target approximation

The same Gaussian pair and three-body force now supplies antisymmetric neutron–triton J=0 elastic amplitudes below its own breakup threshold. The direct folded interaction is retained as an exact radial background; the residual exchange operator and the full norm transformation enter a finite-rank continuum solve. Target and radial refinements give pure-J=0 cross sections 211.574, 208.284 and 201.908 fm² at 50, 100 and 200 keV. Independent real inhomogeneous radial equations reproduce the phases within 2.4×10⁻¹¹ radians. Multiply by 1/4 for this channel's unpolarized contribution; the separately reviewed J=1 extension below supplies the other spin contribution. Frozen-target polarization, rearrangement, breakup and radiative currents remain unfinished. [Derivation and independent review](fourbody-scattering-findings.md).

## CO cooling and background-driven heating

The public LAMDA CO levels, Einstein coefficients and separate para/ortho-H2 collisions now define one consistent statistical-equilibrium energy balance. Of 324 prescribed reservoir points, 81 transfer radiation energy into gas. Populations, stimulated emission, absorption, transients and line opacity pass independent reconstruction. At 300 K the coarse J=30 truncation fails; J=35/38 agree with J=40 within 0.039% at the hardest tested point. The declared column stays optically thin. Fixed CO/H2 and ortho/para ratios are inputs; this is neither a chemical abundance calculation nor cloud fragmentation. [Rates, conservation, failed coarse control and review](molecular-cooling-findings.md).

## Isotropic spatial stochastic controls

Weighted real Fourier-shell waves replace the earlier octree noise for a separate controlled calculation. Their covariance converges to sinc(k r) under unrelated rotations; an expanding shell reproduces analytic free and linear-drift spatial covariance. Four nonlinear conditional ensembles evolve for three e-folds at 125 sample locations with 512 independent realizations and coupled timestep controls. Independent finite-volume probability evolution reproduces their one-point distributions. Sample locations are correlated and do not multiply the realization count. Initial ensembles, constant H and the potential are prescribed; no reheating, gradient-resolved walls or physical population prediction is inferred. [Equations, arrays and independent review](isotropic-domain-findings.md).

## Actual satellite clock products with an explicit significance boundary

Three days of ESA final clock and orbit products support an eight-satellite, training/held-out analysis. Common-reference contrasts, daily differencing, affine nuisance projection and a full covariance model act identically on data and spatial transient templates. Same-epoch orbit-file clocks independently check the clock-file time tags, whose mixed-GNSS header omits the required time-system field. Product-level injections recover their expected increments. Residual lag-two correlation reaches 0.236 and empirical scan tails depend strongly on block length. The resulting conditional scores are not calibrated Gaussian significances, detections or physical exclusions. [Products, filters, diagnostics and limits](gnss-clock-findings.md).

## A count likelihood for the atomic analogue

The published spatial-window and magnetization classifier reproduces all ten probability groups from 68 destructive snapshots. A conditional Weibull binomial model has median probability time 3.370 ms and shape 2.701. Finite-sample bootstrap, classification variants and assumed misclassification scenarios expose the limits of the inference; 16.3% of fitted-family bootstrap draws reach the declared shape bound. These are repeated samples of different clouds, not tracked individual waiting times. A failed tail-rounding implementation is retained and corrected in log space. Neither this probability fit nor the source's initially noisy GPE protocol establishes a microscopic vacuum-decay lifetime. [Likelihood, bootstrap and source-method audit](snapshot-likelihood-findings.md).

## Printed pumping-fit rounding cannot explain the central-value difference

A supporting source audit exhausts every admissible one-decimal rounding of all twelve pumping signals and uncertainties under the stated independent-Gaussian, no-offset model. Two independent directed-interval calculations exclude a stationary fit throughout the printed 45.85–45.95 s interval. The nominal refit is 47.7044 s, only 0.083 of the source's quoted 21.8 s uncertainty away. This narrows the unresolved fitting conventions; it is no evidence that the publication is wrong and adds no independent physical component. [Exact rounding envelope and review](pumping-rounding-findings.md).

## Projected-pair searches remain nonstationary

Six number-projected A/B searches now preserve full terminal pair matrices and normalized sector vectors. Their energies remain valid variational upper bounds, but every search hits its iteration limit with a nonzero physical gradient and a negative curvature direction. Independent review reconstructs the bounds, derivatives and projection conditioning. It also finds a trust-step hard-case defect in the historical proposals; corrected controls pass, while the original searches retain their source versions and nonstationary status. These outcomes establish neither optimized competing minima nor mixed-state entropy. [Frozen attempts, correction and review](projected-pair-stationarity-findings.md).

## Colder paired minima with the volume failures retained

At T=1 MeV, the same L3 A/B Gaussian models retain restricted paired local minima with free energies 2.17907/2.25927 MeV below their respective normal saddles. Four independent complex starts return to those normal saddles rather than finding another paired minimum. A full sixteen-state Fock calculation verifies the stable free-energy-difference identity, and independent hyperbolic-response Hessians verify the selected curvatures. The original L4 extension failed the cold finite-path charge and curvature controls. Later coupled-charge and density-increment calculations repair both local normal-state checks under the original external gates; their independent reviews and preserved arithmetic limitations are detailed below. Larger-box paired-root continuation, volume convergence and a thermodynamic phase remain unresolved. [Exact ensemble, numerical failures and review](paired-volume-findings.md).

## Both frozen-target neutron–triton spin channels

Exact target–neutron triplet coupling supplies the J=1 partner of the reviewed J=0 elastic channel, with the identical force, target and relative L=0 basis. Independent finite-element boundary maps reproduce all three phases within 7.3×10⁻¹⁰ radians. A separate initial-spin density trace verifies the weights 1/4 and 3/4, giving unpolarized L=0 cross sections 180.386, 177.948 and 173.207 fm² at 50, 100 and 200 keV. Target polarization, higher relative orbital waves, rearrangement, breakup and radiative formation remain outside this frozen-target model. [Equations, refinement and independent review](fourbody-triplet-findings.md).

## Scalar–fluid shock heating from one energy functional

A compressible ideal gas and scalar field now exchange work through the same energy functional. The conditional periodic counterflow develops shocks, increases gas entropy and conserves total energy; independently integrated field gain equals the matter work loss. Separate exact-wave, oscillator, Sod shock and smooth-entropy controls distinguish conservation from shock convergence. A second numerical flux and scalar solver reproduce the refined profiles. This dimensionless control has no nuclear EOS, metastable field potential, gravity, microscopic friction or physical stellar history. [Equations, energy decomposition and review](scalar-hydrodynamics-findings.md).

## Conservative spatial transport of released isotope tags

Authenticated wind histories and saved helium-exhaustion shell interventions now supply six single-cohort, ten-channel spatial transport cases. Hot/cold advection and diffusion, interphase exchange, trapping and both boundary escapes close each isotope ledger without clipping. Independent raw-source reconstruction and an interleaved banded-density solver reproduce all five grid/time configurations within 1.2×10⁻¹² relative. Local ratios reflect differently timed inputs under species-blind transport. The imposed source geometry and coefficients are conditional; these are tagged ejecta ratios, not an ambient-gas observation or a terminal explosion yield. [Source adapter, controls and review](material-transport-findings.md).

## Independently normalized higher-doublet breakup

Explicit charge/spin recoupling now supplies neutral spectator waves l=1–4 at 3/5 MeV and cutoffs 600/1200 MeV. The angular-independent LO three-body contact projects only into l=0, retaining the same force. A separate spherical-harmonic spin sum and Gauss–Jacobi phase integral reproduce all 16 selected outgoing fluxes within 1.5×10⁻⁸ relative. The higher-wave pure-doublet sums are 2.73467 and 15.8404 fm²; their unpolarized contributions carry weight 1/3. A rigorous geometric bound applies to the real driving source. Converting it to an infinite-wave flux bound still requires the unresolved open-cut inverse and outgoing-map bounds. Finite l=5–8 elastic-loss diagnostics are not that remainder. [Derivation, all four runs and review](doublet-higher-findings.md).

## Nonspherical local stability of the same dark pockets

The fixed-charge angular Hessian now extends the three saved scalar/Fermi profiles. Integrated finite elements resolve a fermion-edge alias that made an earlier translation control fail. Independent consistent-mass elements and nonlinear fixed-charge energy variations confirm positive l=2–8 spectra and the l=1 translation zero mode. For an exact monotone stationary profile, a ground-state factorization makes the l=1 quadratic form nonnegative; the centrifugal increment then proves positivity for every l≥2. This implication remains conditional on that same Thomas–Fermi functional and exact profile, with ordinary coupling zero. Nonlinear fission, quantum-shell effects, formation and thermal/gravitational dynamics remain open. [Angular derivation, failures and independent review](fermi-angular-findings.md).

## A compatible transfer-only thermal network

The same neutral three-body force now supplies independently solved forward and inverse p+(nn) ⇌ n+d amplitudes at three declared temperatures. Partial-wave unitarity bounds both unsampled energy intervals, separately from the larger numerical quadrature error. Nine four-species histories preserve three reaction invariants and approach independently derived constrained Saha equilibria. A different-author quadrature and analytic reaction-extent solution verify the rates and histories. The force's 11.767-MeV tritium binding is recorded consistently, but tritium remains inactive: its formation edge needs an unsolved four-body rearrangement amplitude. This is an accepted transfer subsystem, not a repaired full BBN model. [Contract, energy coverage and review](matched-transfer-subset-findings.md).

## Charged elastic amplitudes with their asymptotic phase retained

The same LO charged kernels now supply both spin sectors at one below-breakup S-wave point, with the point proton–deuteron Coulomb phase fixed independently by radial scattering. Renormalized composite Coulomb differs from point Coulomb by −0.40818 degrees at the smallest declared screening mass; that finite offset is retained in both channel amplitudes. Independent complex-pole solves, angular/grid refinement and screening-window controls pass. These calculations do not certify an infinite-screen remainder or complete charged breakup. [Derivation and independent review](charged-asymptotic-findings.md).

## A magnetic contact fitted to the same nuclear draws

Seventy-two authenticated joint NN draws now supply their own coupled deuteron and singlet scattering waves. One ordinary cold capture datum fixes two real magnetic-contact branches per draw; the squared rate alone cannot choose their signs. Both branches reproduce that input, but their finite-energy M1 spectra differ. Independent spin/polarization normalization and six consistent-mass finite-element solutions confirm the selected numerical calculation. The aligned branch satisfies the declared current-size diagnostic in all 72 draws; this assumption is not an observed sign or proof of EFT convergence. The newly acquired Suzuki total-capture measurements also contain E1 strength and beam averaging, so they are not treated as an M1-only validation. [Derivation, fits and review](np-m1-calibration-findings.md).

## Finite-level radiative cascades with an explicit metastable boundary

The same ordinary Coulomb Hamiltonian now supplies all bound and continuum E1 dipoles through separately declared principal ceilings 2, 3, 4 and 6. Seventy-two charge-neutral reservoir equilibria obey level-resolved Planck/Saha balance; nine independently integrated transients close the population and energy ledgers. A different-author review checks nonterminating hypergeometric integrals, high-level radial waves and a third population solver. A zero-radiation 3p cascade leaves 0.118335 in 2s because this E1 model lacks two-photon decay. The finite Coulomb partition function has no convergent infinite-level limit without plasma physics. Neutral helium, collisions and weak destruction remain unfinished. [Equations, retained failures and review](hydrogenic-cascade-findings.md).

## Improved projected searches retain unresolved stationarity

The revised projected-pair optimizer preserves full-sector variational bounds and corrects trust-step scaling. Independent 60-digit contractions expose finite-precision errors in its poorly conditioned horizontal Hessian coordinates; all selected terminal states remain nonstationary. A's original derivative-check failure is retained. The unprojected probability of the selected number sector depends on the representative pair scale, even when the normalized projected state is identical. It therefore cannot by itself exclude a limiting Gaussian representation. No candidate is promoted to a competing minimum. [Qualified independent review](projected-newton-review-20261002.md).

## Raw NN observations are now locally available

The official Granada public download form supplied the raw pp/np archive. A lossless parser retains 620 blocks and 6,391 rows, including negative uncertainty markers. Selecting positive-marker rows reproduces the source's 388 observations through 5 MeV and 746 through 15 MeV, with matching observable groups. The half-MeV and 10-MeV counts retain smaller discrepancies, and undocumented normalization header fields remain opaque. A different-author 7,035-check reconstruction verifies all datum identities and declared count groups. Primary PWA and force papers document the profile-normalization algebra; the separately derived Gaussian marginal also requires its determinant term. Availability of the archive does not yet establish the author's exact likelihood, covariance or selection epoch. [Selection audit and remaining conventions](granada-selection-findings.md).

A finite-rule molecular population tube is now certified for the original case-03 initial state, one fixed 10^7-second horizon and the saved narrow temperature/composition rectangle. The three bound families and all eight preregistered relative radii are retained. The first radius passes; radius one fails its temperature-face condition. The exact initial nonunit mass is carried without normalization. This uses the fixed 96-node atomic source graph and 26 heat stages; it certifies neither continuous atomic quadrature nor a computed physical step, entropy history, 121-point history or 68-case census. Six author checks belong to the selected component; independent review checks are reported separately. [Tube findings](molecular-forcing-tube-findings-20261005.md), [independent review](molecular-forcing-tube-review-findings-20261005.md).

The native EOS sidecar review accepts captured algebra and the faithful reproduction of nineteen grad_ad failures. All 64 lnP/E support comparisons pass, while thirteen grad_ad path tests and six span tests fail. Branch support, the older 287/112 derivative failures, 762 whole-matrix references and 72 excluded ties remain unresolved. No new EOS calls or stellar trajectory are admitted. [Native smoothing findings](runs/20261005T082932128050Z-E04-native-EOS-smoothing-analysis/results.json), [review findings](native-eos-smoothing-review-findings-20261005.md).

The free charged-contact certificate is independently verified. A later accepted stage-F run preserves four original cells and completes 137 new committed cells; a fresh 512-bit independent review checks all 141 and the exact commit chain. The earlier whole-method run still failed at the screened 4,097-node solve. Free-only accuracy does not admit the original source/phase gates or a zero-range limit. [Earlier free certificate review](free-contact-certificate-review-findings-20261005.md), [staged free review](charged-doublet-stage-F-review-findings-20261005.md).

The native atomic source backend and independent 3212-check review complete 72 small source controls and 178 native calls. Full 257 and 4224 node quadrature remain unperformed, and the old 0.05347>0.05 timing failure remains. The bound-free method and 6,414-check independent review complete 25 states and 66 spin-resolved Coulomb channels, including all 125 thermal moment sets and 250 absolute model-tail constructions. The reviewer uses independent angular/Laplace-derivative algebra and 512-bit directed tails; its saved-node thermal reconstruction explicitly reuses the producer cross-section samples. Finite interior quadrature, collision joining, two-photon normalization and population evolution remain unaccepted. Earlier exact-zero hypergeometric failures and the reviewer parser failure remain archived. [Bound-free method](helium-boundfree-channels-method-addendum-20261005.md), [independent review](helium-boundfree-review-findings-20261005.md).

Nine author partitions and nine independent full 512 product reviews now support an independently checked synthesis of all 27 pending candidates and 216 fresh Hamiltonian products. The synthesis preserves 135 original numerical-gate outcomes,27 admitted domains,23 accepted signed policies and four rejected policies: historical-retracted and covariant_trust at half 3 and half 4. Its 231 author checks form the selected component; the independent synthesis review and reused partition reviews are counted separately. The synthesis performs no new Hamiltonian products or numerical reconstruction, does not join the cached 13 into a 40-node claim, and accepts neither state propagation nor stationarity. Earlier startup, sealing and cost failures remain archived. [Synthesis findings](pair-pending27-synthesis-findings-20261005.md), [independent saved-synthesis review](pair-pending27-synthesis-review-findings-20261005.md).
## Complete local nuclear response coefficients, with physical matching still open

The source-specific inverse-amplitude study now differentiates all 88 saved roots in three independent coefficient directions. It checks 528 nearby pole supports, ten threshold locations in three directions, the local split of coalescing roots, and 28 regular current supports. All 1,336 author checks and six analytic tests pass. Independent elementary-logarithm formulas at up to 220 decimal digits pass 6,613 further checks. The zero-binding current derivatives remain finite, and coalescing poles use the coupled threshold equations instead of a singular simple-pole formula.

These are transfer coefficients multiplying unknown physical responses. The inverse-slope factor is not a complete amplitude residue. Nucleon-mass, amplitude-numerator and current-counterterm responses remain separate; no missing coefficient is assigned zero, no ordinary anchor is refitted, and no finite-angle matching or joint covariance is supplied. The earlier strict first-order pole-shift diagnostic remains failed. [Complete sensitivity study](quarkmass-sensitivities-findings-20261005.md), [independent review](quarkmass-sensitivities-review-20261005.md).

## Full molecular stationary populations and parameter responses are enclosed

For the same 41-level CO rate model, outward interval arithmetic now encloses the stationary populations and their temperature and collider-fraction responses at the declared point and throughout a small parameter rectangle. The verified inverse has a rectangle contraction factor below 0.000697, and all 41 population lower bounds remain positive, including the rarest level near 5.24e-49. The calculation passes 31,019 checks and 15 tests. Independent source parsing, interval identities and high-precision solves pass 28,944 additional checks.

An independent mutation witness exposed a guard omission when the original method ran as a main module. That initial capture and failed review remain preserved. The corrected source seals the actual function namespace and passes the same witness. The accepted result is a stationary and response enclosure; it does not establish the evolving population tube, slow-variable speed bounds, passage through rate-table knots or a physical cooling history. [Enclosure method and retained guard failure](molecular-stationary-enclosure-findings-20261005.md), [independent review](molecular-stationary-enclosure-review-20261005.md).

## Eight molecular transition bounds control conditional mixing

The same full 41-level rate model now has rigorous lower transition kernels at four declared durations, from 100,000 to 100 million seconds, for both the point and the parameter rectangle. A common lower generator bounds every time-dependent generator inside the rectangle, including noncommuting histories. Positive uniformization terms and repeated squaring use exact integer arithmetic with downward rounding. No kernel is normalized or clipped to conceal lost mass.

All eight kernels pass 16 author checks and 19 tests. Independent rational exponential bounds and a differently ordered integer recurrence reproduce every entry of all 378 saved stage matrices, passing 22,762 checks. The first review stopped on a diagnostic integer-to-text size limit; its failed record remains intact, and the replacement changes only that diagnostic representation. The rectangle mass-zero contraction upper bounds are approximately 0.992543, 0.325690, 0.000834056 and 0.008213772 at the four durations. Their nonmonotonicity reflects the deliberately conservative substochastic lower kernel.

These bounds are conditional on remaining in the declared parameter rectangle. They do not supply the actual temperature and collider-fraction speeds, a population tube, a lag bound or a coupled cooling history. Those require the same-model forcing and invariant-domain estimates. [Eight-kernel calculation](molecular-mixing-lower-findings-20261005.md), [independent full replay](molecular-mixing-lower-review-20261005.md).

## Refined charged-channel toy controls retain the continuous-source requirement

All 18 fixed Gaussian Green-source toy cases pass their original tolerance after the preregistered quadrature refinement. A separate closed-form Gaussian-moment and explicit-determinant review passes 64 checks. The continuous screened ODE comparison and rank-96/192 phase controls are useful numerical evidence, but a rigorous continuous-source error certificate is still missing. Earlier coarse quadrature and admission failures remain in the record; coupled charged production is not accepted. [Refined controls](charged-doublet-green-quadrature-20261005.md), [independent analytic review](charged-doublet-green-quadrature-review-20261005.md), [remaining continuum certificate](charged-doublet-green-outcome-20261005.md).

## Complete cached paired-trial census, with fresh physical cost still pending

The complete controller first finishes nine of thirteen saved nodes before its work deadline; the partial tenth has seven of eight committed transactions. Independent review authenticates that prefix without promoting its partial node. A separately declared four-node job then recomputes the tenth node from scratch and completes the remaining three, using 32 fresh cached transactions. The exact ordered nine-plus-four inventory is accepted by a further 510-check review. The original interrupted run and its partial transactions retain their status.

The new source-file launcher captures its preimport alarm, and independent helper/namespace mutation controls pass at all three callback boundaries. The longest complete cached node takes 52.11 seconds. No fresh physical Hamiltonian product or pending candidate has been evaluated by this completion; a separately reviewed physical driver and one saved-node cost are still needed before scheduling the 27 pending trials. [Complete cached census](pair-pending27-cached-completion-findings-20261005.md), [independent completion review](pair-pending27-cached-completion-review-20261005.md).

## One complete paired-state curvature block excludes its original strict threshold

A bounded physical calculation refines the lower-temperature restricted Gaussian root and assembles all 5,778 columns of its neutral-even curvature block. The root uses 17,388 charge-block-normal coordinates; the enlarged Hessian uses 23,220 coordinates. Independent review reconstructs all 181 saved panels, checks the complete block projection and leakage, and compares six freshly calculated actions against an independently implemented response. Stationarity and mean charges pass the original gates.

A fully bounded quadratic-form calculation places one represented symmetric-block Rayleigh quotient between 1.5811806756e-6 and 1.5811889132e-6. Its upper endpoint is below the unchanged 2e-6 strict minimum threshold, so this represented block cannot pass that threshold. The conclusion does not require the trial vector to be an exact eigenvector. One old soft vector also fails the eigenvector residual gate on the refined operator and remains a diagnostic witness. Other blocks, operator-rounding and exact-root bounds, physical positivity and macroscopic phase competition remain separate open questions. [Full block cost](gaussian-neutral-block-cost-findings-20261005.md), [independent block and threshold review](gaussian-neutral-block-physical-review-20261005.md).

## Full magnetic-component triaxial backreaction passes its finite study

The same sextic field action now evolves a genuinely triaxial quadrupole with all real spherical-harmonic magnetic components, including odd sectors. Both seed amplitudes, 1e-4 and 1e-3, reach T=200 in all six declared angular, radial, temporal and box variants. All thirty shell comparisons pass the unchanged 3% criterion; the largest difference is 0.2601%. Full/core energy imbalances remain below 0.0101%, and tangent and quadratic-feedback controls pass.

Independent review authenticates all 1,206 saved frames, recomputes their full and core Hamiltonians with separately evaluated harmonics, and reconstructs every convergence norm without a denominator floor. The time-step damping and interface-flux integrals remain authenticated producer records, supported by independently reviewed small integrator controls; they are not independently reconstructed from one-second snapshots. This is a finite-time, finite-amplitude and finite-grid result. Arbitrary-perturbation stability, infinite angular resolution, ordinary-matter matching, formation abundance and physical lifetime remain open. [Complete finite study](triaxial-production-findings-20261005.md), [independent complete review](triaxial-complete-review-20261005.md).

## Native equation-of-state composition and packing controls

A native companion captures 25 baseline/direction cases and 400 support paths from the actual electron/ion interpolation routines. All 1,000 native primal coefficient bits match. Independent endpoint-polynomial interpolation and analytic ion derivatives reproduce the diagnostic, including all 287 failed controls. Those failures include interpolation/reference differences, above-table behavior and cancellation-sensitive mixed derivatives. This does not repair the full stellar Jacobian or admit an altered stellar evolution. The first build capture was invalidated because a compiler child outlived its controller; its original manifest, changed-output history and all 40 current partial outputs are retained. A separately captured build used guaranteed child reaping. [Native composition diagnostic](native-eos-composition-bridge-findings.md), [independent review](native-eos-composition-bridge-review-20261005.md).

A subsequent method preserves 3,900 packed value/temperature/density primals and all 216 native blend-polygon values, and tests 800 composition derivatives, 624 simplex-gauge controls and 144 smooth geometry slopes. It uses eight actual FreeEOS table cases. Exact native real and automatic-differentiation evaluation orders are retained at branch decisions; the earlier resource and polygon failures remain visible. Seventy-two vertex cases do not establish global derivatives, and the finite geometry support paths hold the selected segment fixed. The complete packed Skye/FreeEOS smoother and interface contraction, the original 136-call stencil inventory and the 762 unresolved whole-matrix references remain unfinished. Independent review reconstructs the thermodynamic Jacobian with separate Helmholtz formulas, the eight table cases with Newton divided differences, and all 576 smooth moving-polygon supports without branch changes. The first review is retained as an incomplete runtime-provenance attempt; only the fully pinned replacement is counted here. [Packing and table method](native-eos-packing-bridge-findings-20261005.md), [independent native-packing review](native-packing-bridge-review-20261005.md).

## A finite charged-pair contact repair passes unitarity but misses continuum accuracy

The original doublet trial mixes a continuum-renormalized contact with a finite-grid Coulomb background and fails its pair unitarity gate. An independent resolvent identity locates that mismatch. A new energy-independent delta-shell contact, calibrated once to the same Coulomb-modified scattering length, makes all twelve finite-grid unitarity checks pass and agrees with an independent transfer solution. Its rank 48 continuum phase errors at two energies nevertheless remain 2.94e-5 and2.67e-5, above the unchanged 2e-5 criterion. A separate continuous-source boundary-value calculation exhausts its fixed node cap. No coupled charged-doublet amplitude or flux is accepted. [Contact diagnosis](charged-doublet-contact-review-20261005.md), [regulated method](charged-doublet-regulated-results-20261005.md), [independent failed-gate review](charged-doublet-regulated-review-20261005.md).

## Atomic and paired-state execution methods retain their separate admission limits

Independent mutation controls now cover native scalar dispatch for both branches. Atomic cost execution then exposed an additional helper alias and an overly broad seal of the module registry. The repaired controller pins consumed entries while allowing unrelated lazy imports; real 257-entry structural setup and independent mutation controls pass. All seven full-bank operations now finish across three bounded jobs, including an exact fresh legacy comparison. Independent MP130 contraction, saved overlap/Hamiltonian/contact reconstruction, projected-source orthogonality and transaction inventories accept the saved cost census. No new source moments are recomputed by that reviewer. The original 0.05-second guard timing failure remains failed, and all 48 physical quadrature plans remain unexecuted. [Registry repair review](atomic-registry-cost-review-20261005.md), [seven-operation census](atomic-contact-partitioned-cost-production-findings-20261005.md), [independent complete-cost review](atomic-contact-partitioned-complete-review-20261005.md).

The first fixed-27 paired-trial controller exactly replays its thirteen saved candidates and passes small arithmetic controls. Its measured full-input audit overhead gives a 198.19-second complete-node allowance, which does not admit the proposed 14/13 partitions. A separate immutable-input method exposes and then repairs a same-layout receiver-class mutation. Independent review reconstructs every one of its 233,920 coefficient tags from binary64 bits, checks all 392 materialized inputs and reruns the original three receiver-mutation boundaries with durable rejection. Its 60.49-second partial allowance still excludes the complete future physical controller. No new physical Hamiltonian product, pending trial, optimized paired state or propagated endpoint is accepted. [Initial cost disposition](pair-pending27-cost-review-20261005.md), [retained receiver failure](pair-immutable-receiver-review-20261005.md), [repaired immutable method review](pair-pending27-immutable-method-review-20261005.md).


## Direct full-phase quartet integration completes its four finite cases

The neutral leading-order quartet calculation now integrates the direct scattering source over the full declared breakup phase space at both energies and cutoffs, retaining spectator waves l=0 through 4. All five phase, internal-grid and product-rule variants are complete for each case. Independent review replays all 27,904 saved phase rows and 80 wave comparisons and checks 48 reserved direct-source values with a separate equation evaluator. Those samples are not a uniform source-error certificate. The first interrupted calculation is preserved, and its exact unfinished source inventory was completed once; the other cases used six independently authorized partitions.

| Energy (MeV) | Cutoff (MeV) | Pure quartet (fm²) | Two-thirds quartet contribution (fm²) |
|---:|---:|---:|---:|
| 3 | 600 | 0.15555448015 | 0.103702986767 |
| 3 | 1200 | 0.155559061007 | 0.103706040672 |
| 5 | 600 | 1.80635382352 | 1.20423588235 |
| 5 | 1200 | 1.80643272872 | 1.20428848581 |

The last column is only the quartet part of an unpolarized observable. Finite cutoffs and l<=4 remain model restrictions. The comparisons are empirical convergence controls, not rigorous endpoint or omitted-wave remainder bounds. Charged/NLO/field matching and detector predictions remain separate. [Four-case synthesis](quartet-full-phase-synthesis-findings-20261005.md), [independent synthesis review](quartet-full-phase-synthesis-review-20261005.md).

## A reconstructed connected-cutoff resolvent fails on physical three-nucleon packets

The actual antisymmetric Jacobi construction now tests the proposed connected-cutoff pencil on normalized physical packets, including the exterior spectator interval. Sixteen contractions cover two pair cutoffs, two spectator grids, two energy pairs and two spin packets. Independent Cartesian spin tensors, different full momentum quadratures and direct MP65 integrals reproduce the saved results. Both two-energy and derivative identities retain nonzero physical defects exceeding ten times their observed refinement changes. This rejects the specific energy-independent resolvent reconstruction within the declared controls; it does not establish a universal four-body obstruction or identify a new force. The producer's exit audit failed on an inherited path alias. The reviewer authenticates that alias and the saved scientific scope while preserving the failed producer status. [Diagnostic](stm-cluster-resolvent-findings.md), [independent review](stm-cluster-resolvent-review-20261005.md).

## Atomic callback consistency is repaired, while the original timing target still fails

An independently reviewed bounded-memory method retains ordered MP arithmetic and exact cache limits. Subsequent mutation tests found mutable scientific helper bindings in an apparently immutable wrapper. The new callable graph rejects those witnesses and preserves all 36 small source comparisons and metadata. Its original 0.05-second maximum-check target nevertheless fails at 0.05347 seconds. The failed benchmark remains failed. A read-only cross-check also identifies unsealed native MP and matrix lookup slots; the existing witness acceptance does not cover that path. A separate dispatch repair and independent review are required before cost admission. A new proposal divides the seven full-bank cost requests into three explicit jobs, accounting for measured setup and guard overhead; it is not a completed cost calculation. All 48 physical quadrature plans remain unexecuted. [Bounded cache method](atomic-contact-bounded-cache-findings.md), [callable review and timing disposition](atomic-callable-guard-review-20261005.md), [new cost contract](atomic-contact-partitioned-cost-plan-20261005.md), [additional dispatch precondition](atomic-native-dispatch-precondition-20261005.md).

## Paired-state numerical costs and controller failures remain distinct

The physical polynomial pilot completes 200 Hamiltonian actions and reproduces the thirteen saved trial endpoints. Independent higher-precision Fock-space calculations accept its coefficients and endpoint arithmetic; fresh balancing also matches saved native values. Controller reviews then expose mutable clock settings, conversion helpers and actual MP receiver dispatch. The latest candidate passes 69 listed checks, but a separate diagnostic and independent review show that changing the native scalar class lookup commits 6 for an intended value of 3. All three admission checks remain false. Those failures remain explicit and do not authorize the remaining twenty-seven trials. The numerical cost evidence supplies no new paired stationary state or propagated endpoint. [Physical pilot review](pair-curve-physical-review-20261005.md), [conversion admission failure](pair-curve-admission-review-20261005.md), [latest independent dispatch diagnostic](pair-curve-conversion-review-20261005.md).

## Complete Gaussian coordinates retain both physical soft modes

Independent review resolves all 23,220 fluctuation coordinates into four charge/time-reversal blocks, reconstructs 192 sampled full-operator columns and checks separate 16-state thermal traces. Both small-curvature modes lie in the physical neutral-even block and cannot be removed as forbidden directions. Their Rayleigh values remain about 1.58e-6 and 3.79e-7, below the unchanged 2e-6 promotion threshold. Measured block assembly is feasible, but sampled leakage is not a complete operator error bound. No corrected physical root or rigorous full minimum certificate is claimed. [Method and measured cost](gaussian-phase-precision-findings.md), [independent review](gaussian-phase-precision-review-20261005.md).

## Molecular, pion and spectral methods advance without physical promotion

The physical molecular adapter reproduces all seven saved 41-state stages and thirty proposals with unchanged native arithmetic. Separate small stiff-chain controls establish a conservative mixing-and-driving bound and check the public CO rate formulas. A full coupled 41-state temperature corridor and physical trajectory remain unfinished. [Adapter review](molecular-positive-physical-review-20261004.md), [corridor method review](molecular-corridor-bounds-review-20261005.md).

Independent full-pion Hessians and constrained tangent controls exclude the tested off-axis alternatives at leading order and for the declared local quadratic correction. The full NLO determinant is still restricted to positive pion-mass domains. These checks do not supply a viable field, kinetic history or formation mechanism. [Off-axis method review](axion-offaxis-review-20261005.md).

Small spectral tests independently validate the full constrained Hessian, including residual second derivatives, continuum cross terms and priors, and its streamed assembly. The physical optical cost attempt saves 24 of 178 fixed sample keys before its time cutoff; all 24 are in the first line. Independent analytic MP derivatives reproduce every saved jet and all six geometry maps. The expensive nested derivative oracle dominates the cost. A roughly 37.6-hour two-pass extrapolation is descriptive and nonrepresentative, not a time bound or completed certificate. The full physical root and minimum remain uncertified. [Small certificate](spectral-constrained-certificate-review-20261005.md), [streamed Hessian](spectral-enclosure-stream-review-20261005.md), [partial physical cost review](spectral-physical-optics-review-20261005.md).

## A new two-photon source leaves its photon-count normalization unresolved

The acquired CosmoSpec paper prints a HeII spectral fit whose integral is 0.99999559091949, while its text specifies two photons per decay. Independent direct, beta-function and transformed-integral calculations reproduce the discrepancy. Its separately printed neutral-helium profile instead integrates to 2.00039934. The stated HeII rate 526.57 per second also differs from the previous nonrelativistic hydrogenic nominal 526.1184; the source attributes its value to a relativistic correction and hydrogen normalization. This source comparison does not supply an uncertainty distribution or justify changing a coefficient. The closed 25-state census still has its two-photon channel off. The official implementation page supplies only a construction notice; the inspected public hydrogenic package provides further bound-free routes but no adopted profile correction. [Acquisition and source audit](helium-two-photon-source-audit-20261004.md), [376-check independent review](helium-two-photon-source-review-20261004.md).

## Native stellar composition tangents preserve values but retain failed controls

A generated native composition tangent reproduces all 400 original Coulomb free-energy coefficients bit for bit and keeps all forty recorded branch traces. Independent high-precision differentiation accepts 240 required mixed derivatives, with a separate reviewer reconstructing the composition mixtures and moving-temperature boundary chain. The original finite-path tests still contain 112 failed comparisons, concentrated in higher thermal derivatives. The review accepts the diagnostic evidence; it does not accept the replacement method. Actual full EOS tables, packing/blend composition derivatives and the assembled stellar matrix remain unfinished, including 762 unresolved whole-matrix references. [Native diagnostic](offzone-eos-composition-repair-findings.md), [4,184-check independent review](offzone-composition-repair-review-20261004.md).

## Correct local paired-state geometry does not ensure a useful finite step

Independent projected-state jets, five directional controls and the full 92-mode connection and trust-system identities validate the local covariant construction. A bounded run completed thirteen of forty prescribed trials before its cooperative deadline. Large retracted and trust steps still raise the actual energy despite predicting a decrease. Three physical-Cauchy diagnostics lower energy by roughly 1e-12; no state was propagated. Independent MP110/120 calculations reproduce all thirteen raw and balanced endpoints, domains and original signed energy policies. Twenty-seven nodes remain pending, and neither interaction has a newly accepted optimization endpoint. [Findings](pair-covariant-trust-findings.md), [4,366-check review](pair-covariant-trust-review-20261004.md), [separate polynomial proposal](pair-covariant-trust-polynomial-plan-20261004.md).

A separate exact polynomial method then passes small projected-state controls. Independent creation/annihilation expansion in the full 256-dimensional Fock space reproduces every coefficient and structural zero for both 20-state fixtures, including a chart of degree eight. The actual 512-state physical support is enumerated only to establish its degree-six bound; no physical Hamiltonian column or pending node is evaluated. Native scaling retains guard bits returned by the linear solver, and direct endpoint corrections distinguish an unrounded formal curve from the actual MP80 endpoint. Failed premature-rounding controls remain archived. [Small polynomial method](pair-curve-polynomial-findings.md), [3,631-check independent review](pair-curve-polynomial-review-20261004.md).

## Exact atomic templates pass small controls, while full-basis cost remains unresolved

All 2,233 required moment powers pass independent MP150 analytic-moment controls; ordered recurrence values also reproduce the previous arithmetic bit for bit. Complete small signed source vectors and literal-metric response solves pass. The first authorized full257 cost calculation then hits its unchanged 300,000-value cache limit after 11.925 seconds, completing zero of twelve vectors. This retains a failed cost result and establishes no full-basis speedup. A separate bounded-memory derivation preserves ordered sums through deterministic eviction and exact recomputation. All 48 physical quadrature plans remain unexecuted. A later dependency audit also identifies the exact inherited coalescence file through all five authenticated parents, while recording that old template entry/exit checks did not directly rehash that file. [Small-method review](atomic-contact-template-review-20261004.md), [cost failure](atomic-contact-template-cost-findings-20261004.md), [input qualification](atomic-contact-template-input-qualification-20261004.md).

## A serial native matrix kernel reproduces the positive molecular method

The corrected native kernel preserves exact rounded dot products and separate scalar multiply/add rounding. An independent integer-only dyadic oracle checks all operations at five precisions, and fresh augmented matrix exponentials test the nonreversible semigroup. All seven saved 41-state stages reproduce their small and final propagation/occupation matrices bit for bit, including states, scalar preparations and separately checked heats. Seventeen regression tests cover the repaired admission defects and resource/failure controls; the earlier failed candidate remains archived. These 460 independent checks establish a matrix method, with no physical adapter or trajectory admission. Source-panel turning and full coupled histories remain further work. [Native method](molecular-positive-native-findings.md), [independent review](molecular-positive-native-review-20261004.md), [physical adapter contract](molecular-positive-physical-adapter-plan-20261004.md).

## The full declared helium radiative census passes transient controls

All 30 prescribed gas/radiation cases now pass: three source gas temperatures, three electron densities, three radiation-to-gas temperature ratios, and three additional CHIANTI-energy LTE controls. Independent review covers all 900 saved population vectors and 180 initial states at MP80/100, reconstructing the full 25-state kinetics and every one of its 24 relaxation modes. Separate gas and photon heats, strict positivity, raw transient entropy balance and chronological relative-entropy decay pass their original gates. The slowest relaxation time across the declared grid ranges from 1.63 nanoseconds to 2.73 seconds. This completes the fixed Table-2-only transient census; the two-photon channel, charge-state chemistry and a physical reservoir history remain absent. The 12 earlier deep-equilibrium entropy failures retain their failed status, and the source's printed oscillator-strength coefficient discrepancy remains unexplained. [Complete census](helium-radiative-census-findings-20261004.md), [35,490-check independent review](helium-radiative-census-physical-review-20261004.md).

## Three-body matching exposes a separate four-body cluster failure

A new Hermitian conversion regulator reproduces the full negative-energy three-channel STM operator in its declared regulator limit. Independent reconstruction checks all forty finite-cutoff matrices and the limiting operator. The same prescription fails the separated deuteron–dineutron test: its spectator filter still acts on the other dimer's internal momentum when the two clusters are spatially separated. The parent product-state energy defect grows from 2.60 to 10,427 MeV over the declared cutoff ratios and approaches a nonzero linear slope, 0.837835 MeV fm. This is an expectation-value defect demonstrating failure of that cluster prescription, not a calculated new threshold. Reversing the regulator limits changes the result; tied limits also leave a finite defect. Actual attached four-body kernels, ultraviolet cancellations and reaction amplitudes remain unfinished. [Construction and rejection](fourbody-conversion-findings.md), [independent review](fourbody-conversion-review-20261004.md).

## Positive molecular integration passes coupled source-step controls

A positive semigroup retains all 41 molecular populations without floors, clipping or renormalization. A new endpoint composition couples its two heat stages to the spin and atomic reactions and solves the temperature energy equation. Separate matrix-exponential, reaction-integral and entropy controls accept all seven declared physical step tests. MP80/100 agreement is below 1.01e-75 in the scaled state comparison. Full-step versus two-half-step differences pass the original 2e-6 scaled gate, but the largest rare-population relative difference is 2.96e-5; arithmetic precision and time-discretization accuracy remain distinct. The previous complete 68-case integration retains its positivity failures. These method tests do not replace it with a validated full trajectory or supply cloud chemistry. [Source-coupled method](molecular-positive-source-findings.md), [independent review](molecular-positive-source-review-20261004.md).

A separate two-level synthetic adaptive trajectory now saves all 121 declared times and reproduces its native history exactly across 96 restart segments. Independent review checks 4,726 numerical and provenance conditions, including a separate Radau solution and the temperature-knot event. The largest state error is 7.00e-9 and the refinement difference 5.28e-9, below the unchanged 2e-6 gates. Two budget interruptions remain interrupted; a new authorized supplement completes only the remaining refined steps and reuses the exact completed engine records. Sampled event panels do not prove the absence of hidden turning points. This accepted method still requires a budgeted physical 41-level trajectory adapter and independent admission. [Adaptive method](molecular-positive-trajectory-findings.md), [independent trajectory review](molecular-positive-trajectory-review-20261004.md).

## Atomic contact work retains the full finite metric

The atomic calculation now starts its Ritz solve below the point-Coulomb lower bound and retains every basis direction. Independent review verifies the full 233-function metric and excitation operator. All four declared 257-function banks pass their MP80 and MP100 ground calculations, retaining all 256 excitation directions. A separate 300-check review reconstructs their saved-matrix norms, energies, contacts, residuals and Cholesky factors at MP120; it does not repeat the literal moment assembly or eigensolvers. Ordinary-contact changes from the 233-function result range from 2.68e-9 to 1.87e-5; none evaluates the radiative-contact correction. Separate quadrature reviews exposed malformed plan values, noncanonical context types and mutable callback identities; the corrected method wrapper preserves entry identities and passed a separate 797-check review. The earlier failures remain archived. A measured three-screen source probe takes about 40 seconds per pair of nuclear and electron-pair contractions, so the proposed 16-node blocks exceed their time budget. Exact moment precomputation is specified for benchmarking; the full 48 quadrature plans remain unexecuted. [Geometry census](atomic-contact-geometry-census-findings-20261004.md), [independent matrix review](atomic-contact-geometry-census-review-20261004.md), [cost probe](atomic-contact-parent-source-cost-20261004.md), [transaction review](atomic-quadrature-snapshot-review-20261004.md).

## The aligned static axion branch now has a complete root census

Outward interval certificates cover the entire declared LO and NLO aligned branch and establish exactly six stationary points on 0 <= theta < 2 pi. The roots are j pi/3: three have positive static reduced curvature, 3.553408655e7 MeV^4, and three have negative curvature, -5.199622407e7 MeV^4. All nine pion directions are positive. All 18 full ten-coordinate calculations at three scales pass a separate 3,398-check review, including a distinct closed-chain interval proof on every leaf. This completes the finite-order aligned static census; off-aligned SU(2) branches, the complete dynamical kinetic metric, dense-matter matching and formation remain open. [Census](axion-static-census-findings-20261004.md), [independent review](axion-static-census-physical-review-20261004.md).

## Native stellar composition derivatives retain a specific source-level mismatch

An additive capture preserves all 34 old native outputs and both earlier traces exactly, while exposing 136 calls with the ten Helmholtz derivatives and initialized composition rows. The reconstruction passes its thermodynamic and value controls, but 56 of 64 composition-path comparisons and all four original face-support spans fail. A separate 21,189-check review reproduces these failures. The pure Skye source initializes both composition derivative rows to zero, and the blend source omits composition derivatives of its weight. These source facts narrow the diagnosis without establishing a corrected derivative or dividing every error into those causes. All 762 unresolved native Jacobian references and the failed ordinary stellar comparison remain open. [Captured analysis](offzone-eos-tangent-capture-analysis-20261004.md), [independent reconstruction](offzone-eos-tangent-capture-analysis-review-20261004.md).

The independently checked ideal electron reference now covers all 50 distinct captured temperature/density states under both named constant conventions: 100 evaluations and 272 call/convention comparisons. All 272 original value gates and six directional gates still fail. Eight fresh independent MP100 controls agree with the integral jets within 6.97e-74; the review passes 10,020 reconstruction checks and retains all 278 scientific failures. For face535/d5, the reference-only support span passes at ratio 0.8412, while the comparison including candidate uncertainty fails at ratio 2.4822. Thus improved reference stability still does not admit a native replacement. Neither constant convention is established as the table's provenance. [Complete comparison](electron-gas-captured-production-findings-20261004.md), [independent review](electron-gas-captured-production-review-20261004.md).

## The complete molecular schedule retains its strict positivity failure

All 16 main trajectories and 52 controls are saved. A failure-preserving collector retains all 51 original completed states byte for byte and evaluates the remaining 17. Every one of the 49 aggregate solver, level, tolerance and quadrature comparisons passes. Nevertheless, the original 41-level case contains a population of -2.07e-38, making its mixture entropy undefined. Seven fresh independent trajectories agree within 1.04e-9 of the saved paths, but two independent 41-level controls also produce tiny negative populations. The overall calculation remains failed_numerics; no population floor or clipping is applied. A positive integrator is still needed before acceptance. Ground-state hydrogen, a fixed-volume gas and a prescribed Planck bath do not supply complete chemistry, dust transfer or cloud formation. [Complete schedule](molecular-catalyst-census-findings-20261004.md), [independent failure disposition](molecular-catalyst-census-review-20261004.md).

An additional official NN-OnLine format audit distinguishes its upload and browser formats from the acquired Granada files. It does not decode the remaining archive-header fields or authenticate the calibration mask and nuisance contract. The exact independent nuclear likelihood remains blocked on those named inputs. [Source-format audit](granada-nijmegen-format-audit-20261004.md).

Earlier sections retain their historical scopes; this checkpoint supersedes their pending aligned static census and molecular production steps while retaining the failed physical controls.

## Both spin sectors now contribute to four coherent neutral breakup predictions

All four declared energy/cutoff cases pass the original finite-bin grid, product and angular-integration gates. The new quartet calculation retains coherent l=0 through 4 interference, then combines the orthogonal incoming spin sectors as sigma_D/3 + 2 sigma_Q/3 using exact matching doublet parents. Two strict interior bins and four reserved points per case are reconstructed in a separate complete-source review. The largest quartet grid difference is about 5.65e-6 against the unchanged 0.003 gate. The summed weighted refinement changes are observed diagnostics, not rigorous error bars. The infinite-wave remainder, uniform endpoint control, charged and higher-order matching, and laboratory detector likelihood remain unfinished. [Four-case production](quartet-bin-production-findings-20261004.md), [independent review](quartet-bin-production-review-20261004.md).

## Source-matched NLO pion gaps pass the vacuum control

The existing three-sector vacuum candidate now has strictly matched NLO charged and neutral pion pole masses and neutral-angle shifts. Every saved angle, sector-path and sensitivity row passes an independent SU(2) trace and order-expansion review. Central sampled mass minima are 82.7064 MeV charged and 78.8379 MeV neutral; the largest sampled squared-mass correction is 7.7356% of LO. The inherited 139.57 MeV normalization is kept as a model input, not relabeled as the measured neutral mass. The source's printed vacuum-loop sign inconsistency is retained explicitly; renormalization-scale cancellation supports the Lu/Grilli sign already used by the parent. Positive fixed-theta pole masses do not establish the complete coupled static Hessian, an interacting dense EOS or a viable physical history. [Equations and source audit](axion-pion-nlo-findings.md), [independent review](axion-pion-nlo-review-20261004.md).

## The declared Gaussian phase-search schedule is complete

All 30 declared states now have independently reviewed same-Hamiltonian stationarity, charge, curvature, finite-path and free-energy controls. Heating/cooling and both midpoint seeds match after exact lattice translations. The sampled normal pairing-curvature sign change lies in [2.6328125, 2.640625] MeV. Two nonzero paired candidates have lower free energies than their normal comparators, but their lowest reported curvatures, 1.5813e-6 and 3.7924e-7, remain below the original 2e-6 minimum-promotion gate. Both remain inconclusive. The upper candidate is normal-like by the declared pair-norm cutoff; its 6.9594e-8 direct matrix distance does not pass the separate 2e-8 branch-match gate. The extra triplet seed reaches a matched normal saddle. This completes the fixed finite-volume schedule and its qualified comparisons, not a global phase diagram, coexistence result, interface or nucleation calculation. [Production](gaussian-phase-polished-continuation-production-findings-20261004.md), [independent 30-state review](gaussian-phase-polished-production-review-20261004.md).

The two projected-pair physical searches each accept four very small energy-lowering steps, but both still fail the original gradient and curvature gates in occupied and original coordinates. Independent 100-digit replay reproduces all 19 trials, both final state journals and all 96 occupied-Hessian columns in each family. Its 739 checks accept the quantitative failure classification. No stationary or minimum endpoint is promoted. [Production](pair-energy-continuation-production-findings-20261004.md), [independent physical review](pair-energy-production-review-20261004.md).

## The free spectral fit reaches its original stationarity threshold

The exact-parent dense continuation now reaches chi-squared 2292.741782257208 and original-forward KKT 1.84584e-7, below the unchanged 2e-6 threshold. All 21 accepted proposals and complete endpoint derivative controls pass a separate 14,305-check reconstruction. The profile retains four negative full-space modes, while the current 135-dimensional free face is positive. Strict nonzero active-bound multipliers matter to feasible perturbations; the original minimum flag remains false. The final tiny likelihood changes are sensitive to upstream forward/profile arithmetic, so large final acceptance ratios are not precise descent certificates. This is numerical stationarity in the stated bounded coadd likelihood, not an alpha measurement or field bound. [Production](spectral-dense-continuation-production-findings-20261004.md), [independent review and precision diagnosis](spectral-dense-production-review-20261004.md).

Earlier checkpoint sections preserve their historical scopes; the completed Gaussian schedule supersedes the earlier pending physical-continuation step.

The next molecular-method control replaces the prescribed hydrogen ion fraction with evolving ground-state H, protons and electrons, while retaining every CO population mode and the H2 spin/gas energy equations. A separate 174-check speed-integral, mixture-entropy and full-mode integration review passes; the largest independent trajectory difference is 3.59e-10. This accepts the two method controls only. The declared 16-case production, complete chemistry and evolving radiation spectrum remain unfinished. [Method](molecular-catalyst-findings.md), [independent review](molecular-catalyst-review-20261004.md).

## Three particle-pocket fission endpoints cost energy

The same zero-temperature scalar/Thomas–Fermi functional now compares an N=30000 pocket with three asymptotically isolated daughter pairs: (3000,27000), (7500,22500) and (15000,15000). Their endpoint energy costs are 588.4354, 925.4043 and 1098.0537 GeV. All exceed ten times the summed observed refinement and independent-seed energy envelopes; those envelopes are empirical numerical controls, not rigorous error bounds. Independent reconstruction of the sixteen new daughter profiles and the frozen anchors reproduces every endpoint sign. Exact volume-preserving affine deformations also increase the scalar gradient energy, and a connected dilation trial improves on two identical isolated copies. These are finite comparisons within one declared dark-sector model. Other partitions and branches, finite-separation barriers, evaporation, formation and lifetimes remain unresolved. Ordinary-matter coupling is still zero. [Derivation and controls](fermi-fission-findings.md), [endpoint calculation](fermi-fission-production-findings.md), [independent production review](fermi-fission-production-review-20261004.md).

## Full projected-state derivatives and energy decisions are independently checked

The projected-pair method retains all 512 amplitudes, all 96 real affine derivatives and all 92 directions after removing exact gauge coordinates. Independent 100-digit differentiation checks both complete Hessians, including nonstationary terms, and replays all 24 archived step decisions with separate state construction and energy quotients. Twenty-two formerly rejected terminal lines pass the declared energy/model criterion even though a gradient merit increases. This validates an energy-based method; it does not accept a physical optimizer endpoint. Both frozen families still fail their stationarity gates. A lossless multiprecision continuation and complete diagnostics in both coordinate conventions remain required. [Method](pair-energy-trust-findings.md), [independent Hessian and decision review](pair-energy-trust-review-20261004.md).

## A direct electron reference preserves the stellar discrepancy

Independent momentum quadrature and thermodynamic identities validate the direct ideal electron–positron reference. Comparisons at four captured native stellar cells retain the original ion, radiation, Coulomb and blend contributions. Both separately declared constant conventions fail all four original value gates: eight failures remain. The older Timmes constants reduce the discrepancy, but this does not identify the table generator or justify a source replacement. No native EOS patch, complete third-derivative path or validated stellar trajectory follows from this diagnostic. [Reference equations](electron-gas-reference-findings.md), [independent reference review](electron-gas-reference-review-20261003.md), [captured comparisons](electron-gas-comparison-findings-20261004.md), [independent comparison review](electron-gas-comparison-review-20261004.md).

The reviewed dense Gaussian continuation adapter recovers the complete pairing-curvature substep and advances the bidirectional search. At 2.6875 MeV the two roots differ by 2.7882e-8 after the best exact lattice translation, exceeding the original 2e-8 matching gate. The run stops with that failure preserved. Its schedule-termination flag does not mean that the refined branch or physical phase comparison is complete. A separate stricter internal normal solve matches the saved pair to 7.8e-15, with fresh curvature/path audits and all scientific gates unchanged. Independent response, correction and restart controls pass. The local error estimate does not replace direct matching, and the full physical continuation still needs its own result and review. [Dense adapter](gaussian-phase-dense-continuation-findings.md), [partial production and mismatch](gaussian-phase-dense-continuation-production-findings-20261004.md), [polishing method](gaussian-phase-polished-continuation-findings.md), [independent polishing review](gaussian-phase-polished-continuation-review-20261004.md).

For the spectral fit, a complete dense active-face quadratic method passes independent multiprecision sphere/box controls and full-kinematic synthetic recovery. A separate Cholesky solve reproduces the saved physical model step. Every proposed displacement, continuum profile, energy decision and trust-radius update in both synthetic recoveries is independently reconstructed. This is a method benchmark; a production adapter and a new physical fit remain required. The prior failed stationarity result is unchanged. [Dense method](spectral-dense-trust-findings.md), [independent review](spectral-dense-trust-review-20261004.md).

Earlier checkpoint sections below preserve historical scopes. The new results supersede only the identified numerical steps.

## Closed helium collision dynamics retain every rare population

The complete published HeII table supplies 300 fine-structure transition pairs at ten printed temperatures. Independent extraction and quantum-label matching resolve the paper's non-energy-ordered indices and stale CHIANTI text labels. A closed 25-level collision model now evolves all 24 relaxation modes at 80 digits, with selected 100-digit controls. All twenty temperature/energy-convention cases, three preparations and eight times pass separate 120-digit full-generator evolution and bath-heat integration. Positivity, mass and the original raw entropy identity pass on the saved decimal generator; binary64 exports remain separately labeled. High working precision does not improve the physical precision of the atomic inputs. Slow modes remain unequilibrated at the last saved time: the largest final L1 distance from equilibrium is 1.22381. Radiation, ionization, recombination, neutral-helium collision kinetics and environmental reservoirs remain absent. [Source and rate controls](helium-collision-findings.md), [precision backend](helium-collision-precision-findings.md), [population results](helium-collision-population-findings.md), [independent production review](helium-collision-precision-production-review-20261003.md).

## A covariant channel cutoff still fails to define a physical projection

An explicit finite source-mode calculation reconstructs the four physical pnnn spin-one states from 840 auxiliary channel coordinates. Exact coefficient algebra proves physical rank four and channel nullity 836. Although the proposed pair filters transform covariantly under permutations, they send a zero physical state to a nonzero state: the retained null witness acquires norm 0.66096 after filtering. The filter also fails to commute with the induced physical metric. It therefore cannot be promoted as a physical orthogonal projection. With the literal Hamiltonian unchanged, complete static and Feshbach resolvents and their different norm metrics agree with direct finite-space calculations. Dropping the Feshbach self-energy changes the projected resolvent by roughly 6e-5 relative. This rejects one candidate and verifies the exact finite algebra; it supplies no globally matched STM regulator, continuum amplitude or reaction rate. [Finite derivation and witness](fourbody-projection-findings.md), [independent review](fourbody-projection-review-20261003.md).

## Derivative repair and optimization remain separate

The stable Voigt-tail derivative implementation now passes all original scalar and full/profile Hessian gates. Independent review includes the complete 699-point union, supplying 147 points omitted from the revised producer benchmark; the earlier coverage overclaim is preserved with its correction. The new fit wrapper preserves the coadd objective, all 154 full and 142 profile coordinates, nuisance bounds and native feasible-path checks. A bounded continuation lowers chi-squared to 2292.74269, but its scaled KKT residual remains 0.02011 against 2e-6. Passing the derivative controls does not accept that nonstationary endpoint as a minimum. [Tail controls and coverage correction](spectral-tail-findings.md), [independent derivative review](spectral-tail-review-20261003.md), [continuation method](spectral-tail-continuation-findings.md), [bounded production](spectral-tail-continuation-production-findings-20261003.md), [independent production review](spectral-tail-continuation-production-review-20261003.md).

The same-H Gaussian phase method and its bidirectional translation-aware continuation driver also pass independent controls. Ten coarse normal roots cover 2–4 MeV in both directions, but the first midpoint's 2916-dimensional pairing eigensolve fails to converge. A coarse curvature-sign change is not a refined bifurcation, a paired free-energy comparison or a phase boundary. A separate full 2916-column matrix calculation and independent reconstruction recover positive midpoint curvature 0.01747765 at 2.75 MeV. The faster scalar Krylov candidate misses members of repeated eigenspaces at all three control temperatures and remains rejected. The accepted dense calculation retains every direction and the original residual and finite-path gates. Resuming the physical temperature search still requires an explicit reviewed adapter for the changed curvature method. [Gaussian method](gaussian-phase-competition-findings.md), [driver and restart controls](gaussian-phase-continuation-findings.md), [partial production and failure](gaussian-phase-continuation-production-findings-20261003.md), [dense recovery](gaussian-phase-curvature-recovery-findings.md), [independent recovery review](gaussian-phase-curvature-recovery-review-20261003.md).

For the stellar equation of state, the author's direct Timmes and Fermi–Dirac source archives have been acquired. Their older constants and ionization/positron conventions must be reconciled before an ideal-electron reference comparison. No downloaded code has yet been executed, no native source has changed, and the original value and uncertainty-inclusive derivative failures remain. [Source inventory and required controls](electron-eos-source-audit-20261003.md).

The following earlier-checkpoint sections retain their historical scopes; later controls supersede only the numerical substeps identified above.

## A second odd pp wave changes the finite-screen breakup calculation

The charged quartet calculation now solves pp pair/spectator angular momenta 1 and 3 coherently at 5 MeV, cutoff 600 MeV and screening radius 2 fm. The independently normalized final breakup cross section is 0.0554739885 fm². Its increase over the same-grid l=1 calculation is 1.90667e-6 fm², larger than the saved paired refinement spread of 2.55006e-8 fm². Most of that change enters through the coupled incoming solution; direct pp3 outgoing intensity is tiny. The held-out elastic-loss comparison differs by 0.000609 relative, within the original 0.003 gate. All declared spectator, radial, product/angular and outgoing-volume refinements pass. A timed-out operator run preserves 331 rows; exact assembly/resume controls precede the authenticated final 31-row continuation. The coarse 120-panel optical discrepancy remains failed and unselected. This is a finite-screen, finite-wave quartet component, with no bound on omitted odd waves, unscreened limit or charged doublet. [Operator and results](charged-odd-wave-findings.md), [independent review](charged-odd-wave-review-20261003.md).

## Threshold arithmetic distinguishes three pole prescriptions

An integral form removes the magnetic-current formula's zero-binding cancellation while retaining the source effective-range pole in that formula. Independent literal-amplitude calculations verify the inverse-NLO denominator, its derivative, ordinary matching and the finite-amplitude unitarity convention. A bounded scalar survey retains all 88 roots across 110 cases, including negative-residue roots, with multiprecision brackets and original binary64 residual checks. Ten continuous zero-binding or double-root thresholds are solved separately. For the nominal triplet inverse-amplitude prescription, zero binding occurs at pion mass 105.19908 MeV and the positive-residue branch meets a negative-residue root at 142.02597 MeV. This explains the sampled integer window within that declared prescription; finite grids do not prove global root completeness. At the ordinary mass, inverse-NLO, effective-range and strict first-order binding estimates are 2.36117, 2.22799 and 1.88716 MeV respectively. Their difference is not erased by a refit; the first-order relative pole shift exceeds the declared 0.1 small-shift diagnostic. The raw truncated amplitude retains its leading pole structure. No unique higher-order physical window, joint lattice posterior, finite-angle matching or new physical current uncertainty is supplied. [Threshold controls](quarkmass-thresholds-findings.md), [bounded survey](quarkmass-bracketed-survey-findings.md), [independent survey review](quarkmass-survey-review-20261003.md).

## Accurate projected energies retain the optimizer's terminal rejection

All 24 saved projected-pair cases are replayed with complete 512-component even-block states, independently checked in the full 14,400-state fixed-number sector. The reference occupancy gives amplitude polynomials of degrees one, two and three; none is omitted. Multiprecision stable energy differences, all 96 coordinate gradients and the full 92-direction tangent frames agree with a separate 120-digit construction. All 22 terminal proposals lower energy but increase the gradient merit when baseline and candidate use consistent balancing. Both previously accepted controls lower both quantities. Thus energy cancellation alone does not resolve the stalled acceptance rule. The amplified directional curvature discrepancy of 0.9573 MeV at a roughly 670-MeV scale passes its original relative 0.002 gate; this is not a full multiprecision Hessian certificate. No new optimizer or stationary physical state is selected. [Precision replay](pair-precision-findings.md), [independent review](pair-precision-review-20261003.md).

## Electron interpolation is located, but its proposed replacements fail

The native stellar capture records every interpolation cell, coefficient and weight for the same 136 electron free-energy calls while preserving all original state binaries and transport traces. Independent 80- and 100-digit polynomial reconstruction and a different-author Bernstein/de Casteljau evaluation agree. Re-forming Hermite weights at high precision reduces the disputed derivative's support spread enough to pass a comparison that omits candidate uncertainty, but the full original uncertainty-inclusive comparison still fails by a factor of 2.48. All four proposed interpolation substitutions also fail the original thermodynamic value gate; the largest discrepancy is about 10.25 times that gate. Those four failures remain explicit in both analysis and review. This identifies weight formation as the dominant arithmetic layer without establishing the underlying table's physical accuracy or repairing the derivative. All 762 unresolved full-matrix references remain; no corrected stellar trajectory is selected. [Native interpolation and rejected substitutions](offzone-electron-interpolation-findings.md), [independent review](offzone-electron-interpolation-review-20261003.md).

The full published HeII collision table has also been acquired from its author's institution: 300 fine-structure transition pairs at ten temperatures. The preprint provides only excerpts, while the separate CHIANTI file omits the near-degenerate 2s–2p transitions. Extraction and level-convention reconciliation precede any collisional population calculation. [Source acquisition and limits](helium-collision-source-audit-20261003.md).

The following earlier-checkpoint sections record the evidence available at those checkpoints. The continuations above supersede their pending numerical substeps only within the scopes stated here.

## Electron free-energy inputs retain the stellar derivative failure

A new native capture preserves all 34 original perturbation states and the transport trace exactly while recording 136 Skye jets and 68 actual FreeEOS–Skye blends. Independent multiprecision reconstruction checks the thermodynamic formulas, blend weights, all original support uncertainties and an ordered-permutation attribution. The face-535 composition derivative still fails: its span is about 9.44e-9, giving a gate ratio of 4.54 or 5.41 with the retained candidate uncertainty. The captured electron free-energy jets carry nearly all the spread; final thermodynamic arithmetic and FreeEOS do not explain it. This locates the precision layer without establishing the upstream interpolation cause. All 762 unresolved full-matrix references remain, and no derivative repair or corrected stellar trajectory is selected. [Captured inputs and attribution](offzone-eos-precision-findings.md), [independent 12,213-check review](offzone-eos-precision-review-20261003.md).

## Faster full spectral curvature retains the original fit failure

Cached adjoint assembly computes the complete residual Hessian in about 0.55 seconds at the benchmark states while retaining the original likelihood and all 142 profiled directions. Independent full-coordinate calculations pass the original matrix, finite-path, transpose and known-minimum gates. A stricter residual-only precision diagnostic remains failed and is recorded separately. One strictly bounded continuation lowers chi-squared from 2295.716 to 2293.483, but the original projected gradient is still 0.193335 against 2e-6 and negative curvature remains. At the new endpoint, a stable-tail independent reference finds one full-Hessian column error of 3.68e-10, exceeding the original 2e-10 gate; endpoint curvature precision is therefore rejected as well. The cooperative stop occurs at 510.157 seconds, inside the strict 520-second optimizer bound. Lower objective values do not establish a stationary fit or an alpha estimate. [Method and bounded fit](spectral-adjoint-findings.md), [benchmark review](spectral-adjoint-review-20261003.md), [independent physical-path replay](spectral-adjoint-production-review-20261003.md).

## Attached four-body coefficients expose a regulator mismatch

The static-action calculation now derives its actual spin factors, normal-ordered two-, three- and four-nucleon coefficients, and finite-mode attached Born terms. Independent occupation-state and determinant calculations reproduce them. An additional antisymmetry control caught an incorrect intermediate-state exclusion; the failed attempt is retained and the corrected product-space propagation passes the original gate. The resulting contact-only spectator regulator nevertheless fails complete matching to the frozen three-body equation: a positive pure-exchange contribution survives outside the parent's global integration interval. Independent 80-digit integration confirms the missing contribution at four negative energies. This rejects that realization; it does not establish that every four-body extension is impossible or demand a new fitted force. A consistent global projection across embedded partitions and its physical reconstruction remain unfinished. No continuum amplitude or rate is selected. [Derivation and failed matching gate](stm-attached-contact-findings.md), [different-author review](stm-attached-contact-review-20261003.md).

## A source-specific mass continuation fixes one leading current response

The Chen et al. source equations now reproduce the declared ordinary and lattice scattering anchors and determine a leading magnetic-current mass continuation after its ordinary normalization. Rendered primary equations distinguish the barred intercept from the fixed loop parameter; independent amplitude and multiprecision current calculations check that distinction and the source coupling conventions. The calculation retains scale/scheme controls, explicit pole exclusions and one-at-a-time input scenarios. Its truncated effective-range poles are not the paper's full-amplitude bound-state window. The binary64 current formula also loses cancellation very near zero binding, outside its six declared finite-current supports; the independent integral stays regular, and that implementation extension remains unfinished. The lattice triplet interval crosses zero, so neither a Gaussian inverse-scattering-length likelihood nor a joint posterior is supplied. This is a fixed-order, isosymmetric quark-mass calculation. It does not supply finite-angle CP-odd, isospin or three-nucleon matching for the cyclic field model. [Equations, conventions and limitations](quarkmass-matching-findings.md), [independent review](quarkmass-matching-review-20261003.md).

## The remaining declared larger-volume starts reach normal states

The cold family-B curvature failure is repaired by deriving the charge solver's stopping tolerance from the actual seven-point curvature stencil. The original interaction, charge offsets, chemical-work term and external gates remain unchanged. Independent 80-digit small-Fock and scalar-tail controls, plus all 33 saved large-state nodes and nine stencils, reproduce the accepted final error and span. A later production run completes both original B-at-1-MeV starts; independent terminal maps and enlarged Hessians classify both as normal local minima. Alongside the earlier A-at-1-MeV and B-at-2-MeV results, all six declared L4 starts are now classified. Reusing the normal path requires the original density-distance tolerance; the terminal arrays are not bitwise identical. The separately recentered old-mode diagnostic remains uncertified. Broader disconnected condensates, exact-number thermal ensembles, full eigensystem precision bounds and thermodynamic limits remain unfinished. [Stopping rule and census](paired-curvature-budget-findings.md), [independent benchmark and production review](paired-curvature-budget-review-20261003.md).

## Earlier full spectral active-bound search

An exact residual Hessian retains every one of the 142 profiled directions in the same 45-component spectral likelihood. Independent full-coordinate differentiation and Schur elimination verify it, including the nonzero-residual terms. A cancellation-resistant wavelength reference passes the original derivative controls; the earlier rounded-path failures are retained separately. A single bounded active-face fit lowers chi-squared from 2298.407 to 2295.716, but the original full-coordinate projected gradient remains 1.34655 against 2e-6. Its time limit leaves unfinished optimization, not a stationary solution or an alpha estimate. Measured exposure LSF and calibration covariance remain separate missing inputs. [Method, controls and failed fit](spectral-active-face-findings.md), [independent review](spectral-active-face-review-20261003.md).

## Earlier isolated two-dimer and conditional-block controls

An exact spectral calculation shows that multiplying two already-renormalized pair amplitudes into an isolated, energy-independent two-dimer propagation produces a logarithmic divergence. A finite energy difference exists, but does not determine its additive constant. Independent continuum quadrature verifies the spectral density, signs, pole residue and tail bound. A separate finite canonical block model verifies reconstruction, derivative norms and all ordered mixed/contact Born terms. These mathematical identities do not establish the actual static auxiliary-field conversion maps or the complete four-body interaction. The divergence alone does not imply a new physical four-body force. The subsequent attached-action calculation above derives finite coefficients but rejects the contact-only regulator realization. The same frozen STM subcluster still requires a compatible global projection and full physical reconstruction. No physical rearrangement amplitude or thermal rate is supplied. [Derivation and actionable next step](stm-fourbody-contact-derivation-20261003.md), [independent 198-check review](stm-fourbody-contact-review-20261003.md).

## Coherent differential breakup at four declared cases

The same neutral LO doublet amplitudes now combine spectator waves l=0 through 4 coherently in differential points and two finite center-of-mass bins, at 3/5 MeV and cutoffs 600/1200 MeV. Each new observable uses the directly evaluated cut-split, pole-subtracted source equation. Independent spin contraction and angular integration reproduce every bin; grid, product and bin-quadrature refinements pass. Cross-wave interference changes the finite bins relative to an incoherent wave sum. The separately interpolated full-phase flux is only a normalization regression. Exact phase-space endpoints, an infinite-wave remainder, quartet differential completion, charged/NLO/field matching and detector likelihoods remain unfinished. [Equations and four cases](coherent-breakup-findings.md), [independent production review](coherent-breakup-review-20261003.md).

## Physical-metric pair coordinates still fail stationarity

Retaining all 92 resolved nongauge tangent directions, a physical-metric trust step and admissible Cauchy comparison pass their derivative and known-minimum controls. Both finite physical searches still fail the original stationarity and curvature gates. Independent 80-digit polynomial-state evaluation finds a terminal binary64 energy-sign error for one rejected proposal; tiny-step gradient resolution also remains limiting. The projector endpoint itself was not archived, so its reported convergence cannot be independently replayed. These controls and failures supply no optimized paired component. [Method and saved paths](pair-metric-trust-findings.md), [independent precision diagnosis](pair-metric-trust-review-20261003.md).

## Atomic vacuum-polarization energy, with unresolved contact response

The static electronic vacuum-polarization operator is evaluated on the same ordinary finite-mass correlated helium states, retaining both attractive nuclear pairs and the repulsive electron pair. Independent polynomial moments and a 50-digit infinite-loop integral reproduce the full energy shifts. The resulting neutral and ionic threshold components are +18.86 and +14.10 micro-eV. These are individual corrections, not complete threshold accuracies. The separate helium contact response changes by 13.55% under the last basis refinement, failing its 1% gate; independent review also identifies a non-identity overlap metric in the larger response spaces. Neither the controlled hydrogenic response nor the accepted energy contraction rescues helium contact convergence or supplies a corrected weak rate. [Equation, units and failed attempts](atomic-vacuum-polarization-findings.md), [independent review and conditioning diagnostic](atomic-vacuum-polarization-review-20261003.md).

## Four larger-volume starts reach normal states

Two declared starts each for L4 family A at 1 MeV and family B at 2 MeV converge under the original stationarity gates. All four endpoints are normal states with positive full enlarged Hessians; no paired minimum is found in this finite census. Independent free-energy and charge-path calculations verify the accepted results. Family B at 1 MeV fails the original three-support curvature-span gate before either seed starts: its last span is 3.90e-7 against 2e-7. The independently replayed failure is dominated by amplified residual chemical work at the smallest support. That failed run remains archived; the subsequent curvature-aware calculation above completes its two original seeds. Broader searches remain executable work. No canonical thermal phase, global minimum or thermodynamic limit follows. [Finite census](paired-volume-continuation-findings.md), [independent review](paired-volume-continuation-review-20261003.md).

## Finite-amplitude angular backreaction

An even-axisymmetric scalar evolution now lets the quadrupole perturbation feed back into the spherical background and higher angular modes. Two prescribed amplitudes resolve leading quadratic l0/l4 responses over 200 inverse-mass time units. Separate angular, radial, timestep and box comparisons pass the original waveform and energy gates; independent full-field evolution reproduces the difference-coordinate calculation. The energy ledger includes measured damping and a canonical core flux, without identifying that flux as outgoing radiation. Only the stated finite interval, amplitudes and even m=0 sector are tested. No general three-dimensional stability, formation abundance or asymptotic lifetime follows. [Nonlinear equations and trajectories](axisymmetric-backreaction-findings.md), [independent review](axisymmetric-backreaction-review-20261003.md).

## Molecular populations, spin conversion and gas heat evolve together

Sixteen fixed-volume trajectories couple every retained CO population to two-state hydrogen spin conversion, gas temperature and a prescribed Planck reservoir. Four controls turn off conversion, and two cases receive higher-level refinements. Independent full-mode integrations agree within 1.07e-9 in the declared scaled norm; high-precision calculations also reproduce tiny near-equilibrium heat currents. Energy, positivity and entropy gates pass. Fixed molecular and catalyst abundances, an approximate source rate and a two-state hydrogen truncation remain explicit assumptions. Acquired thermal tails do not bound arbitrary pumping, and these conditional trajectories supply neither formation chemistry nor cloud fragmentation. [Equations, source discrepancies and trajectories](molecular-kinetics-findings.md), [independent review](molecular-kinetics-review-20261003.md).

## Seven fresh clock-product days reject the frozen noise model

The model selected on January 28–29 is held fixed for seven preregistered dates, January 31–February 6. All fourteen archived products pass input eligibility, and independent complete parsing, alternative spline construction and symmetric-metric filtering reproduce all 1,008 templates and 28 injections. Every day fails the fixed lag-correlation gate; five also fail the moment interval. No date, satellite, coefficient or gate is changed. The largest conditional score is therefore not a calibrated event significance. These new observations reject the tested residual model's adequacy while leaving receiver transfer, clock-species and physical-trajectory calibration absent. [Fresh observations and per-day failures](gnss-fresh-findings.md), [independent review](gnss-fresh-review-20261003.md).

## Improved clock filtering still fails noise adequacy

Four training-only folds select an order-16 vector autoregression from eighteen fixed candidates. An independent augmented-SVD implementation reproduces all 72 fits, 144 templates and 100 saved-origin parameter refits. On the reused third day, the maximum tested cross-channel lag correlation falls from 0.22248 to 0.069964 but remains above 0.05; one centered second moment is 1.20325 against an upper gate of 1.2. Both failures remain. Parameter resampling now refits the filter but conditions on its selected order and penalty, with fewer lag rows per replicate. Neither those sensitivity summaries nor the separate block-resampled scan tails establish event probabilities, fresh blind validation or a receiver-level exclusion. [Equations and diagnostics](gnss-temporal-findings.md), [independent review](gnss-temporal-review-20261003.md).

## Same-interaction four-body operator controls

Fermionic pair kernels in the 3+1 and 2+2 partitions now retain the sharp spectator regulator, spin/charge permutations and exact embedded three-body interaction used by the thermal subset. Independent integrations, full antisymmetrization and pole controls pass. A candidate auxiliary contact also passes finite-vector integration and local weighted-reciprocity checks, but the complete four-body derivation remains unverified: mixed contact/pair diagrams, auxiliary two-dimer propagation, full state reconstruction and channel residues have not been established. No rearrangement amplitude, thermal rate or active tritium edge is supplied. [Operator derivation](stm-fourbody-findings.md), [independent 919-check review and missing proof](stm-fourbody-review-20261003.md).

## Gradient-resolved conditional wall dynamics

Five canonical scalar-field evolutions now resolve gradients on 24³, 48³ and 96³ periodic grids for two declared smooth initial fields, two tilts and one prescribed expansion. Independent discrete-wave, kink, ODE and Fourier controls verify the operator and refinement. The finest phase-space differences are below 1.2%, with about fourfold improvement under refinement; energy balances include the explicitly supplied background work. Nine area definitions and three near-vacuum energy cuts retain their definition dependence. Near-vacuum harmonic excitation is not an outgoing radiation flux in this periodic box. These are conditional finite-time trajectories, with no matched stochastic formation law, reheating, domain abundance, infinite-volume limit or stellar-site prediction. [Equations and runs](domain-wall-findings.md), [independent review](domain-wall-review-20261003.md).

## Two-photon relaxation in the finite hydrogenic network

A source-normalized approximate two-photon spectrum adds 2s→1s relaxation, stimulated emission and directly integrated inverse absorption to the same finite-level E1 model. Independent high-precision frequency integrals, all 72 equilibria and 12 pairs of transient histories verify balance and energy accounting. The nominal infinite-mass Z=2 vacuum lifetime is 1.9007 ms. The hot-bath population corrections are small; some transient differences remain below numerical resolution. The spectral-fit error is not bounded by the much smaller quadrature error. Collisions, neutral helium, plasma dissolution and closed radiation transport remain absent. [Derivation, source normalization and limits](hydrogenic-two-photon-findings.md), [independent review](hydrogenic-two-photon-review-20261003.md).

## Larger-volume coupled charge response

An analytic two-charge susceptibility repairs the L4B normal-state curvature plateau under the original gates. Independent full Fock matrix-exponential controls verify the Nambu factor, partition constant, coupled chemical-potential solve and constrained response. The colder L4A absolute-subtraction calculation still fails, but a subsequent density-increment calculation passes the unchanged external gates at local supports: conservative span 6.54e-9 below 2e-7 and error 5.17e-9 below 2e-4. Independent 75-digit small-Fock exponentials and all 17 saved large-state contractions retain baseline chemical work and bound the observed raw/projection stencil correction. The old relative-zero diagnostic remains failed; this budget does not certify all binary64 eigensolver error. Neither result completes the paired-root census or supplies a canonical thermal phase. [Charge derivation](paired-charge-response-findings.md), [charge review](paired-charge-review-20261003.md), [cold increments](cold-charge-increment-findings.md), [independent cold review](cold-increment-review-20261003.md).

## Occupied-reference pair coordinates and a failed search

An occupied-reference Gaussian chart reproduces the original physical charge projection and all 14,400-state variational bounds. Independent 70-digit operator-exponential derivatives confirm the improved arithmetic. Both 120-step searches remain nonstationary with negative curvature, and an explicit failed classification accompanies the historical scoped recording status. A later exact charge-fugacity balance preserves the projected state and repairs the trust-radius update: 339 of 400 accepted steps use their full proposal. Both 200-step continuations still fail original and atlas stationarity/curvature gates. A separate 2,826-check review verifies charge-polynomial balancing, terminal derivatives and all 468 proposal identities/radius recurrences. These are mathematical controls and failed optimizations, not accepted minima or new production components. [Charts](pair-occupied-chart-findings.md), [chart review](occupied-chart-review-20261003.md), [balanced continuation](pair-gauge-newton-findings.md), [independent review](gauge-newton-review-20261003.md).

## Free spectral fitting remains nonstationary

Exact bounded elimination of twelve continuum coefficients preserves the full 45-component coadd likelihood and leaves all kinematics free. Independent QR and finite-profile controls confirm the derivative, including its nonzero-residual term. A single 520-second reflective fit lowers chi-squared from 2311.515 to 2298.407, but the original scaled projected gradient is 1.54119 against 2e-6. All 581 saved contact masks reproduce the original convention after a helper correction; the failed endpoint remains failed. Measured exposure-specific LSF and calibration covariance are separate missing inputs. [Conditioning and failure](spectral-conditioning-findings.md), [independent review](spectral-conditioning-review-20261003.md).

## An ordinary stellar microphysics interface with explicit missing joins

Native replay reproduces 1,300 EOS states and 45 corrected network states. Thirty fresh fixed-composition supports test the first-law Maxwell identity with positive heat capacity and compressibility. Independent isotope mass-excess sums reproduce nuclear heating plus nuclear-neutrino loss, while thermal neutrinos remain a separate term. The saved diffusion opacity includes electron conduction. Seven concrete component joins fail species, domain, force/current, mass-reference or field-work compatibility. These local ordinary controls provide no altered stellar endpoint, complete nuclear derivative matrix or common finite-field action. The native binaries and declared sources are pinned; runtime dependency provenance remains partial. [Interface and rejected joins](stellar-microphysics-interface-findings.md), [independent review](stellar-microphysics-interface-review-20261003.md).

## Ordinary weak capture: currents as well as recoil kinematics

Exact external spinors and two-body phase space now determine tree-level electron–proton capture for a declared point current, including weak magnetism. Independent covariant traces, a different gamma-matrix basis and Gordon-form current, lab/centre-of-mass integration, and coherent spin projectors agree. At total electron energy 12 MeV, the current calculation increases the kinematic-only response by 4.90%. A separate ordinary muonic-hydrogen benchmark reproduces the source singlet/triplet formulas and its rounded radiative factor; that factor is never transferred to electronic capture. Electronic QED, finite-q form factors, atomic screening/exchange and trinucleon recoil operators still gate a corrected atomic lifetime. [Equations, source reproduction and independent review](ordinary-weak-capture-findings.md).

## Nonspherical perturbations and tensor-source controls

One canonical phi-six oscillon benchmark now evolves seeded l=1,2,3 linear modes for 200 inverse-mass time units, with separate grid, time and outer-box refinements. The selected l=2/3 core norms do not exceed their initial values during the sampled interval. Translation, work and flux controls pass; an independent adaptive integrator confirms second-order convergence. Absorption depends strongly on distance to the boundary, so this is not a radiation lifetime or a nonlinear/Floquet stability certificate. Separate analytic ellipsoidal stress, transverse-traceless projection and forced-wave controls fix tensor energy/work normalization and preserve the spherical null. No dimensional coupling, population or gravitational forecast is inferred. [Dynamics, units, limitations and review](nonspherical-control-findings.md).

## An actual coupled screened charged three-body pilot

The quartet total-orbital-L=0 model now couples np S waves to pp pair/spectator P waves through reciprocal Jacobi permutations and an iterated off-shell screened Coulomb operator. At 5 MeV, cutoff 600 MeV and fixed screening radius 2 fm, the independently normalized outgoing flux is 0.05547208 fm², compared with held-out elastic loss 0.05543833 fm². The 0.0609% balance difference passes the unchanged 0.3% gate after grid, rank, angular/product and phase refinements. A coarse 0.477% failure remains rejected; its discrepancy tracks the independent neutral grid error. This is a finite-screen, finite-angular-space charged Hamiltonian. Unscreened convergence, higher pp waves and charged doublet/three-body-force sectors remain uncompleted calculations. [Amplitude, controls and independent review](charged-threebody-findings.md).

## The complete public exposure set, with conditional registration

All 17 extracted ESPRESSO exposures are acquired from the pinned author repository and match their upstream Git blobs. Native-pixel wavelength integrals, all 45 fixed Fe II components and independently profiled trace/order continua give 102 conditional line shifts, spanning −1.385 to +1.023 km/s. The median χ² per fitted degree of freedom is 1.223. Independent Voigt normalization, raw-array, wavelength-Jacobian and nuisance-projection checks accept the calculation. Conditional Fisher information is explicitly separated from the residual-dependent observed Hessian. The independent freely fitted coadd solver still fails its stationarity gate after 1,500 iterations; neither branch establishes a new alpha interval or calibration covariance. [Exposure acquisition, equations and failed free fit](native-exposure-findings.md).

## Ordinary oxygen ordering beyond the restricted-singlet control

Spin-pure valence CAS(12e,8o)/cc-pVDZ calculations now follow the triplet, both Δ-singlet components and the next Σ singlet over five O–O separations. The triplet remains lowest, with fitted minima at 1.220673, 1.234675 and 1.247507 Å and corresponding singlet gaps 0.946669 and 1.451121 eV. A fresh-geometry anchor detects and fixes an earlier frozen-core transport error; the rejected curve is retained. Independent all-electron minimal-basis FCI and explicit spin-orbital determinant matrices reproduce the ordering. Reduced-active-space gap shifts remain significant, so dynamic correlation and basis convergence are still required for quantitative chemistry. No field response or biological inference is supplied. [Hamiltonian, root tracking and independent review](oxygen-multireference-findings.md).

## Same-Hamiltonian comparisons at exact particle numbers

Four-particle Pfaffians project pure Gaussian pair trials onto Np=Nn=2 in the L=2 box. The same full A/B Hamiltonians define their energies, exact ground projectors and complete canonical traces. Best sampled pair energies are −94.952913 and −125.389246 MeV, above exact ground energies −115.214665 and −144.223972 MeV; ground-projector probabilities are 0.767819 and 0.737981. Independent amplitudes, Hamiltonians, gradients and canonical sums verify the variational bounds. The pair trials remain nonstationary and have S=0; a rejected ill-conditioned refiner is preserved. No Gaussian thermal entropy is borrowed after projection, and this box is not compared as if it shared the L=3 mean-number ensemble. [Variational bounds, failed optimization and limitations](projected-pair-competition-findings.md).

## A charged outgoing-cut building block

An independently driven pp Coulomb wave, its outgoing current and the spectral cut agree within 2.87×10⁻¹⁰ relative for compact l=0/1 sources at 0.1 and 1 MeV. Screening removal through 320 fm controls flux, but a stricter phase test initially fails and requires 2560 fm to pass its unchanged 0.001-radian criterion. Radial-to-momentum normalization and the off-shell resolvent identity are separately checked. This is a two-body building block with no pp strong interaction; a separate finite-screen three-body pilot is reported above, while screening removal and missing spin/angular channels remain open. [Derivation, phase failure and independent review](charged-breakup-findings.md).

## Independently normalized doublet breakup

The neutral LO doublet spectator S wave now has explicit antisymmetric spin/isospin outgoing amplitudes with the same calibrated three-body force in source and kernel. Two cutoffs, two real-axis grids, contour and quadrature controls pass 83 production checks. At cutoff 1200 MeV the pure-doublet flux is 4.999752 fm² at 3 MeV and 9.537036 fm² at 5 MeV; multiply by 1/3 for its unpolarized contribution. Independent spin sums and direct phase-space quadrature reproduce all eight saved-grid fluxes within 1.18×10⁻⁷ relative. The elastic-loss comparison was held out and never fitted. Higher doublet waves, unscreened spin-complete charged flux, NLO currents and a rigorous full-wave remainder remain unfinished. [Derivation and independent review](doublet-exclusive-findings.md).

## Stationary paired local minima in the cold finite box

An analytic constrained BdG response resolves the earlier stalled roots: almost all stalled gradient lies in soft, nongauge pair-orientation modes. Spin-aligned A/B starts at L=3, T=2 MeV and mean Np=Nn=3.375 converge below the unchanged stationarity gate, at 4.39×10⁻¹⁰ and 2.19×10⁻¹⁰. Their Helmholtz free energies lie 0.358239 and 0.395101 MeV below the corresponding normal roots. A positive enlarged normal+pp/nn/np Hessian, with a zero interaction extension in excluded pn-normal directions, supplies a sufficient certificate for the original restricted local minima. The smallest nongauge curvatures are 7.87×10⁻⁶ and 6.47×10⁻⁶, reproduced by independent finite paths and a different eigensolver start. These are finite-box Gaussian local minima; exact number projection, broader disconnected starts, the colder grid, thermodynamic limits and interfaces remain separate work. [Solver, stability proof and provenance](paired-stationarity-findings.md).

## An explicit discrete-sector vacuum through NLO

The cyclic QCD-sector action now supplies an exact LO sum and its complete cited NLO vacuum correction, with no freely adjusted tilt or cancellation. For N=3 the exact LO curvature is 1.67596 times the prior cosine approximation; NLO increases it by a further 2.43%. At N=5 and 7, cancellation amplifies the NLO/LO correction to 22.5% and 47.4%, so a small single-sector correction cannot be used as their precision budget. Independent literal-equation, scale, symmetry, LO pion-mixing and planar-wall controls pass. A common ideal-neutron energy functional supplies conditional pressure/source identities; it is not a matched interacting nuclear EOS. Formation, finite-field interactions/currents, higher-order transverse stability and applicable observations still gate physical use. [Action, quantum corrections and independent review](discrete-axion-findings.md).

## Released posterior transformation: a specific diagnostic candidate

An independent spin-algebra and distribution audit now covers all 16 released central-coefficient table pairs. Source Eq.11 requires row samples X to transform as X Mᵀ, where M=[[1/4,3/4],[−1/4,1/4]]. The released standard-basis columns instead closely resemble X M: the maximum two-sample distance across four fixed projections is 0.01608, compared with 1 for the correct map in every table. This supports a transpose-inverse candidate explanation. It does not establish upstream code behavior, row correspondence or full multivariate equality. The accepted nuclear calculations retain intact channel-basis samples. A fresh public repository check finds the same archived commit and no issues; original transformation code or documented paired chains remain the precise unsent request. [Audit, equations and limits](basis-audit-findings.md).

## A compatible formation subset with explicit mass gates

The new observable contract rejects the old full-network tritium mass: the selected force changes its binding by 3.285354 MeV. A four-species p+nn ⇌ n+d subset instead uses compatible frozen LO forward rates and recomputed reverse rates, with Galilean kinetic masses explicitly separated from rest thresholds. Six constant-temperature/tail cases pass conservation, positivity, equilibrium and independent tolerance controls. The zero and S-wave unitarity tails remain conditional closures. This calculation supplies a usable subset and a rejection gate; it does not repair the complete BBN network, derive omitted charged/weak/radiative amplitudes or match a physical field. [Contract and results](network-contract-findings.md).

## Higher spectator waves in exclusive neutral breakup

The independently normalized neutral quartet flux now includes spectator l=0…4 at 3 and 5 MeV, with two cutoffs and independent grid, product and phase-space quadrature checks. The maximum difference from separate contour elastic loss is 7.85402e-06 relative. At cutoff 1200 MeV the finite-wave pure-quartet sums are 0.1555591 and 1.8064327 fm²; l=1 dominates, so retaining only the earlier l=0 channel omits most of this calculated contribution. Multiply by 2/3 for the unpolarized quartet contribution. Reserved pointwise pair-amplitude controls and a finite l=5…8 diagnostic are separately preserved. They do not bound all higher waves or validate a coherent differential spectrum. The subsequent doublet S-wave component is reported separately; higher doublet waves, charged three-body and NLO corrections retain their independent gates. [Derivation, controls and remaining scope](exclusive-partial-findings.md).

## Inclusive atomic spectator recoil with an explicit current boundary

The finite-mass spectator calculation now sums excited bound states through n=800, a true delta-k-normalized Coulomb continuum and boost-generated angular channels through l=6. Independent radial Coulomb integrals, reduced-mass and zero-boost limits, spectral closure, energy moments, threshold onset, and separate angular/series/grid refinements pass 63 checks. No missing spectral strength is renormalized away. The reported quantity is a conditional MeV² phase response with neutrino recoil and the energy-delta Jacobian; consistent weak recoil currents and external-state normalization are still needed before it becomes a corrected lifetime. Sub-ppm isolated boost ratios lie below the 2-ppm response gate and remain unresolved. The empirical ordinary capture remains closed. [Derivation and limits](atomic-inclusive-findings.md).

## A colder anomalous instability, without a paired-phase claim

At L=3, filling 0.25, T=2 MeV and fixed mean proton/neutron numbers 3.375 each, both A and B normal roots retain positive charge-conserving normal curvature and pp/nn pairing curvature, but their lowest np anomalous curvatures are -0.13444260 and -0.15995355. Independent fixed-mean-charge Helmholtz paths reproduce these negative directions. Two separately validated 27-check audits underlie a two-check frozen synthesis; those supporting checks are not counted twice as completed components. The initial finite-amplitude searches lowered free energy but failed the unchanged stationarity gate. A subsequent analytic-response solver and coupled audit, reported above, resolve restricted paired local minima. The original result still identifies the normal-state saddle; neither calculation establishes exact canonical pairing, phase boundaries or interface dynamics. [Searches, independent audits and remaining work](pairing-continuation-findings.md).

## Public-rate helium populations under prescribed cooling

Published ground-state ionization and radiative/dielectronic recombination coefficients now drive charge-neutral He/He⁺/He²⁺ kinetics. Sixteen constant-temperature transients agree with an independent electron-exposure propagator; three prescribed cooling histories pass positivity, conservation, tolerance and density/time controls. Fast cooling retains a bare-ion fraction 0.3058 where instantaneous coronal balance gives approximately 1.54×10⁻⁵. This conditional lag is not a field history or a population-weighted capture lifetime. Escaping recombination photons give coronal equilibrium; a separately constructed three-body inverse checks Saha algebra without treating radiative coefficients as LTE inverses. Source-fit differences and omitted metastables, photoionization, density effects and thermal balance remain explicit. [Coefficients, population histories and limitations](atomic-populations-findings.md).

## Charged quartet S/P/D/F waves below breakup

An independent spin/charge projection supplies the quartet scalar kernels, including the photon bubble and exchange box, with one strict range/gauged-range insertion. The 128/192-node refinement passes 1,328 unchanged checks over three momenta, four spectator waves, two cutoffs and four photon masses. Raw full/pure amplitudes and all 24 zero-screen intercepts pass independent recomposition. The maximum grid difference is 0.01571 degrees and the screening-window difference is 0.002735 degrees. Four earlier failures, including a corrected principal-phase subtraction convention, remain visible. This adds below-breakup elastic phases; finite-screen fitting, S-wave pair forces and small regulator spread do not establish EFT precision or charged breakup flux. [Equations, gates and independent review](pd-quartet-findings.md).

## Exact fixed-charge thermal correlations

The complete 14,400-state L=2, Np=Nn=2 traces now supply density correlations and local four-mode quartet occupancy at 2, 4, 8 and 16 MeV. All eight momentum blocks and spin sectors use the unchanged A/B Hamiltonians; complete spectra reproduce their frozen parent. An independent free-fermion wedge exponential, fixed-number covariance and ordered-pair sum rules pass. Direct re-evaluation from every saved Fock probability independently reproduces all twelve observable sets within 2.82×10⁻¹⁴. The local quartet projector is an occupancy diagnostic, not a physical alpha-state overlap or species abundance. One four-particle box does not supply a correlated multi-alpha gas or a thermodynamic phase boundary. [Observables, derivation and controls](canonical-correlations-findings.md).

## An allowed finite-photon dineutron current benchmark

Pauli antisymmetry and a singlet-S final state forbid the zero-photon M1 and spin-independent charge E1 routes, but they do not forbid all capture. A finite-photon one-body neutron magnetization current supplies transverse spin-E1 and M2 contributions from odd triplet incidence in an explicitly projected Gaussian model. Independent photon normalization, antisymmetric spin averaging, Fourier/radial multipoles, radial refinement and P-wave threshold checks pass 180 gates. At 10 keV and R=1 fm the conditional σv/c is 4.09705×10⁻¹¹ fm². This is a fixed-energy current benchmark, not a matched physical or Maxwell-averaged rate. Relativistic electric spin-orbit terms can enter at the same order as the isolated spin-E1 piece; nn odd-wave interactions, two-body currents and field responses remain unresolved. The model is separate from the STM and four-body Hamiltonians. [Derivation, normalization and limits](nn-magnetic-capture-findings.md).
## Independently integrated neutral quartet breakup

The exclusive-flux gap is now closed for one channel: the **LO neutral quartet spectator l=0** contribution. This uses the Faddeev breakup operator and Jacobi permutations in [Witała et al., equations 2 and 15–17](https://arxiv.org/abs/nucl-th/0604033), reduced to the previously specified zero-range STM model. The spin-maximal quartet has two interacting np triplet pairs and no allowed nn triplet S pair. Exchanging the identical neutrons subtracts their two components. With D(q)=[√(3q²/4−E−i0)−γ]⁻¹, the final amplitude is proportional to D(q₁)t(q₁,k)−D(q₂)t(q₂,k).

The absolute normalization follows from the normalized deuteron pole residue, incident relative velocity, plane-wave phase-space measure and final neutron factor 1/2!. It gives σ=4γ/(3k)∫dq q²p∫dx|D(q)t(q,k)−D(q₂)t(q₂,k)|². **No elastic loss is used to normalize or evaluate this integral.** A naive reconstruction from the old deformed contour failed contour independence above breakup. The new real-axis solver explicitly resolves the moving exchange logarithms and subtracts the elastic pole analytically. Its elastic amplitude and independently integrated three-body flux are checked against the separate contour solution.

At cutoff 1200 MeV, the pure-quartet results are:

| Collision energy (MeV) | Exclusive flux (fm²) | Independent contour loss (fm²) |
|---:|---:|---:|
| 2.3 | 1.45474621e-07 | 1.45480372e-07 |
| 2.5 | 2.31573441e-05 | 2.31577112e-05 |
| 3 | 0.00107384004 | 0.00107384848 |
| 5 | 0.0553452211 | 0.0553453879 |

Across both cutoffs and all four energies, the largest flux/comparator difference is 3.95322e-05 relative. Multiply these values by **2/3** for their contribution to an unpolarized cross section. The 240/720-panel, phase-space quadrature, product quadrature, conjugation, deuteron normalization and neutron antisymmetry controls pass. Two unsuccessful attempts remain preserved. The near-threshold real-axis elastic loss suffers greater cancellation: its absolute balance gate does not imply subpercent relative precision there. The distribution is integrated over common orientation, rather than a laboratory detector cross section; integrated-flux checks do not establish pointwise convergence. Numerical agreement and cutoff sensitivity do not constitute an EFT uncertainty estimate. Exclusive doublet/higher waves, charged breakup, NLO, matched field response and a full reaction network remain unfinished. [Figure, tidy data and provenance](/research/metastable/full/public-inputs/E34-exclusive-breakup.json).

## Anomalous pairing stability of the finite-temperature roots

The 32 previously selected normal-density thermal minima now receive **96 complete anomalous pp, nn and np channel audits** at the same two mean fillings, boxes L=3,4 and temperatures 4,8,12,16 MeV. No pairing strength or orbital cutoff is fitted. Every tested channel has positive local curvature; the softest is np at L=3, interaction A, mean A=6.75, T=4 MeV, with dimensionless curvature **0.151780325**. Consequently there is no negative mode to follow on this grid, and no paired endpoint is claimed.

The finite-dimensional HFB construction follows [Lin and Wu, sections 3–4](https://doi.org/10.1051/m2an/2020074). The anomalous susceptibility is χᵢⱼ=(1−fᵢ−fⱼ)/(ξᵢ+ξⱼ), with limit f(1−f)/T when the denominator vanishes. The Hermitian congruence I+√χ Vpair√χ tests the constrained quadratic free energy. Complete complex antisymmetric same-charge pairs and proton-first np pairs retain both anomalous quadratures. Normal and anomalous variations decouple at zero pairing by charge symmetry. The physical doubled-Nambu entropy contains a factor **1/2**; an independent 16-state Fock-space calculation checks density, anomalous contractions, Wick energy and entropy, with largest discrepancy 6.68e-16. Four paths enforce both mean charges at every finite step and agree with the constrained curvature within 1.09e-06.

Coincident pp/nn curvatures follow the retained strong Hamiltonian's charge symmetry, not a new Coulomb calculation. These are dimensionless local-stability signs, not excitation energies, rates or error bars. Positive curvature excludes infinitesimal anomalous instability only at the retained roots. Disconnected finite-amplitude paired minima, colder pairing, proton–neutron coherent normal densities, exact number projection and correlations beyond quasifree states remain possible. Two finite boxes do not determine a pairing critical temperature, thermodynamic limit, phase coexistence or transition lifetime; the changed-field Hamiltonian is still unmatched. [Figure, all channel data and provenance](/research/metastable/full/public-inputs/E35-thermal-pairing.json).


## Neutral higher partial waves and breakup

Ordinary neutron–deuteron scattering now extends across the three-body threshold. The [primary partial-wave equations](https://arxiv.org/abs/nucl-th/9911034), especially equations 3.2–3.4, define Q_l with an additional (−1)^l relative to the usual Legendre-function convention. Independent complex angular integrals verify that sign and the branch convention. Pair interactions remain S wave, while spectator orbital waves l=0 through 4 are solved in both doublet and quartet spin channels. The ordinary triton fixes the doublet S-wave force once at each cutoff; no continuum datum is fitted.

The contour q=x−id x(1−x/Λ) passes below outgoing singularities and retains the same real cutoff endpoint. The on-shell amplitude is reconstructed with real external momenta. Upper-contour amplitudes reproduce the complex-conjugate advanced solution, and all twelve comparisons with the independent below-breakup principal-value solver agree within 4.82e−13 fm. The maximum tested grid/contour amplitude difference is 2.431e−6 fm. The first 112-node attempt failed its fixed elastic-unitarity gate; 176/256-node refinement passes without relaxing any tolerance.

| Ecm (MeV) | Elastic (fm²) | Inclusive breakup (fm²) | Total (fm²) |
|---|---:|---:|---:|
| 2.3 | 107.651087 | 0.097213 | 107.748300 |
| 3 | 91.543538 | 2.681848 | 94.225386 |
| 5 | 64.760151 | 9.663449 | 74.423599 |
| 10 | 38.168770 | 13.650648 | 51.819418 |

These sums retain l=0..4 and both spin weights at cutoff 1200 MeV. Inclusive breakup is inferred from **elastic loss**, πw(2l+1)(1−|S_l|²)/k². Agreement with total minus elastic is an algebraic normalization check, not an independent integrated exclusive three-body flux. Below threshold |S|=1 within the declared tolerance; above it inclusive loss is nonnegative. The threshold is bracketed by ±0.001 MeV. At 10 MeV, l=4 still contributes about 0.1014 fm² of inclusive loss; this does not bound omitted waves. Cutoff variation is a diagnostic, not a complete LO uncertainty. A later quartet l=0 component independently integrates exclusive flux. Charged breakup, doublet/higher-wave exclusive amplitudes, higher-order corrections and finite-field matching remain unfinished. This ordinary-neutral calculation does not replace the hypothetical-dineutron BBN transfer model or remove its tail uncertainty. [Amplitudes, figures and data](/research/metastable/full/public-inputs/E30-neutral-breakup.json).

## Bound electrons and atomic capture

The frozen trinucleon masses and one-body weak currents now also drive an atomic capture calculation. A symmetric, explicitly correlated helium Rayleigh–Ritz basis supplies the electron contact density and the conditional spectator wavefunction. Perimetric coordinates turn its matrix elements into polynomial Gauss–Laguerre integrals. Neither the atomic energy nor the contact density is fitted. The final energy is -2.903724300960 Hartree, against the independent -2.903724377033 reference; the one-electron contact density is 1.810347269 a0⁻³, against 1.810429317. The [primary helium reference](https://arxiv.org/abs/1912.05276) and [correlated-wavefunction study](https://arxiv.org/abs/1705.03417) are frozen inputs. Basis, exponent, quadrature and metric controls remain distinct from the absent relativistic and finite-mass corrections.

For bare-nuclear Q = M(He nucleus) + me − M(triton nucleus), bound-electron capture also pays the initial-to-final atomic energy difference. In the nonrelativistic infinite-mass model, hydrogenic capture opens at 54.42277 eV; neutral capture into the hydrogen ground state opens at 65.40867 eV. The remaining electron can occupy bound or continuum final states, whose inclusive contribution is calculated rather than replacing the spectator by its ground state. Statistical spin averaging is explicit; polarized or selectively populated hyperfine levels require a different rate.

The spectator now uses analytic hydrogen bound states and exact Coulomb continuum waves, with independent direct radial projections and energy-moment checks. Its physical continuum probability is 0.066258284; summing through n=1600 leaves positive omitted bound probability 2.47e-08. Refining both that sum and continuum quadrature tests the inclusive rates. The earlier finite box misplaced the near-threshold onset and assigned about 0.06536 to continuum, despite nearly unchanged large-Q rates. Its moment agreement was a discrete-representation identity, not a physical continuum check. The new exact-channel onset agrees with the independent ground-state overlap times excess energy squared; both old diagnostics and the superseding run remain preserved.

| Conditional Bnn (MeV) | Neutral-atom EC rate (s⁻¹) | Positron-decay rate (s⁻¹) | EC / positron |
|---|---:|---:|---:|
| 0.1 | 1.172734e-07 | 0.001226154 | 9.564326e-05 |
| 2.5 | 1.616436e-06 | 6.273587 | 2.576574e-07 |

The nn-perturbed cases already permit much faster positron emission. Atomic capture therefore adds a channel without providing longer survival in those two examples. The empirical ordinary helium-3 channel remains closed. The local nuclear model's R=2 and 2.5 fm ordinary cases instead predict reversed capture because they miss the small empirical nuclear mass difference; those rates are **model-discrepancy diagnostics**, not a prediction about observed helium.

An independent dilute limit of the existing Fermi–Dirac conversion integral checks the pi² density normalization. Initial neutral, singly ionized and bare populations are solved with ground-state Saha equilibrium and charge neutrality over three densities and 31 temperatures; all pass the nondegenerate check. Bound-electron and bare-ion free-electron contributions are recorded separately. Free capture on populated atoms needs screened/exchange continuum states; instantaneous LTE is not an ionization-kinetics calculation or a complete reacting environment. [Atomic figures and data](/research/metastable/full/public-inputs/E27-atomic-capture.json).

## Finite-temperature ensembles and complete canonical nuclei

The unchanged finite-range A/B Hamiltonians now enter actual thermal density operators. For the saturated Gaussian mean-field ensemble, E = 4 Tr(t rho) + 2 Tr(Gamma[rho] rho), S = −4 Tr[rho log rho + (1−rho) log(1−rho)] and F = E−TS. Self-consistent Fermi occupations fix **mean** A through a chemical potential. This implements the finite-temperature variational framework described by [Keller et al.](https://arxiv.org/abs/2011.05855); it is not an exact particle-number projection.

All 480 starts converge at L=3,4, mean A=4,8,12,16 and ten temperatures from 0.05 to 24 MeV. Each point retains a uniform start and two archived localized starts. The selected root has the lowest free energy among those starts; this original scan does not prove global minimization or unrestricted thermal stability. The later matched-density calculation below tests the full normal-density curvature for its own stated subset and follows selected instabilities. Independent free-Gaussian partition products and cold-energy limits pass. Warm branch continuations check dF/dT = −S with maximum error 5.69e-07. Particle-number variance is saved, and is substantial in the warm ensembles. Two volumes at fixed A vary density as well as box size; this is not thermodynamic finite-size scaling.

Separately, complete diagonalizations sum the **canonical** trace at fixed proton/neutron counts. All spin components and all eight momentum sectors are retained. Through A=4, this includes every one of the 14,400 states of the small-box four-nucleon Hamiltonian, without a CCSD or low-level spectral truncation.

| Family | A | States | T (MeV) | F (MeV) | E (MeV) | S |
|---|---:|---:|---:|---:|---:|---:|
| A | 2 | 256 | 8 | -32.926407 | -19.457220 | 1.683648 |
| A | 3 | 1920 | 8 | -62.991186 | -53.295883 | 1.211913 |
| B | 2 | 256 | 8 | -37.997104 | -24.228740 | 1.721046 |
| B | 3 | 1920 | 8 | -77.211756 | -66.483872 | 1.340986 |
| A | 4 | 14400 | 8 | -116.755094 | -111.066154 | 0.711117 |
| B | 4 | 14400 | 8 | -147.006776 | -138.525237 | 1.060192 |

Complete free spectra match independent one-particle occupation enumeration. Full eigenvector residuals, saved exact ground energies, dF/dT, dE/dT and low/high-temperature limits control the traces. The 10^6-MeV check only tests finite-Hilbert-space normalization and lies outside the nuclear model's physical domain. An earlier cold-limit check failed because 0.001 MeV still resolves a nearly degenerate multiplet thermally; the corrected check uses T no larger than the computed gap/50 and preserves the failed attempt.

These exact small-box traces and Gaussian ensembles have different constraints and are not subtracted to estimate a many-body error. Neither a heat-capacity feature nor the loss of a localized mean-field root establishes alpha-gas/liquid coexistence. Correlated multi-alpha ensembles, controlled thermodynamic extrapolation, interfaces and dynamics remain unfinished; the subsequent two-size comparison holds mean density fixed. [Mean-field figures](/research/metastable/full/public-inputs/E28-thermal-ensembles.json) and [canonical traces](/research/metastable/full/public-inputs/E29-canonical-nuclei.json) retain those distinctions.


## Charged doublet higher partial waves below breakup

The ordinary proton–deuteron calculation now includes spectator P, D and F waves (l=1,2,3) in the doublet spin channel, with LO and a single perturbative NLO insertion. Pair interactions remain S wave, so this is not a new NN P-wave force or a resolved spin-orbit/total-J splitting. The extension projects the angular Coulomb bubble, box and triangle kernels of [König, Grießhammer and Hammer](https://arxiv.org/abs/1405.7961) before their S-wave average. The original charged sources did not print these higher-wave predictions. Momentum-independent three-body contacts project to zero at l>0; no higher-wave datum is fitted.

A separately solved composite-deuteron Coulomb amplitude is subtracted at the same grid and screening mass. This is the existing source convention, rather than point-Coulomb subtraction. All momenta remain below breakup, and the above-breakup guard is retained. Results at cutoff 1200 MeV are:

| k (MeV) | l | LO phase (degrees) | LO + strict NLO (degrees) |
|---:|---:|---:|---:|
| 20 | 1 | -0.806764 | -1.102840 |
| 20 | 2 | 0.056485 | 0.079774 |
| 20 | 3 | -0.003742 | -0.005267 |
| 35 | 1 | -2.957560 | -3.980296 |
| 35 | 2 | 0.492915 | 0.698907 |
| 35 | 3 | -0.071943 | -0.101240 |
| 50 | 1 | -4.423657 | -5.779442 |
| 50 | 2 | 1.230346 | 1.753078 |
| 50 | 3 | -0.256574 | -0.360753 |

The table uses linear zero-screen estimates from photon masses 0.5, 0.25 and 0.125 MeV, with the 1-MeV point testing the fit window. The largest grid difference is 0.010568 degrees and the largest window difference 0.003141 degrees. Independent angular photon projections, neutral-amplitude controls, S-wave recovery, kernel reciprocity and LO/first-order NLO unitarity test the implementation. These numerical differences are not a complete EFT uncertainty, and small cutoff differences should not be interpreted more precisely than them. Charged quartet higher waves, charged breakup, doublet/higher-wave exclusive neutral breakup flux and finite-field matching remain unfinished. [New charged partial-wave data and figure](/research/metastable/full/public-inputs/E31-charged-partial-waves.json).


## Matched-density boxes and unrestricted thermal Gaussian stability

The next A/B calculation fixes **mean particle number per lattice site**, 0.25 or 0.5, across L=3 and L=4. These correspond to 0.03253723 and 0.06507446 fm⁻³, with mean A=6.75/13.5 and 16/32 respectively. The fractional mean numbers belong to the Gaussian thermal ensemble; this is not exact canonical number projection. Uniform, localized heating, cooling and two frozen-parent starts are kept at each temperature. The parent starts matter: a narrower initial search missed an already known lower root, and its complete earlier run remains preserved rather than silently replaced.

The free-energy Hessian now covers every complex Hermitian charge-block density fluctuation, including occupations, spin mixing and imaginary matrix elements. In a Fock eigenbasis, the positive entropy susceptibility is χᵢⱼ=(fᵢ−fⱼ)/(εⱼ−εᵢ), with diagonal/degenerate limit fᵢ(1−fᵢ)/T. The congruence I+√χ K√χ preserves stability signs. Fixed mean proton/neutron number requires **Σᵢ√χᵢᵢYᵢᵢ=0**, not an unweighted trace after the transformation. This differentiates the finite-domain functional discussed by [Dolbeault, Felmer and Lewin](https://arxiv.org/abs/0802.1577), with independent free-gas and interacting complex-density controls.

The 160 converged branch records classify as **144 local Gaussian minima and 16 saddles**, including repeated roots. The real tangent dimensions are 5,830 and 32,766. Independent finite Fermi-density paths check curvature while preserving positivity and charge. Thus a converged self-consistency iteration is no longer sufficient for a candidate to enter the selected table. The weighted curvatures are dimensionless signs of a variational quadratic form, not dynamical rates.

| Followed uniform saddle | Mean A | F decrease (MeV) | Final minimum weighted curvature |
|---|---:|---:|---:|
| L3/B, T=8 MeV | 13.5 | 30.100462 | 0.258915 |
| L4/A, T=8 MeV | 32 | 25.590476 | 0.098403 |
| L4/B, T=8 MeV | 32 | 280.616099 | 0.542703 |

Each followed root passes full stationarity and curvature again. The unrestricted solver permits spinor changes, but these particular endpoints return to the saturated subspace: they show a spatial/occupation instability, not a new polarized phase. Other starts can find still lower roots, so this is not proof of global minimization.

At mean A/L³=0.5, the lowest computed locally stable candidates give:

| Family | T (MeV) | L3 F/mean A (MeV) | L4 F/mean A (MeV) |
|---|---:|---:|---:|
| A | 4 | -7.862920 | -7.191623 |
| A | 8 | -9.690507 | -9.032766 |
| A | 12 | -15.069251 | -14.461925 |
| A | 16 | -23.485445 | -23.220899 |
| B | 4 | -15.095407 | -21.015275 |
| B | 8 | -16.703623 | -21.996460 |
| B | 12 | -21.249069 | -24.609818 |
| B | 16 | -29.825234 | -29.572989 |

This resolves the earlier fixed-A density mismatch, while retaining substantial low-temperature volume dependence. **Two boxes are only a matched-density finite-volume comparison, not a controlled thermodynamic extrapolation.** Correlated multi-alpha/paired ensembles, more volumes and regulator controls, global free-energy competition, interfaces, transition dynamics and a matched field response remain separate work. Numerical heating/cooling branches are not evidence of physical hysteresis or macroscopic phase coexistence. [Matched-density and saddle diagnostics](/research/metastable/full/public-inputs/E32-thermal-stability.json).

## Finite nuclear motion and atomic thresholds

The atomic component now solves H = H∞ + (p1+p2)²/(2M), including mass polarization after removing center-of-mass motion. This follows the nuclear kinetic operator in [Yerokhin, Patkóš and Pachucki, equations 62–63](https://doi.org/10.3390/sym13071246). In the same correlated Hylleraas basis, independent gradient forms agree, the nuclear kinetic matrix is positive, and its expectation agrees with the derivative of energy with respect to inverse nuclear mass. Omitting mass polarization deliberately recovers the exact reduced-mass scaling control; that simplified control is not used as the full calculation.

With CODATA nuclear/electron mass ratios, the calculated nonrelativistic helium-4 ground energy is -2.903304481724 Hartree, compared with the independent rounded -2.9033045577 value in [Table I of the primary ground-state study](https://arxiv.org/abs/1807.02457). The approximately 7.6×10⁻⁸-Hartree difference is consistent with the retained variational basis error; older source mass constants are negligible at this tolerance. Basis, exponent, quadrature and metric variations are retained. This is a numerical atomic benchmark, not a fit to its energy or contact density. Helium-4 supplies only an electronic benchmark; no alpha-to-triton capture channel is proposed.

For ordinary helium-3/tritium masses, the neutral ground-spectator capture threshold moves from the prior infinite-mass 65.40867 eV to 65.39598 eV: a -12.687-meV change. The one-electron contact density changes by -554.81 ppm. Thus the earlier sub-millielectronvolt numerical onset controls did not establish physical threshold precision. Relativistic/QED, nuclear size and hyperfine terms remain to be included before interpreting such fine energy differences. The empirical helium-3 capture channel remains closed.

The frozen nuclear-model cases retain their own masses, rather than substituting the ordinary measured values. For an exclusive hydrogenic daughter ground state, the neutrino has k = Q(2Mfinal+Q)/[2(Mfinal+Q)]. Its recoil also boosts the spectator wavefunction; angular projection gives j0[k r/(α(Mdaughter+me))] with r in ordinary Bohr units. The overlap is checked against exact exponential form factors, independent reduced-mass hydrogen projection and quadrature refinement.

| Conditional Bnn (MeV) | Ground-channel neutrino energy (MeV) | 1s projection / zero boost | Neutrino k²/Q² only |
|---|---:|---:|---:|
| 0.1 | 1.838464 | 0.994645 | 0.999346 |
| 2.5 | 6.930100 | 0.926935 | 0.997532 |

These columns isolate two effects. The final column is **not a rate correction**: recoil phase-space normalization and weak-current corrections are still absent. The overlap is for **one exclusive final state**, and recoil transfers amplitude to excited, continuum and higher-angular-momentum spectator states. It cannot be applied as a correction factor to the earlier inclusive capture lifetime. Screened/exchange free-electron capture and ionization kinetics also remain unfinished. [Figure, data and provenance](/research/metastable/full/public-inputs/E33-atomic-mass.json).


## What remains, and why

- **Physical matching:** ordinary scattering fits do not determine how two-, three- and many-body forces or weak/EM currents respond to a different field. The earlier local matched trajectory did not cross a robust threshold. A new trajectory must also satisfy its appropriate spatial, temporal and observational constraints.
- **Public-equation solver extensions:** unscreened/spin-complete charged three-body breakup, higher doublet waves and rigorous remaining-wave bounds, four-body reaction amplitudes, inclusive atomic recoil/current and screened capture corrections, a complete modified stellar reaction network, broader finite-amplitude or colder paired searches, projected thermal pairing, correlated multi-alpha ensembles and controlled thermodynamic extrapolation remain uncompleted calculations. They are not falsely labeled inaccessible data; the completed few-body and dilute calculations do not substitute for them.
- **Source/data questions:** the precision of the Bonn paper's approximate scattering-length target, separate Lee OBE convention, historical nonresonant coefficients, original matched alpha–alpha inputs, event covariance and isotope closure chronology retain their documented status.
- **Integrated history:** crossing a static nuclear boundary does not determine field persistence, formation, abundance, stellar ejecta, interface propagation or biological consequences. The original field-energy and history checks remain applicable. The prepared source-input packet remains **unsent**.

## Reproduction and integrity

Run `python run.py full STAGE` for any stage in the coverage table; the expensive BBN likelihood runs 59 independent network cases and the contraction stage runs four real MESA tracks. The NuLattice stage also uses the pinned optional dependency in `config/public-lattice-requirements.txt`; the original base lockfile stays unchanged because previous frozen records hash it. `python run.py full public-report` regenerates this report and its figures from validated frozen evidence. `python run.py smoke`, `python run.py validate`, and `python tools/verify_release.py --output reports/public-input-integrity.json` check numerical controls, run integrity and publication artifacts. Passing those checks does not remove physical limitations.

Each selected run includes its input hashes, exact configuration, source snapshot, raw numerical arrays and logs. Source download provenance is included in the [acquisition ledger](spin-isospin-acquisition.json). The [recent method-control history](public-method-controls.json) also exposes failed and superseded exact-nucleus, isomer, atomic, neutral-continuum and thermal-ensemble attempts, their manifests, implementations and solver logs. Failures do not enter the completed-component count. Public exports omit platform-specific binaries/restarts/caches and the third-party Bonn core; all omissions are explicitly listed and verified locally. The original core is retrieved using its recorded source URL and hash. Any historical capture errata are documented separately without rewriting old run records. The [validation record](public-input-validation.json) distinguishes the new checks from prior browser audits.