Complete note archive

Priority 3 · result · proved-unrefereed

A pinned selected-source peek motivates an exact quartic connected-response test: m4-3m2^2=-(b/hbar)(m2 m6-m4^2)<0. The equal-cell cumulant is asymptotic to -3 lambda hbar^3 T^5/(256m^4), while A_hbar fourth derivative divided by w tends to 6 lambda. A full Gaussian exponential-observable reflection test admits the zero-action member.

Scope: External RP1, QED3 and YM manuscripts remain unaudited; reading is passage-level or theorem statements as recorded. The scalar cumulant sign differs from the fermion determinant probe. A shrinking-cell cumulant is not a macroscopic continuum witness; positive hbar is supplied. No Yang-Mills result, quantum identification or independent Newton action necessity is established.

Priority 3 · result · proved-unrefereed

For arbitrary unequal time grids, the exact Dirichlet Green row sum gives a contraction factor Lambda T squared over 8, polygon convergence and a uniform inverse-Hessian/source-response bound. Gaussian fluctuation widths retain an explicit hbar factor.

Scope: Short two-endpoint intervals and stated global curvature bounds or invariant local tubes. Gaussian covariance is a quadratic reference, not a full nonlinear measure estimate. Harmonic boundary caustics do not obstruct initial-value uniqueness; positivity of hbar and physical weighting remain unproved.

Priority 3 · result · written proof; standard polygon methods and exact conservative examples; internal checks only

Arbitrary positive partitions and mixed kick/drift orders construct global-Lipschitz Newton motion with explicit mesh error. Conservative energy gives local continuation and magnetic global examples; exact logarithmic arrival criterion has threshold beta=2, vanishing annular action and local WKB residual. Unbounded slope is not a full-state merger; no action floor follows.

Scope: Global ODE construction assumes a globally Lipschitz acceleration; local forces require continuation control. Weighted-law and Euclidean/real-time identification remain separate physical premises. This lesson supplies no independent positivity argument for hbar.

Priority 3 · conditional · proved-unrefereed

Copy-index symplectic mixers and weak or W2 closure admit a Gaussian with each admissible covariance; a positive infimum determines the entire Gaussian sector. The older density ceiling fails a two-copy mixer. A supplied entropy floor supports actual cusp transport and mixing with W2 closure but fails weak closure and, as a subsequent note proves, selective Gaussian-meter closure.

Scope: Copy, mixing and closure are explicit hypotheses; the entropy model supplies its floor, is not weakly closed, and fails the specified selective Gaussian-meter test. Physical record closure, scale selection and identification with quantum hbar remain open.

Priority 3 · conditional · written classification, covariance and additivity proofs; borrowed probability inputs with stated reading levels; internal review only

One nonnegative additive-function theorem controls mass-only independent fluctuation composition, including continuous Brownian free paths and Green-Kubo plateaus. A finite-rate reference supplies positivity; matched-scale ordered streams retain a zero plateau. Additive force-independent Brownian noise gives the sharp pinned-path record floor; symmetric Levy laws obey phi(m,k)=m Gamma(k/m).

Scope: Mass-only all-positive-mass admissibility, independent composition and whole-body preparation equivalence are premises. Positive reference clocks do not give a uniform floor across classes. The record bound needs additive force-independent position noise; Nelson reconstruction and phase laws are additional, and a Euclidean relativistic jump generator gives no causal path-speed bound.

Priority 3 · result · proved-unrefereed

Two Gaussian families have identical position densities and the same limiting incoming trajectory, yet converge to reflection or transmission through the C1 cusp; their superpositions yield any mixture of the two outgoing branches.

Scope: Specified logarithmically shrinking preparations and a finite local time interval; the result does not classify all preparations or waiting times or derive a positive action scale.

Priority 3 · result · written energy, Weyl and conditional Gaussian information proofs; historical interpretations explicitly qualified; internal reviews only

One canonical Dirichlet action treatment: orthogonal arbitrary-cut hat energies and mesh exhaustion; exact Gaussian bridge KL/testing and sharp finite-grid invariant mark information; Weyl polygon phase, positive ordering holonomy and partition identity. General force bound retains one-sign attainability. Historical Mohist/Archimedes sources and conditional Opticks fit-count proof distinguish source reading from quantum assumptions. No floor, Weyl law or universal physical hbar is derived from fits.

Scope: Proposition 7's sharp value is a supremum, never attained by finite end marks. Cell floors and cut-share floors differ; noise/action scales remain supplied.

Priority 3 · result · proved-unrefereed

A single classical Gaussian meter with a supplied minimum-action pointer takes an entropy-floor input to arbitrarily small posterior action on a positive-probability coarse record. A Gaussian-mixture repair preserves the record and copy mixers but fails a nonlinear canonical shear. corrects: central-limit-canonical-gaussian-sector.

Scope: Zeta is supplied and the initial preparation aperture diverges along the postselection limit. The result assumes access to the specified Gaussian pointer and shear; the quantum-incompatible examples do not exclude all classical completions. The nonlinear kernel witness moves into suppressed tails as hbar/S0 tends to zero.

Priority 3 · conditional · proved; repository-internal written review

In the non-adaptive Gaussian mark model with preparation-invariant tests, optimal force discrimination is continuous at zero force iff κ>0; finite schedules realize spent cut action/κ. Historical passages support an explicitly modern record-level reading of continuity.

Scope: The optimized Gaussian-family iff requires saturation/dense-protocol availability and a balanced zero-noise family at zero floor. The historical record-level reading is an additional interpretation; all quantum aperture bounds require hybrid-state envelopes.

Priority 3 · result · bounded theorem proved in writing with named analytic inputs; universal lifting remains conjectural

Defines the universal constructive lifting conjecture with fixed certificate schema, noncommutative target, toolkit and joint observable limit; Koopman unitarity alone is insufficient. Proves the conservative global-Lipschitz C1,1 instance using graph estimates and Nelson: fixed-hbar product convergence and sharp mesh-uniform sqrt(hbar) classical recovery. Universal lifting, singular completion and positive-action necessity remain open.

Scope: One-dimensional conservative globally Lipschitz class and supplied h; fixed-h product bounds need the stated operator toolkit. No universal lift, action necessity or Euclidean measure for arbitrary unbounded-below potentials is claimed.

Priority 3 · conditional · mixed

Classical independent-box protocols obey the 9/36 conditional bounds, while Gaussian invariant tests have sharp constant 48. Supplied quantum probes give the standard-deviation cost hbar/2; correlations and non-Gaussian probes qualify it. Body-independent-noise classification and unread Gaussian channels are integrated. is superseded by: record-distance-path-length.

Scope: Hard widths differ from standard deviations and hybrid-state spreads. A zero infimum of mark products alone does not supply balanced protocols. Paley–Wiener does not prove a universal arbitrary-instrument recoil-width claim. General refinement arrays require weighted covariance convergence, not total precision alone. Newton fits are not transverse quantum uncertainties.

Priority 3 · conditional · written unit/record theorems and classical countermodels; response and calibration assumptions explicit

Normalized radiation spectrum defines an action unit under explicit scale and integrability assumptions, not universal quantum structure. Cross-field thermal extension remains conjectural. Reversible two-pointer records defeat a prior density cap; exact Rutherford, regulated radiation-response, pumped-depot and thermodynamic-work countertests retain their distinct hypotheses. Lorentz spectral form and any Planck calibration are conditional; work lower bounds are not attainable optima.

Scope: The radiation unit is enclosure-independent under its premises, not a quantum recording law. Resonant SED calibration is asymptotic and conditional on a disputed Planck derivation. Viscous and third-order response have distinct ultraviolet behavior. A work lower bound alone proves neither attainability nor an optimal posterior area.

Priority 3 · conditional · proved-unrefereed

A double-well time-cell response ceiling gives a uniform positive lower bound on the supplied action multiplier using an exact variance identity. The small-action source transition has a tanh profile. A specified global-minimum zero extension is discontinuous, while classical branch memory and a thermal weight are explicit controls.

Scope: Euclidean positive-weight family at fixed cell parameters with d>0. The response ceiling and physical branch law are independent premises; alternate zero-action extensions are not excluded. No universal quantum identification or Planck value is proved.

Priority 3 · conditional · written conditional floor theorems, exact classical countermodels and Gaussian limit proofs; no physical selector or universal hbar identification

Bounded C1 cusp potential has full point-state mergers and global classical solutions. Conditional positive covariance floors follow either reversible instantaneous phase evolution plus canonical/weak closure, or actual Gaussian cusp transport plus W2 copying/continuity. Full-trajectory group is a distinct zero-floor classical completion; Gaussian branch limits have nonuniform zero-width continuity. Sharp density-cap witness 1/(4piM), entropy witness and fine joint-record contradiction delimit claims. Quantum dynamics at supplied hbar and quantum identification are separate.

Scope: Weak preparation closure, affine canonical covariance and point-trajectory compatibility are extra premises. The point condition is vacuous if no points are admitted; the density ceiling fails two-copy mixer closure. Records, universal calibration and quantum identification remain open.

Priority 3 · conditional · conditional

A conditional classical statistical-speed theorem reproduces the exact sagitta/impulse disturbance bound; Gaussian ensembles with affine generators satisfy its premise with h*=2ζ. Background and Newton inflexion candidates leave record closure and scale matching open.

Scope: covariance alone fails for fine nonlinear generators, and indirect records can sharpen stationary priors. SED recording tests physical instruments and later nonlinear counterexamples.

Priority 3 · result · mixed

Canonical Galileo/Newton geometry, source paragraph map, matched-endpoint action and arbitrary-partition defect, receding-centre correction and exact controlled F,−F,F reunion. Conditional copy thresholds are kept separate from action quantization.

Scope: Spatial areas are frame-dependent, endpoint choices change coefficients, and deterministic work differs from energy uncertainty. Quantum phase, coherent arm control, copy count and measurement rules are supplied; no universal action floor or established fluxional-time interpretation follows.

Priority 3 · result · written conditional cell proofs and selected source passages; no selection of physical hbar

Convex Euclidean one-vertex integration has normalized tail, variance and curvature bounds; quartic integration generates positive stiffness at supplied hbar while minimization has zero central quadratic curvature. Retains finite constants, one-halving bad-event bound and precise Dimock/Balaban source evidence: noisy block average, generated terms, large fields and earned locality do not follow from a density-tower identity.

Scope: Finite Euclidean cell and one halving of nonadjacent sites under convexity. Positive action weighting is supplied; the coarse potential is not automatically a physical Newton force or a real-time reconstruction. Further halving closure and quantum identification remain open.

Priority 3 · conditional · mixed

Integrated mathematical and historical manuscript: conditional Gaussian and quantum disturbance bounds, exact polygon phases and the source-based Newton comparison. Historical M3 is an interpretation; operator spreads, hard widths and posterior inference are distinguished.

Scope: Supplied physical hbar, Gaussian/correlation and nuisance-test hypotheses, domains and hybrid-state resource envelopes remain explicit. Newton fits supply neither transverse quantum variances nor a theorem negating every recording restriction. Coherence and relativistic operator-domain premises remain separate.

Priority 3 · conditional · mixed

Added covariance classifies the Moyal algebraic route and a Gaussian state-restriction route, both retaining an unselected zero branch.

Scope: Both families retain the zero branch; interpreting Newton’s ultimate velocity as a record postulate is separate from the mathematical classification. Newton-indeterminacy-routes supplies the commutative alternative.

Priority 3 · conditional · mixed

One Bures-angle proof gives supplied-hbar aperture, general-disturbance and recoil-only bounds, with explicit hybrid-state envelopes and distinct preparation quantifiers. Separated packets and modular grid probes retain the resource counterexamples; force-split minimization needs envelopes uniform over splits. corrects: mark-cost-and-statistical-floor.

Scope: Physical hbar is supplied. Variance bounds must cover hybrid and translated conditional states and differentiable generator paths. The disturbance theorem uses a composite test for every pair of initial states. Pointwise force-split minimization requires common envelopes, not automatic split-independent spreads; general attainment remains open.

Priority 3 · result · proved; repository-internal written review

Constructive quartic and angular-phase tests retain zero-action sequences despite valid refinement and cycle consistency.

Scope: The failure concerns extending the original single-critical-point prescription; WKB/trivial-holonomy premises do not select a positive universal value. Rotation composition tests universality separately.

Priority 3 · result · written mechanical, canonical-algebra and spin-patch proofs with separate premise sets; standard source geometry

Reciprocal spring responses fix compatible inertia ratios through a connected-graph cycle test. Newton derivation compatibility and mechanical rotation closure equate constituent algebra constants, admitting zero. A supplied spin phase imposes 2s/K integrality while common normalization and arbitrarily small local cap actions remain free.

Scope: Reciprocal dynamics, canonical product structure, measured generator identification and U(1) phase transport are distinct premises. Equality or integral ratios do not identify a nonzero universal physical hbar; a reference calibration or positivity law is additional.

Priority 3 · result · mixed

Shared-score preparation and canonical copies lead to a restricted Fisher action, explicit positive forward orbit and switching representation. Terminal conjugate records and ordinary transport fail the restriction; branch controllers preserve unread energy while closure selects no kappa.

Scope: Supplied constrained dynamics, not invariant Newtonian free transport. The fixed-shape theorem needs zero-gradient preparations and compact fixed perturbations. Positive completion is forward only, with a width/phase linear tangent; the nonlinear-copy obstruction concerns finite Hermite–Gaussian families. The realization needs ρ>0, is not continued through nodes, gives no packet crossing on the even orbit, and nonexplosion for the quartic-tail posteriors over an interval is not claimed. Closure selects no scale; terminal access, phase reunion, positivity, universality and calibration remain open.

Priority 3 · result · mixed

Canonical classical recording treatment: regulated common-bath readout escape, terminal Gaussian closure, complete-record adaptive information, physical copy/refinement and continuum recoil, nonlinear-copy failures, and local phase/unread-energy tests.

Scope: Joint preparation, declared terminal coordinates, precision density, bounded feedback and deterministic schedules are explicit. Classical ideal conjugate access and nonlinear copies are different tests. Supplied quantum-reference curvature/phase calculations derive neither quantum kinematics nor a positive scale.

Priority 3 · result · mixed

Classical Kepler collision flow is incomplete while non-relativistic Coulomb dynamics is global for every h>0. The October 2 written derivation proves (L)=O(h^(3/2)) and position/momentum centroid errors O(sqrt(h)) on fixed collision-free intervals. Proposition 2 gives selection-dependent Dirac and Klein–Gordon h-sets, with explicit endpoint rules.

Scope: Fixed orbit, coherent preparation, T and smooth bounded V_delta, with delta below the minimum radius; constants depend on separation and derivative norms. Dirac finite-potential selection gives (k/c,infinity), critical Schur/cutoff selection [k/c,infinity), and essential self-adjointness [2k/(sqrt(3)c),infinity); extra boundary data permit every h>0. Klein–Gordon strict energy/charge construction gives (2k/c,infinity); its principal radial Friedrichs endpoint is included, but full critical dynamics is not proved. Case body unavailable; replacement claims use read primary passages and direct derivations. October 1 acceptance covers the original theorem only; October 2 additions are unrefereed self-review. No universal positive-action necessity or long-time limit. The critical Dirac domain uses the Esteban–Loss form closure; a written angular-channel counterexample prevents using the literal compact rewrite in Esteban–Lewin–Séré (1.32).

Priority 3 · result · written stationary-phase and exact quadratic Gaussian-transport arguments; corrected branch/record scope; repository-internal review only

Coherent finite-partition point records converge iff every nonzero action-frequency sum vanishes; equality to the classical value additionally requires cancellation of equal-action cross terms. Gaussian record laws vary continuously with the supplied action, and incoherent narrowing packets recover the classical swarm for quadratic forces. Includes local visibility, two-ended imprecision, free-fold and radial-angle constructions. The radial-angle model requires signed interval widths: positive Kepler action gradients compress before folding; T²/DeltaT is a dephasing scale, not an unconditional first wrapping time. corrects: score-constrained-ensemble.

Scope: Requires specified coherent amplitudes, nondegenerate illuminated arrival branches, fixed finite partitions or exact quadratic models, and explicit record/error regimes. Equal-action and frequency cancellations are allowed. No universal three-stream or late-time theorem, necessary resolution threshold, positive-action necessity, physical hbar calibration or exclusion of classical completion. The angle observable is stipulated and physical angular quantization restricts h subsequences; the continuous-parameter criterion must be retested there. Single-packet dephasing estimates require their stated width/chirp assumptions.

Priority 2 · result · written dimensional, dilation, orbit and countermodel proofs with exact hypotheses

Dimensional and similarity tests distinguish action units from floors. Sharp excitation and singular-Coulomb bounds retain their premises; the integrated C130 radial-action calculation gives the supplied-quantization Sommerfeld energy correspondence.

Scope: Each theorem states its observable, preparation and admitted class. A dimensional unit supplies no positivity; fixing a Coulomb coupling requires no charge quantization. Classical thresholds do not determine quantum operator domains or identify a universal floor with physical hbar.

Priority 2 · result · mixed

Exact force lenses and clipped-cut areas, a symmetric-fibre minimax theorem, all three finite-horizon phase-risk regimes, independent composition and shared/block budget laws are proved in one information model.

Scope: Known initial state, arbitrary bounded force, nondisturbing records and supplied deterministic error bands are premises. No whole-class speed ceiling or back-reaction is included. Precision/delay refinement closes the action-risk products; a shared norm budget differs from independent apparatus budgets.

Priority 2 · result · proved; repository-internal written review

Endpoint-matched constant-force and impulsive propagators differ by the scalar phase Kτ/ε, yielding an exact independent-copy discrimination law. Refinement erases the signal; classical-first coherence oscillates, while relative-resolution averaging or calibrated attenuation gives specified noncommuting limits.

Scope: coherent control, Born rule, supplied ε and resource budget are essential; this does not select positive action. See polygon phase.

Priority 2 · result · proved-unrefereed

Actual two-dimensional degree-one sigma solutions keep energy 4πrho while their radius-crossing action vanishes. A three-dimensional quadratic/quartic trial family fixes radius but exposes the supplied action coefficient b/c and a free common stiffness normalization.

Scope: The 2D family consists of actual static solutions; the 3D minimum is only within an explicit trial family. Degree restrictions, stiffnesses, time convention and crossing diagnostic are supplied. No full minimizer, global nonlinear dynamics, quantum gap or action selection is derived.

Priority 1 · result · proved-unrefereed

A return with endpoint velocities ±u and acceleration ceiling a exists iff T≥2u/a and has unique minimum kinetic action mu³/(3a). Chord-action defect is bounded sharply by ma²T|π|²/24.

Scope: Endpoint-conditioned cost and refinement excess differ; decreasing u closes the cost. External bounded control supplies the dynamics, not an autonomous conservative receiver.

Priority 1 · result · mixed

An affine-test orientation ball and its minimal composite have finite operational capacity, but a complete reversible spin interaction fails moment descent and admits no finite bounded-observable repair for all Borel preparations. Source reconstruction premises and the verified seven-dimensional generator restriction are integrated.

Scope: All Borel preparations and the specified restricted product tests are essential. Infinite observable dimension establishes neither infinite distinguishability capacity nor quantum structure. The complete reconstruction theorem is borrowed with additional operational premises; only its initial generator restriction is proved here. Spin action magnitudes remain supplied.

Priority 1 · result · mixed

The zero-mass continuous-spin label is a transverse boost moment, realized by a signed-mass dipole; elementary orbit curves have fixed physical time, and nonzero momentum precludes an energy lower bound. A supplied Poincaré energy gap gives a boost–momentum uncertainty floor proportional to ℏΔ/c², while a positive continuous-spin label still permits energy approaching zero.

Scope: Signed masses extend ordinary positive-mass mechanics; the pair realizes a family of charge orbits, not one elementary orbit. The elementary orbit's spacetime curves have fixed time. Quantum action and the spectral gap are supplied in the uncertainty theorem; no universal action selection, energy gap from the continuous-spin label, or Yang–Mills continuum construction is proved. Written derivations are not independently refereed.

Priority 1 · result · proved-unrefereed

An explicit ideal string network produces mechanical work fringes whose normalized autocorrelation survives attenuation, while energy and canonical action shrink quadratically. The finite-pulse construction distinguishes canonical action from the traveling wave’s zero Lagrangian action.

Scope: ideal junctions and sinks are supplied; deterministic work fractions are not exclusive click probabilities. Interference alone selects no action unit; threshold events tests detector additions.

Priority 1 · result · known

CHSH bounds arbitrary shared-readiness local models under setting independence but cannot select an action normalization.

Scope: Setting-dependent censoring changes the tested ensemble. The witness is dimensionless and leaves every common action normalization admissible.

Priority 1 · conditional · conditional

Exponential mixing of a connected Anosov setting device bounds hidden-variable/setting correlations for Lipschitz-resolved conditional preparations, with a logarithmic resolution-dependent onset.

Scope: A supplied isotropic preparation floor turns this into an Ehrenfest-type lifetime; it neither derives that floor nor covers integrable devices, exact automaton labels or singular ensembles. Event bounds also require regular setting cells and suitable delays.

Priority 0 · result · written event, optimization and countermodel proofs under specified mechanical or stochastic receiver laws

Fixed-split conditionally independent monotone receivers obey the sharp coincidence-product bound. Passive barriers, exact capped-Arrhenius retention/deadline optimization and shared-release countermodels retain their separate premises, energy accounting and causal timing. None selects a universal physical action.

Scope: Fixed routing, conditional independence and all-gate sampling are essential to the coincidence bound. Thermal costs require fixed prefactor, temperature and transduction; the exclusive shared-release model changes the independence premise and supplies its hazards.

Priority 3 · conditional · written exact Gaussian identities and regulated norm bounds; interacting one-loop matching formal or conditional as stated; internal checks only

Exact directional Gaussian composition preserves harmonic flux normalization but sharp blocked covariance grows with depth. Internally checked determinant-jet estimates and conditional telescoping give universal average running; §7 supplies formal finite one-step vertices/matching constants.

Scope: Depth-uniform continuum-subtracted generated contractions, infrared limits and interacting remainders remain open. §7's review accepts formal structure while some Schur-test bounds are taken on trust.

Priority 3 · conditional · written formal one-loop identities and convergent coefficient integrals; conditional matching theorem with internal checks only

Computes formal one-step four-dimensional anisotropic shifts and conditionally recovers the universal inverse-coupling change −2b₀ log 2 from matched background determinants. Exact generated-action recursion retains every induced interaction.

Scope: one-loop running requires matching hypotheses; sublinear endpoint corrections and controlled accumulated remainders suffice for its average. Uniform nonlinear iteration remains open in composition.

Priority 3 · result · proved; repository-internal written review

Exact Gaussian product blocking multiplies the small-momentum heat-time coefficient by 2^(4−D), preserving it in four dimensions at every finite depth. Centred nearest-plaquette input retains an O(K²) error.

Scope: covariance must remain bounded near nonzero aliases; remainder constants need not be depth-uniform. Sharp-surface ultraviolet shape growth and interacting matching remain in composition.

Priority 3 · result · written established matrix spectral proofs, explicit conditional classical bounds, labelled semiclassical estimates and corrected extension/domain statements; internal review only

Scalar and SU(2) bosonic matrix oscillator bounds prove compact resolvent, a simple ground state and a supplied-hbar gap with exact hbar^(4/3)g^(2/3) scaling. Classical area restrictions instead bound sublevels and their volume; the Weyl leading count and semiclassical channel levels remain distinct. Includes field-unit conversion, torus/invariant-sector limits, free/compact/2D flux and corrected singular-domain examples.

Scope: Quantum Hilbert/CCR structure and physical hbar are supplied. A classical area floor gives no quantum spectrum and is not shown dynamically invariant; its pointwise admitted energies still approach zero. Gauge-invariant and unrestricted first gaps can differ. Supercritical inverse-square extensions are unbounded below; subcritical scale-preserving extensions and the free delta limit defeat a general consistency-forces-stable-scale claim. No continuum construction or physical YM gap follows.

Priority 3 · conditional · proved conditional transfer; source applicability and physical hypotheses open

H1 multiscale fluctuation/quasi-local Gibbs control and H2 neighborhood mixing imply a volume-uniform Wilson-transfer gap, with continuum transfer conditional on existence and dense observables. Finite certificates, gauge marginal reduction, interaction robustness and numerical bands retain their distinct burdens.

Scope: No physical trajectory, applicable finite-size certificate or continuum theory is constructed. Compactness, a scalar coupling, gauge-invariant expectation or numerical meeting scale does not prove the full interaction/mixing hypotheses; KS and Wilson thresholds differ.

Priority 3 · open · finite-lattice and conditional spectral proofs; linked strong-coupling theorems; weak trajectory, uniform estimates and continuum goal open

One canonical lattice-to-continuum spectral treatment contains T1 finite compact-resolvent positivity, strong-coupling and Wilson/KS distinctions, finite-volume analytic coupling dependence, conditional limiting spectral thresholds and lost-weight counterexample. Integrates former position map and exact Wilson sufficient constants. All-coupling positivity/phase equivalences, large-field rarity, relative flow truncation and exhaustive route closures remain unproved; continuum construction and physical spectral transfer stay separate.

Scope: Finite-lattice existence, strong-coupling positivity, all-coupling stability, continuum scaling and reconstruction are separate. The small-volume field asymptotic and positive SU3 matrix gap are conditional; Lambda_lat is inverse length and m is an energy. No monotone crossover is proved. The small-volume scaling statement names a centre sector; it is not automatically the full finite-box gap.

Priority 3 · result · mixed

Directional heat-kernel halving closes in series but leaves an exactly defined coupled mid-plane parallel theory.

Scope: Nonabelian normalized estimates and stability for generated actions remain conditional/open; fixed-torus holonomies require qualification. The full-plane calculation and small-field notebook continue the live gauge route.

Priority 3 · conditional · formal

Formal full-plane one-loop coupling shifts have explicit signs and SU(N) scaling, with a separate finite-torus winding obstruction.

Scope: Flat-holonomy winding disproves literal fixed-torus flux-only control. Volume-uniform normalized interacting estimates and iteration remain open in the small-field continuation.

Priority 3 · conditional · written estimates and conditional implications with incorporated counterexamples and repository-internal checks; no independent peer review

Covariant inverse decay, bridge tails and local convexity coexist with the flat-holonomy failure of the original normalized comparison. Exact barriered responses, large-field insertions and weak-recoupling dressed activities give (112)/(116); a centered quadratic moving-mean model gives (177), while nonlinear (113), physical recoupling and iteration stay open.

Scope: Original flux-only Proposition 4 is rejected. Physical recoupling, nonlinear background transfer, other subtractions, late SU(3) estimates, full-integral comparison and iteration remain open; continues series/parallel.

Priority 3 · result · exact

Exact bridge character formulas cover every spin; spin-½ is scalar, while internally checked spin-1 matrix entries retain axial anisotropy and explicit image bounds.

Scope: Isolated-cube factorization uses independent bridges. Fixed-spin formulas do not control the full heat-kernel character sum or shared-edge mid-plane; image separation degenerates at the cut locus. Supplies curvature inputs to the full-plane calculation.

Priority 3 · result · mixed

Exact single/multiple-cut character moments classify central image phases by q-torsion; SU(3) trisections see triality while halvings reach only the identity centre element. Fundamental trisection shells are explicit.

Scope: Signed polynomial amplitudes are not probabilities or homotopy sectors. Matrix orientations, normalized uniform image truncation and repeated RG stability remain open; uses SU(2) midpoint.

Priority 3 · result · written scalar benchmark theorem; specified one-loop calculation and conditional field transfer

The one-loop SU2 valley has bounded periodic full potential and an unbounded nonperiodic subtracted correction. For the declared flat 4pi holonomy operator, centre-odd and cubic-invariant trials give an explicit g^(8/3) exp(-c/g) upper gap, so local g^(2/3) spacing is not its global gap. corrects: weak-coupling-feshbach-reduction.

Scope: Flat measure, scalar kinetic operator, all centre characters and the specified one-loop potential are benchmark inputs. No identification with PHP, full field-theory splitting, matching tunnelling asymptotic, SU3 theorem or continuum gap is proved. Gauging centre translations or restricting their character changes the admissible trial; neutral local cyclic space differs from the full finite-box Gauss space.

Priority 3 · result · proved; repository-internal written review

Exact Villain monopole/flux decomposition gives a D=3 full-measure TV bound 2(L/a)³e^(−π²/(6λ₃a))/(1−e^(−π²/(2λ₃a))) and removes compact defects at fixed coupling and volume.

Scope: The same D=4 TV upper bound is useless at fixed coupling; an extensive lower bound proving TV separation is only expected. Blocked Gaussian fields still need smooth-observable comparison, not TV closure.

Priority 3 · conditional · mixed

Proves abstract Schur gap-transfer lemmas, including a constant-shift/relative-error version. Physical H1–H3, UV-uniform domains and bounds remain open, and H3 must identify the centre sector and observable algebra before a local matrix spacing can represent its compressed gap.

Scope: H1–H3, projection domains and uniform constants remain open. Bare sharp-cutoff control fails; linear valley confinement does not automatically control quadratic multiplication globally. Formal SU2 cubic slope and Berry terms are not certified SU3 or continuum bounds. The scalar centre-sector countertest is not a construction or identification of PHP.

Priority 3 · result · proved-unrefereed

The all-representation activity rho=e^(6/g²)−1 yields a volume-uniform Wilson-transfer gap at the sufficient integer bound g²≥1059, with rate 4 log[1/(176 rho)] in units ℏc/a. Exact sufficient KP convergence uses rho≤21/(500e²); the unrounded rate is positive only at a strict margin.

Scope: Spatial tori have sides at least four. The exact sufficient KP condition is g²≥6/log(1+21/(500e²)); its unrounded lower rate is zero at equality. The weaker 176 rate is positive for g²>6/log(177/176), with 1059 sufficient. The number 1056 is linearized and 176 is an unproved leading-activity threshold for general polymers. Wilson and KS are distinct operators; no continuum limit.

Priority 2 · obstruction · mixed

Surface-to-gap criteria scale unfavourably in specified regimes.

Scope: All-coupling monotonicity requires a comparison of block gaps. Connected polymers alone do not prove the proposed Schur-variation inequality; boundary counting and sectors require checking.

Priority 2 · obstruction · mixed

Derives the explicit one-link Wilson sufficient criterion g²>444 for SU3 and its transfer-gap rate. The proposed single-link block-marginal test is empty in the gauge interior; a joint invariant certificate and weak-trajectory control remain additional obligations.

Scope: A block marginal criterion based only on interior single-link distributions sees Haar marginals and certifies no required joint mixing. No box size or monotone gapped interval is established; operators and continuum limits remain distinct.

Priority 2 · conditional · written variational and positive-measure proofs; free Gaussian calibration; continuum and spectral insertion hypotheses explicit

One variational and positive-spectral-measure treatment gives intensive susceptibility bounds, conditional moment ratios and exact spatially flowed Maxwell constants. Temporal-flow admissibility and infrared moments remain assumptions. All finite inverse moments can coexist with a zero edge; complete moment data may determine the observed channel, which need not be the global gap.

Scope: Four-dimensional flow UV finiteness supplies neither a Hamiltonian insertion nor infrared integrability. A single observed edge can miss lower sectors; finiteness of every inverse moment gives no positive lower edge. The conditional continuum construction and weighted derivative/susceptibility limits remain open.

Priority 2 · obstruction · written finite-lattice invariances, gauge variations, auxiliary kernel estimates and stationary-background calculation

Finite-lattice flow preserves spectra and multiplication norms and has a local first-derivative path bound. Exact continuum variation distinguishes the true Jacobian from an auxiliary Gaussian propagator with an absolute tail budget. Landau growth is gB-k_parallel^2; inverse metrics and relative Hamiltonian truncation remain open.

Scope: Auxiliary bounded-coefficient heat evolution is not identified with every gauge-flow Jacobian. Absolute kernel errors and free variances do not establish interacting suprema, inverse-metric control, relative spectral truncation or iteration.

Priority 2 · conditional · written identities and conditional tail/transfer theorems; local and continuum inputs open

Exact magnetic Laplacian/gradient identities and variance sum rule feed finite-volume global Agmon tails with explicit all-path constants. Perron slab and anisotropic temporal limits conditionally identify ground-state slice probabilities; normalized local rarity and operator comparison remain separate.

Scope: Global thresholds and prefactors depend on volume; local large fields need not force a global excess. Temporal limits are iterated at fixed spatial cutoff; no uniform local probability, signed operator inequality, continuum construction or physical gap follows.

Priority 2 · result · written continuum inequality, equality conditions, constrained-minimum counterexample and specified spectral model; no uniform measure theorem

Flowed-block continuum action floor eta_F^2/(4g^2) has exact equality accounting and no proved positive-threshold saturation. Bianchi failure defeats the old localized-constant example. Finite-regulator minimum need not have nonnegative tangent-cone Hessian. Charged-background phase-space and exact box counts retain real/complex SU3 multiplicities and norm-dependent cost ratios. Normalized rarity, entropy and polymer convergence remain unproved.

Scope: Continuum flow and boundary assumptions; distinct eta_F L2 and eta_b spectral thresholds. Sharpness, normalized local weights, arbitrary-background entropy and polymer convergence are unproved.

Priority 2 · obstruction · exact perturbation and IMS identities; lattice localization coefficient conditional, local spectral comparison open

Exact positive-perturbation min-max bound and negative-well ground-state trial identity distinguish measure from signed spectral control. IMS coefficient 32pi^2 c0^2/3 requires extra link-response, pointwise-field and locality assumptions. Local energy excess and vacuum surface subtraction are unproved targets for a specified construction; no exhaustion of Hamiltonian methods or automatic Euclidean closure.

Scope: Exact abstract perturbation identities; IMS lattice coefficient assumes link-response, field and locality/tail bounds. No general Hamiltonian route exhaustion or proved physical spectral comparison.

Priority 2 · result · written regulated variational, Gauss-law and Polyakov bounds; constant-mode substitution conditional

The KS Feynman-Bijl identity gives plaquette strong-coupling order, an abelian structure-factor gaplessness test and an exact averaged-Polyakov susceptibility bound. Constant-colour electric trials vanish for semisimple gauge algebras; a universal string cost for all field-dependent dressed trials is unproved. An exact SU2 adjoint-loop bound targets the neutral centre sector; the one-patch fundamental variance is not a global vacuum variance.

Scope: Nonzero variances and stated domains are required. The Gauss-law exclusion is semisimple, not every nonabelian group; a nonvanishing upper bound proves no converse gap criterion. Small-volume expectation transport and any universal dressing cost remain unproved.

Priority 2 · result · written support-preserving reduction; locality/clustering theorems cited with metadata-level coverage; numerical prefactor not certified

The support-preserving interaction picture reduces commuting unbounded single-link electric terms to bounded plaquette locality, with volume-uniform coefficient of order Nc/g². Numerical prefactors, clustering lengths, physical propagation and stability need their separate hypotheses.

Scope: The old 32e coefficient is not certified from the cited theorem conventions. The dynamical quotient hbar v/Delta alone is not a general clustering length; a diverging upper estimate proves no actual superluminal speed or gap stability.

Priority 2 · result · written Gaussian, tube and soft-resolvent estimates; nonlinear expansion conditional

Exact linearized four-dimensional tube blocking gives the 4/9 constrained gap and Gaussian covariance. A distinct soft constraint has gap 2/9 and a coefficient-specific decay criterion. Correct two-link normalization and infinity-norm shell sum replace the old threshold table; nonlinear KP control remains conditional.

Scope: Hard conditioning and the soft operator differ; no finite-remainder hard numerical rate, normalized interacting expansion, optimality barrier or physical continuum gap is proved.

Priority 2 · result · borrowed local stability theorem with written hypothesis checks; finite-volume and restricted-count proofs; general-history and RG transfer conditional

Yarotsky stability gives an existential volume-uniform KS gap and a specified translation-invariant perturbed target region. Exact finite-volume electric-loop bounds, corrected all-root overlap obligations and restricted continuous-time counts share one proof structure; numerical uniform thresholds remain withdrawn.

Scope: Existential constants are not numerical thresholds. Restricted anchored histories do not control late roots, general shapes or normalized true-vacuum insertions. Physical continuum transfer and thermodynamic consistency retain separate hypotheses.

Priority 2 · conditional · written dimensional and truncation identities; source-dependent strong-coupling theorem; continuum and small-volume transfer conditional

Conditional physical 3D theory has gap E3 f(lambda3 L), with E3=hbar c lambda3 and lambda3=hbar g_cl². Exact SU2 constant-mode scaling and explicit nu=2 strong-coupling lattice constants control different regimes; nonzero-mode transport, construction and a positive finite spectral limit remain unproved.

Scope: Physical construction, nonzero-mode relative errors, sector matching, spectral visibility and compatibility of volume/cutoff limits are required. The SU2 cross-product proof is not a general SU(N) proof. All-coupling positivity is an optional sufficient strengthening; the Euclidean compact U1 source is not automatically a KS theorem. The former dimensionally inconsistent Debye prefactor is withdrawn.

Priority 1 · exploratory · source-based scoped mechanisms and written comparison identities; proposed bridges conditional

One gauge-mechanism comparison keeps curvature, centre deformation, three-dimensional scaling and valley lifting beside correlation inequalities and finite certification. A weak scaling trajectory needs less than all-coupling positivity; temporal operators, rank and physical-hbar limits remain distinct.

Scope: Curvature normalization and physical spectral transfer are unchecked; deformed and lower-dimensional theories need their own continuation. Loop monotonicity is not vacuum-gap monotonicity. Large-N factorization is not the physical hbar-to-zero limit at fixed rank, and no exhaustive route closure is established.

Priority 2 · result · mixed

Finite conservative harmonic receivers have zero long-window response; the prepared increasing-chain plateau depends on limit order and closes under its hard speed-support condition. Exact two-body cut/reset and three-body elimination/observability proofs retain the missing classical memory.

Scope: Receiver energy and modal preparation supply all action scales. Resets alter the experiment, exact two-time recovery becomes ill-conditioned, and the apparatus/calibration/minimax proofs now reside in their separate canonical treatments.

Priority 1 · result · mixed

One autonomous finite-mass Hamiltonian apparatus has vanishing-error/disturbance quiet-preparation limits; exact nonlinear calibration and complete pointer/clock records give bounded-domain recovery. Impulsive repeated-cut readout has separate growing-probe resources.

Scope: Preparation precision, exact model, known incoming probe momenta and joint final record access are supplied. Delayed known-flow reconstruction differs from causal unknown-force prediction; energy-assisted recovery is local unless final clock records are included.

Priority 1 · result · mixed

A common inverse apparatus chart proves shell saturation, energy/calibration ambiguities, rank-controlled global recovery, matched tolerance/coupling crossover and directional calibration fibres. Signed-coupling symmetry cancels the proposed canonical tangent; finite-width quadratic quadratures retain a conditional curvature test.

Scope: The calibrated global inverse uses a reduced fixed box and nonzero displacement; shell, full-state, coordinate-risk and projected-area claims have distinct hypotheses. Correlated estimates concern one local fibre; both projected curvature coefficients remain unproved nonzero. No universal action is selected.

Priority 1 · historical · primary passage interpretation and scoped elementary modern comparisons; compatibility proposal unproved

Unifies Aristotle and Plutarch cut evidence with Indian contact, Avicennian, Chinese dissection/statics and cohesion witnesses, preserving editions and reading limits. Separates sampling, re-solving, marginalization and measurement, and retains fixed-time compatibility premises without an action-selection claim.

Scope: No historical transmission, ancient path-integral theorem or Newton quantum anticipation is established. Unproofread and incomplete witnesses retain their access limits; contextuality is a source lead, not an applied theorem, and compatibility patterns select no dimensional scale.

Priority 1 · historical · reading

Passage-level Newton Project audit separates retrospective analytical discovery from synthetic Principia demonstration and documents generated quantities, limiting ratios and priority/publication revisions.

Scope: the dossier supports no causal claim that shrinking-area doubts delayed publication. Draft layers, deletions and individual dating remain incompletely checked; selected diplomatic/TEI anchors carry the evidence. Book II methodological scholium and Book III Classical Scholia are distinct witnesses.

Priority 3 · result · proved-unrefereed

Assembles the known quartic oscillator into a continuous full-time Euclidean process with a unique vacuum, positive gap, bounded-cylinder reflection positivity and a non-Gaussian one-time marginal; ground-state boundary-weighted lattice measures converge for arbitrary time partitions. The construction applies to the prior SU(2) matrix gap and its singlet sector.

Scope: Supplied hbar>0 and established Feynman–Kac and spectral inputs; bounded-cylinder reconstruction only. Does not derive a full unbounded-field axiom hierarchy, 4D Yang–Mills, an SU(3) matrix gap, relativistic covariance or positive-action necessity.

Priority 3 · result · written packet estimates and weak-measure limit for specified Dirichlet/Neumann states; Robin scattering comparison only

Imposed elastic bounce and quantum boundary choice are distinct. Explicit Dirichlet odd packet has uniform position recovery through impact, current recovery away, and positive boundary-force measure converging to 2|p0| delta. Full Neumann stress has same weak impulse; Robin phases and bound state retain boundary information absent from classical bounce. No self-adjoint half-line momentum is assumed.

Scope: Specified free half-line coherent packets; general Robin wavepacket impulse and nonlocal boundary Hamiltonians are not proved. The full Neumann boundary stress is signed.

Priority 3 · result · written dilation, local heat-kernel and normalized-bridge proofs plus conditional state theorem; established analytic inputs; no universal positive-action selection

Exact interacting dilation commutes with kernel composition and normalized bridge marginalization. Matched hbar=epsilon S0 and time/distance epsilon give a local heat expansion with explicit second-order potential terms and a refinement-uniform Gaussian bridge limit. The shifted kernel product retains insertion centres lost at zero. Row-normalized harmonic transitions fail the semigroup law; well-resolving scaling differs. Retains the O(hbar+mesh) coherent-state field theorem, free zero-action control and bridge action defect.

Scope: The heat-kernel rate assumes nonnegative C3 scalar potential with bounded second and third derivatives and compact q,v,T ranges with T>0. Resolvent and heat convergence under dilation are strong, not operator norm. The separate monomial well scaling is exact, but no general long-time or global uniform estimate follows. Physical hbar and positive action remain supplied; the coherent-state rate has its earlier smoothness/preparation hypotheses.

Priority 3 · open · maintained dependency synthesis; exact identities, formal scaling and open estimates distinguished

Exact series/parallel operations, arbitrary cuts and group image sectors organized by dimension and scientific dependency. Retains exact circle/rotor identity and one-loop scaling, separates depth/volume uniformities, fine-measure construction and spectral visibility. Corrects unsupported renormalon identification and stable-quartic non-Borel claim. No scale-to-gap or confinement implication is assumed. is superseded by: mass-gap-openings.

Scope: it adds no theorem; read linked derivations for hypotheses. Sharp Gaussian fixed-point assertions require the later composition correction.

Priority 3 · result · mixed

Precise classical, operator, packet, sharpness, wall, cusp, relativistic and Euclidean Lean targets with close prior art and a finite rational drift-kick versus kick-drift comparison for newtonlean; exact constant-force overlap exposes joint-limit ray separation and a variable-mass example shows factor-ordering ambiguity.

Scope: Specifications are not compiled Lean proofs; mathematical claims retain the linked notes' unrefereed status. New-to-repository does not establish literature priority. Newtonlean remains core-only and requires an explicit model and completion bridge.

Priority 3 · result · mixed

Normalized supplied-hbar Euclidean blocks and coherent unitary products construct the unique smooth Newton initial-value trajectory with O(hbar+mesh) mean error and O(sqrt(hbar)) widths. Projective Gaussian preparation, raw kinetic divergence and its scale estimator are integrated.

Scope: Quantum action, Wiener reference and operator theorems are supplied. Smooth mean rates differ from the separate C1,1 lift; quartic trajectories use cutoff quantum families, not an uncut uniform quantum estimate. Projective consistency and kinetic estimators do not select a positive scale.

Priority 3 · result · written finite composition and harmonic estimates, supplied-action Gaussian and 2D law constructions; higher-dimensional transport conditional; internal review only

Exact constant-force elimination and supplied-action Gaussian composition accompany sharper harmonic insertion/mixed-kick bounds and an all-partition trajectory/force-law construction. Exact 2D gauge subdivision and a sufficient summable-defect law require different inputs from a classical initial-value theorem; higher-dimensional interactions and physical limits remain open. corrects: low-dimensional-mass-gap. corrects: three-dimensional-gap-one-function.

Scope: Classical normalization remains free; one-loop running and graph projective consistency supply neither OS construction nor physical gap. Live nonlinear recording and weak-strip control remain separate obligations.

Priority 3 · result · proved-unrefereed

Bounded third derivatives give O(h+mesh+c^-2) mean and smooth Weyl convergence to Newton for the positive-energy relativistic particle; fixed h has strong-resolvent convergence, and a separate spectral Dirichlet wall has uniform position convergence through impact.

Scope: One-particle external potential in a fixed frame; action is supplied, time and initial momentum are fixed, c is bounded below, and the spectral wall differs from killing the jump process. No interacting relativistic QFT or independent positive-action necessity is established.

Priority 3 · conditional · source-based comparison, conditional chiral statements and written spectral-limit proofs; main goals open

Physical comparison retains a pure-SU3 gap, massless/massive two-flavour QCD pion channels and Newton record costs as different targets. Axial Ward/pole identity and two positive spectral-measure limit criteria include surviving weight and counterexample to eigenvalue-only convergence. Coherence, singular-domain selection and physical action necessity remain separate premises; chronological queue and repeated domain tables removed.

Scope: Both main goals remain open; pion symmetry breaking and surviving residues are conditional inputs, without a QCD construction. Kepler localization fixes the orbit, interval and smooth regularization. Coherent records require preparation coherence and a specified topology; the incoherent classical countercase is proved for quadratic forces. Coulomb intervals depend on domains and use reduced hbar; full critical KG evolution remains open at the singular-multiplier term. Wilson explicit uniform bounds, KS finite-volume bounds and existential uniform statements concern distinct operators and hypotheses. Weak-trajectory arrival, uniform matching and continuum reconstruction are unresolved. New integration has no independent verification or novelty claim; the dated H3-first plan is historical and STATE controls the queue.

Priority 3 · result · mixed

Two-dimensional infinitesimal symmetric class-function refinement with finite representation exponents yields exactly central Lévy semigroups; a Lindeberg condition selects the Gaussian Yang–Mills member.

Scope: Simple groups give a scalar diffusion constant; nonsimple groups need an invariant tensor. D≥3 error budgets and the strict 4D threshold 2b₀c>4 require normalized density, entropy, blocking stability and reconstruction; they do not prove universality.

Priority 2 · result · proved; repository-internal written review

An exactly unitary normalized checkerboard walk converges strongly on fixed L² wavepackets to Dirac evolution with supplied K; the corresponding real transition rule gives telegraph dynamics. Measuring direction every step yields a ballistic fixed-frequency limit.

Scope: Complex amplitudes/Born probabilities are assumptions; action dimensionality alone selects neither composition rule nor positivity, and branch separation is not a field-theory vacuum gap.

Priority 2 · result · exact

Complete observable frames with bounded total susceptibility lower-bound a finite reversible chain’s relaxation gap; one positive velocity plateau only upper-bounds it. Explicit hidden/weakly observed slow modes preserve the plateau while closing the gap.

Scope: Uniform calibrated coverage and physical clock normalization matter; interaction benchmarking uses Ising gap. This is not automatically a reconstructed quantum-field gap.

Priority 2 · result · written exact path, support and moment derivations; classical models with explicit conditioning

Exact elastic equal-mass receiver and sharp kappa_mid<=mT(u-|v|)^2 support benchmark precede the velocity-resolved telegraph return bridge. Count/simplex law includes midpoint atom, Beta law, arbitrary-cut action error, exact Bessel crossover and small-mass/window nonuniformity. Scalar convolution rigidity is kept in cut-paradox-two-faces. Speeds, reversal rate, apparatus and limits are supplied.

Scope: The right-continuous bridge differs from shrinking endpoint-window velocity conditioning; unscaled midpoint bias survives. Positive mass-universal plateaux need nonuniform small-mass windows, and the reversal rate still sets their value.

Priority 1 · result · source passage restatement; elementary identities proved under stated smoothness; bridges conditional

States the exact Clay data, forcing and quantifiers for fluid A–D and the axiomatic quantum Yang–Mills existence/finite-gap target. Derives scaling and dissipation identities, distinguishes physical from auxiliary clocks and states the obligations for each proposed bridge.

Scope: No theorem transfers fluid regularity or auxiliary mixing to quantum existence or a physical gap. Lower-dimensional SPDE statements retain cited coverage; announcements are attributed artifacts, not independent verification.

Priority 1 · result · written scalar derivations and model identities; matrix conjecture open

Scalar ballistic-support probability convolution laws are deterministic drift, and scalar translation-invariant unitary groups with light-cone kernels are translations up to phase. Internal-state identities, the matrix conjecture, hidden slow mode and exact additive-readout double limit retain separate scope.

Scope: Written scalar proof imports Paley–Wiener–Schwartz and harmonic estimates; no independent peer-review or novelty claim. Matrix classification is unproved, positivity permits killing, and readout bias is not intrinsic action or physical hbar.

Priority 1 · result · known

A depth-two controlled-Z circuit conjugates independent spins to a local interacting cluster Hamiltonian with unique entangled ground state and exact volume-uniform energy gap J. Removing endpoint terms yields four ground states, with gap J above the ground space.

Scope: Quantum kinematics, energy normalization and time/action conversion are supplied; fixed-spacing volume uniformity establishes neither continuum construction nor generated gap.

Priority 1 · result · mixed

The exact interacting heat-bath gap, its explicit positive three-site square-root Gibbs parent and logarithmic global mixing-time comparison are proved or source-matched together. Refresh-clock and supplied-action normalization remain independent of equilibrium weights.

Scope: Finite-volume sequence, finite ferromagnetic coupling and the specified per-site refresh convention. Mixing-time asymptotics are a borrowed primary-source result. Physical state/clock/action identification, infinite-volume representation and continuum construction are not supplied.

Priority 0 · historical · reading

Exploratory dictionary compares Euclidean ground-state marginals and segment factorization with low-dimensional holography, then distinguishes gapless Schwarzian/SYK scales from a confining infrared cap.

Scope: Low-dimensional boundary/semigroup identities and source-based interpretations are not a four-dimensional duality theorem. The entropy expansion assumes a simple vacuum, isolated finite-degeneracy first excitation and a finite heat trace; kernel duration T gives physical inverse temperature 2T/hbar.