Complete note archive

Priority 3 · result · proved-refereed

Any cut spends 3s(1−s)K_τ in orthogonal hat energy; shares add and exhaust K_τ exactly when mesh vanishes. Gaussian bridge information and finite-grid optimal mark discernibility equal corresponding action/scale ratios.

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 · conditional · proved-refereed

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: each fixed protocol is continuous even at zero; static cases and unrestricted prepared apertures differ. The record-topology premise is additional physics, not Leibniz’s own theorem.

Priority 3 · result · proved-unrefereed

Position-coupled quantum probes have exact error/recoil commutator and uncertainty cost ℏ/2. Independent Gaussian marks with preparation-invariant tests yield the sharp τΔE≥48z²κ bound; correlated probes lower its effective cost and can defeat it.

Scope: arbitrary probes and prepared-body protocols escape the Gaussian floor. Recoil and disturbance provide the wider instrument statements; supplied ℏ remains a premise.

Priority 3 · result · proved-refereed

Radiation universality/scaling, finite positive integrated energy and low-frequency normalization define a positive enclosure-independent action unit. A specified classical two-pointer protocol defeats a prior-only density ceiling by sharpening the completed posterior.

Scope: unit, posterior restriction and operational disturbance bound are distinct; the instrument-access premise is essential. Exact Rutherford scattering refutes the earlier universal Coulomb impulse floor. Continue in indeterminacy routes.

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

Establishes the geometric anchor (3F/v)A=τΔE and distinguishes it from matched-endpoint chord and force-centre sectors. Explicit F,−F,F return forces reunite both arms and give an exact scalar quantum phase for any body state.

Scope: phase discrimination is supplied-ℏ/resource-dependent, and kinetic gain is not energy uncertainty. Independent positive-action necessity remains open; Galileo interference sharpens the controlled comparison.

Priority 3 · result · exact

Forgetting a Gaussian position record of width σ gives exactly momentum convolution with variance ℏ²/(4σ²); concurrent mark precisions add, and linear dynamics transports the resulting covariance.

Scope: The continuous-measurement limit needs a summable noise budget; this standard channel supplies a Newton parallel-insertion analogy, not an independently selected positive ℏ.

Priority 3 · conditional · mixed

Main synthesis combines sharp Gaussian recording bounds, general disturbance costs, state-independent polygon phases, and edition-labelled Newton/ancient division sources.

Scope: Gaussian correlation restrictions, hybrid-state resource suprema and controlled-pass budgets matter; Newton’s fits give per-colour premises with a modern extension. Independent positivity/universality of the action unit remains open: consult necessity-unit-and-indeterminacy and the live SED notebook.

Priority 3 · result · proved-unrefereed

Arbitrary adaptive quantum instruments obey JL+sP≥ℏ arcsin(1−2ε) when position/momentum resources are bounded on the required hybrid states. Separated packets permit arbitrarily small Galileo action with unbounded preparation aperture.

Scope: Instrument universality is preparation-relative; an area bound LP alone fails without aspect-ratio control. Record-distance-path-length supplies the sharp geometric formulation.

Priority 3 · result · mixed

Endpoint-matched polygon and forced motion differ by the exact scalar phase ΣKstep/ℏ; constant force gives F²Στj³/(24mℏ), with exact insertion and ordering identities. The same Dirichlet functional bounds Gaussian distinguishability by K/κ.

Scope: Phase uses supplied Weyl relations; Gaussian sharpness requires one-sign force and uncorrelated marks. Newton fits interpretation needs a modern continuous-speed extension and remains per-colour.

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 · result · proved-unrefereed

General unitary recording pays in momentum or position disturbance, evaluated on the hybrid states required by the proof.

Scope: Spreads are over conditional/interpolating hybrid states, not merely observed preparations; infinite spreads make the bound vacuous. Supplied canonical kinematics sets the exchange rate. Record-distance-path-length sharpens the original total-variation constant and combines accounting.

Priority 3 · result · proved-unrefereed

Arbitrary momentum-transfer probes retain a recoil bound, while grid probes show why Gaussian action-error bounds are not universal. corrects: mark-cost-and-statistical-floor.

Scope: Adaptivity requires resource suprema; Gaussian three-mark comparison uses infinite endpoint recoil. Record-costs-disturbance extends beyond this coupling class.

Priority 3 · result · proved-unrefereed

Bures-angle paths sharpen protocol constants and combine aperture and disturbance into one force-split accounting. corrects: planck-gap-probabilistic.

Scope: Hybrid resource bounds and hypothesis preparation quantifiers matter; sharp single-shot constants do not prove protocol attainment or independently select ℏ.

Priority 3 · result · proved-refereed

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 · proved-refereed

Interacting Newton derivations propagate canonical action-constant equality but preserve the common zero branch.

Scope: Rotation covariance alone permits unequal constants; closure identifies the measured mechanical sum with one rotation generator. Connected interactions propagate equality, but the common zero branch and absolute scale remain unselected.

Priority 3 · result · mixed

Restricted classical sheet action gives the Fisher action and reservoir-aware population momentum. Fixed compact shapes fail, but a width/phase mode has a positive non-Gaussian forward completion. Nonlinear copies generate quartic exponential tails outside finite Hermite–Gaussian modes, with exact unread recoil energy. §§6–8: the two-sheet switching realization exists on every positive solution and for all forward time on the explicit orbit; its controller is the field pair and the record history; branch controllers pass the two-unread-width test with a variance witness against a marginal score; closure holds at every κ≥0, so the apparatus-closure route to positivity is closed.

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

Shared-score classical preparations realize nonlinear coordinate-copy correction but fail full readout and free transport. A stationary switching repair matches the Gaussian Fisher target with exact mean-energy balance; its force requires preparation or apparatus memory.

Scope: The correlated score preparation and Gaussian actuators are supplied, not a universal recording law or energetically closed apparatus. Ideal classical pointer access permits recovery; phase reunion remains unexplained. The repair preserves mean kinetic energy, not pathwise total energy. Static bridge results remain unrefereed; physical closure, positivity, universality and calibration remain open.

Priority 3 · result · mixed

For Kepler's force the classical flow is incomplete (explicit collision time, invariant null collision set, no continuation from the axioms) while the quantum Hamiltonian is self-adjoint with global unitary dynamics for every h>0 (Kato, cited); the quartic shows the same contrast; the classical trajectory is the derived centroid (exact for quadratic forces, Hepp cited for Kepler under a stated localization hypothesis). With finite c the self-sufficient theories have h ≥ k/(α_c c): a gap in the admissible values of h.

Scope: (b), (d2) and Proposition 2 rest on cited theorems read at metadata level with their standard statements; hypothesis (L) for the Coulomb evolution of a coherent state is unproved; (4) with (L) gives norm approximation, so centroid language for Kepler is shorthand pending moment bounds; the Klein–Gordon branch selection and the endpoints of (5) are additional premises and the gap is shown in two models only; the planar operator is only remarked; no value of h is fixed; no long-time statement; the nonexistence of completions without h is not addressed.

Priority 3 · conditional · proved-refereed

Specified Brownian free-motion and composition laws force one nonnegative action constant and conditionally yield a sharp Galileo record floor.

Scope: κ>0 remains assumed; Nelson dynamics needs additional state/phase laws. Uses cut measure within the fifth-postulate programme.

Priority 3 · result · proved-refereed

States the gap thesis as a claim about limits and locates it. With Gaussian preparations, quadratic forces and minimal Gaussian marks the record family is continuous at zero (no gap on Galileo's comparison). For determinate preparations whose streams cross, the complete record has an h→0 limit only if signed cross amplitudes cancel, while smeared records converge to the classical branch sum; exact for inertial motion with a graded velocity inside its fold interval and for a Kepler swarm whose windings overlap. corrects: score-constrained-ensemble.

Scope: Thm 2 needs illuminated nondegenerate critical points, a cutoff equal to one near them and bounded branch data on compact record sets; the resolution and thickness statements are sufficient conditions with stated error terms, and kernel Fourier zeros can remove contrast at finite ratios. Thm 3 is the free particle with a compactly supported WKB amplitude; no universal post-fold persistence (affine profiles never fold). Thm 4 is a stipulated angle-record model with the Keller–Maslov spectrum; lapping requires actual winding overlap and monotone I₀, and no universal late-time statement is made. The visibility is an interference contrast, not a spectral gap, and generates no scale; the second premise is an h-independent coherence length between the launch points of crossing streams (partial contrast e^{−Δq₀²/(8σ²)} for any fixed σ, none for the √h mixture), which remains a premise. Refereed by GPT-6.1 Sol (Codex) with corrections applied.

Priority 2 · conditional · mixed

A supplied transverse orbit-area floor confines the x²y² classical valley and reproduces the quantum-mechanical gap's energy unit; a finite gap reconstructs an action unit in the mechanical model and dimensionful lower-dimensional cases.

Scope: Semiclassical counting/levels are estimates; dimensional equivalence fails in 4D. Positivity of the floor is assumed; see low-dimensional-mass-gap.

Priority 2 · obstruction · proved-unrefereed

Space–time contraction q_a(t)=a q(t/a), V_a(Q)=V(Q/a) preserves mass, speed and energy while scaling orbit/Hamiltonian/covariance actions by a, giving zero infimum in contraction-closed classes.

Scope: Couplings and force ceilings may break closure; fixed softened potentials still admit small actions. Read fixed-force-small-circles before treating broken similarity as sufficient.

Priority 2 · obstruction · mixed

Consolidates classical vanishing-action families, conditional excitation/composition bounds, finite-apparatus ambiguity/recovery, and the distinction between relaxation and physical Hamiltonian gaps.

Scope: Premises remain model-specific; no universal selection theorem follows. Apparatus continuation is parked. Later necessity-unit-and-indeterminacy and refinement-results provide the current frame.

Priority 2 · result · mixed

Dimensional homogeneity and universality require a finite positive floor to be a fixed-constant action product; admitted action-rescaling similarities force zero infimum. Classical k_e,c,G yield unique mass-independent unit k_e/c=αℏ.

Scope: Unit existence does not select positivity or quantum normalization; singular Coulomb assumptions matter. Later necessity-unit-and-indeterminacy expands the unit/indeterminacy separation.

Priority 2 · result · proved-unrefereed

Apparatus-only reading error and disturbances force a canonical error–impulse pair; position-undisturbed marks reduce to von Neumann momentum-transfer couplings, and the recoil bound extends to position-displacing marks.

Scope: Strong Weyl commutation and unbounded continuous pointers are essential; general body-dependent noise requires record-costs-disturbance. Quantum action scale is supplied.

Priority 2 · result · proved-refereed

With exact past state but unknown future |f|≤F over delay ℓ, exact minimax errors are Fℓ²/(2m), Fℓ; the curved reachable lens has canonical area 2F²ℓ³/(3m).

Scope: Delay/force resources set the scale; both vanish when refined. reachable-cut-composition composes the full phase-space region across cuts.

Priority 2 · result · proved-unrefereed

Regular closed Newtonian trajectories with speed ≥v_* and force ≤F_max obey J≥m²v_*³/F_max; relativistic momentum adds (1−v_*²/c²)⁻¹. A smooth force-bounded central potential attains equality on a circle.

Scope: Positivity is conditional on persistent excitation and a selected inertial frame; mechanical/canonical momentum identification needs care with vector potentials. Decreasing v_* closes the bound.

Priority 2 · conditional · conditional

Nonnegative mass-only fluctuation coefficients preserved by independent centre-of-mass composition over every positive mass are constant K≥0. A supplied nonzero-speed finite-rate telegraph reference makes K positive within that preparation class.

Scope: Whole-body versus independently composed preparation equivalence is essential; common reservoirs can violate it. Neither physical action selection nor universality across preparation classes follows.

Priority 2 · result · proved-unrefereed

Exact bounded-force position/momentum minimax risks have acceleration, two-arc and saturation regimes, with momentum saturation at T=4√(mε/F). Product fibres compose exactly; averaging records can lose both coordinates during mixed regimes.

Scope: Known initial state, deterministic error band and complete-history access are premises; no speed ceiling or back-reaction is included. Refining ε closes the action-risk product.

Priority 2 · obstruction · proved-unrefereed

One fixed smooth relativistic confining potential with globally bounded force supports radially stable circular orbits whose normalized angular actions behave as √(mK)R²→0. A supplied positive speed floor instead gives a conditional force-limited action bound.

Scope: Binding, regularity and upper speed/force ceilings do not select positivity; closed-orbit-force-action extends the excitation-dependent bound beyond circles.

Priority 2 · result · proved-refereed

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-refereed

Convex symmetry and an explicit force-controlled pair give sharp coordinate minimax risks hidden by an entire bounded-noise position history, matching two-record recovery. At zero delay their product is 2√(mF) ε^(3/2).

Scope: sufficient preparation time, deterministic error bands and known initial state are required; precision/delay refinement closes the bound. Short horizons are solved in finite-horizon minimax.

Priority 2 · obstruction · conditional

Conditional finite-block mixing certification would furnish a volume-uniform Wilson transfer gap and an open target for an RG trajectory in full interaction space. No certificate or reach estimate is supplied. is superseded by: reasons-to-stop-as-research.

Scope: all-coupling coverage and two scalar thresholds are unnecessary; oscillation bounds worsen with blocking. Physical-coupling certification is subsequently rejected as infeasible in reasons to stop.

Priority 2 · conditional · conditional

In an early worst-case mark model, added irrecoverable-recoil premise M3 yields a protocol-dependent Galileo action floor; determinate fits plus later screen records defeat it. Mathematical subdivision remains unrestricted. is superseded by: mark-cost-and-statistical-floor.

Scope: M3 is an added physical premise, not derived from Newton’s optics; finite support widths have no nonvacuous quantum instance. Statistical mark cost supplies the corrected ℏ/2 conjugacy and Gaussian bounds.

Priority 2 · result · exact

Exact Galileo areas give an endpoint-matched action defect and a carefully scoped Newtonian source anchor.

Scope: Deterministic work differs from quantum energy spread; the action defect survives boosts and endpoint gauge changes. Read English witnesses anchor the geometry; supplied phase resolution neither selects ℏ nor prohibits shrinking paths.

Priority 2 · obstruction · conditional

Linear radiation balance cancels probe parameters in a regulated resonance limit but cannot select the reservoir's action amplitude.

Scope: K remains freely rescalable, including zero; removing the cutoff first causes ultraviolet energy divergence. This closes linear reservoir amplitude selection, not causal nonlinear field–particle models; SED-zeta-radiation-link keeps its physical assumptions explicit.

Priority 2 · result · proved-refereed

Relativistic Coulomb binding has a positive angular threshold only with the singular external core.

Scope: This continuous admissibility threshold depends on the singularity, coupling and domain; it is neither action quantization nor a universal floor. Rivero’s consistency test keeps the angular-state premise separate.

Priority 2 · conditional · conditional

A supplied resonant zero-point amplitude can be calibrated to the radiation unit only under a disputed SED thermal-spectrum premise.

Scope: Lorentz invariance leaves κ free; full momentum variance diverges, and a fixed finite linewidth window captures only part of the resonance. Recording closure separately tests retained records rather than assuming equilibrium survives observation.

Priority 2 · exploratory · known

Spin-sphere phase patching quantizes 2s/K only after a phase unit K is supplied.

Scope: Common rescaling of spin, Hamiltonian and K preserves motion and phases, while arbitrarily small trial-loop actions remain possible. Topology selects a ratio, not an absolute unit or dynamical action floor.

Priority 2 · obstruction · conditional

For a uniform prior and bounded posterior support, the stipulated measurement-plus-erasure law gives W≥kBT log(A0/η); classical finite-precision protocols retain arbitrarily small positive η at finite increasing cost.

Scope: This closes thermodynamic record pricing as a universal floor selector within the specified classical class, without proving sharp attainability of the work bound. Uses unit and indeterminacy.

Priority 2 · result · known

Smooth degree-one O(3) sigma-model static solutions of every radius saturate a fixed sector-energy floor, while their action magnitude over one radius-crossing time tends to zero. Vacuum waves remain arbitrarily soft.

Scope: The specified energy-time action is not every action observable; topology fixes neither radius nor a unique gapped ground state. Stabilized topology tests added competing terms.

Priority 1 · result · proved-refereed

For a phase-uniform circular orbit, 2√det Cov(x,p_x)=RP equals normalized orbital action, while displacement-response coefficients vanish at short and long windows.

Scope: Equality depends on preparation; point phase gives zero covariance despite nonzero orbit action. Invariance is affine symplectic in the selected plane; action-scale-dilation contracts between models.

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 · proved-unrefereed

Restricted affine measurements of classical random orientations yield ball states, capacity two, continuous reversible local motion and multiplicative composite dimensions/capacities. Minimal classical composites fail purification and Hardy's subspace requirement.

Scope: Measurement/operation restrictions are stipulated; no apparatus realization or action unit follows. reversible-interaction-premise tests the additional genuine-interaction premise.

Priority 1 · obstruction · proved-unrefereed

The interacting two-spin flow generates arbitrarily many independent time-translated observables, excluding every finite bounded-observable repair containing the original effects. Nonlinear updates of finite expectations cannot evade the obstruction when all preparations are admitted.

Scope: restricted preparations or effects change the premise; infinite observable dimension implies neither quantum structure nor a universal action. Extends moment descent.

Priority 1 · obstruction · proved-refereed

Two classical spin preparations share all retained first/cross moments and microscopic energy but evolve to different allowed product probabilities under H=J u_z v_z. Smooth microscopic reversibility therefore fails to descend to the operational quotient.

Scope: supplied spin magnitudes fix action units; reconstruction needs a stable preparation/effect restriction or state enlargement. Finite closure excludes finite repairs for all preparations.

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 · proved-refereed

Cartesian bounded-error experiments compose exact scalar minimax radii by sums of absolute weighted radii. Identical constituents make the canonical risk product extensive; discarding constituent records can strictly increase momentum risk.

Scope: these are separate coordinate risks under product admissibility, not variance coefficients or simultaneous per-trajectory errors. A mass-independent positive product fails this law. Uses indistinguishable pair.

Priority 1 · obstruction · mixed

Conditional classical bounded-error/bounded-recoil protocols require F²τ³>9mκ; a balanced two-mark construction succeeds above 36mκ. is superseded by: planck-gap-probabilistic.

Scope: The quantum compact-support instantiation is vacuous and does not derive κ>0. Use planck-gap-probabilistic and record-costs-disturbance for valid resource-relative quantum statements; mark availability and incoming momentum assumptions remain explicit.

Priority 1 · result · proved-unrefereed

A genuine receding-centre family retains the matched-endpoint lens but cancels the apparent tangent-sector remainder.

Scope: Subtract divergent references before limiting; comparison-path action has fixed normalization. A cell threshold does not imply exact action quantization without an additional tiling premise. Principia-constant-force-action supplies the maintained Galileo target.

Priority 1 · exploratory · proved-unrefereed

Reciprocal spring networks fix compatible inertia ratios while leaving one common action multiplier free.

Scope: Overall action rescaling and small-amplitude orbits still reach arbitrarily small action. The conclusion concerns the chosen spring completion, not arbitrary mechanics; composition-universality separates relative universality from positivity.

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 · exploratory · proved-unrefereed

Skyrme-type size stabilization exposes a coefficient with action dimensions but leaves common normalization arbitrary.

Scope: Common scaling of both coefficients preserves classical equations while scaling all actions, and vacuum excitations remain gapless. The calculation does not construct a full minimizer or prove dynamical well-posedness.

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 · proved-unrefereed

An explicit dissipative oscillator/depot model loses all action with finite fuel; replenishment above P_c gives attracting J_*=(P−P_c)/(2γω) for positive seeds.

Scope: The power/frequency law supplies the scale, and exact rest persists. Equal power gives equal energy rather than universal action; the model is effective dissipative dynamics, not a closed Hamiltonian field construction.

Priority 0 · obstruction · mixed

Corrected exploratory draft retains exact two-state commutator scales and a specified additive-noise double-limit calculation, while withdrawing general zero-response and classification claims.

Scope: the matrix finite-propagation classification is still conjectural; Markov conservation is an additional premise, and commutator norms can miss slow modes. Noise-selected limits do not force a physical floor. Two faces holds the scalar result.

Priority 0 · result · proved-unrefereed

Independent monotone detectors fed fixed fractions of one scalar pulse energy obey p₁₂≥p₁p₂, with sharp efficiency/balance trade-offs. Saturation and dark counts preserve the premise package.

Scope: fluctuating routing, anticorrelated readiness and outcome-selected gates explicitly evade it. This dimensionless exclusion witness selects no action unit; shared readiness CHSH tests the broader local model.

Priority 0 · result · proved-refereed

Routes to the reviewed collision manuscript: a stationary ordered two-stream preparation makes tagged motion periodic and kills its long-window response while preserving mass, speed, density and mean collision frequency.

Scope: Joint position–velocity correlations change; this tests preparation dependence, not a general action-floor theorem.

Priority 0 · result · proved-unrefereed

Independent equal-threshold detectors cannot maintain both exclusive clicks and unit efficiency over an open energy range; an explicit conservative quartic barrier realizes the crossing threshold.

Scope: Near-saddle preparation makes additional signal energy arbitrarily small, and Δ/ω is detector-dependent. Conclusions concern the stated accumulator/impulse model; conservative crossing alone supplies no permanent latch.

Priority 3 · conditional · mixed

Exact directional Gaussian composition preserves harmonic flux normalization but sharp blocked covariance grows with depth. Refereed 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 · mixed

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-refereed

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 · mixed

Dimensional and spectral scaling organize supplied versus generated gaps; written scalar/SU(2) matrix-model proofs explain transverse zero-point confinement and its ℏ^(4/3)g^(2/3) scale. Includes finite-volume, two-dimensional flux and scale-anomaly examples.

Scope: zero-mode gaps decay with box size; two-dimensional absence of local particles is dynamical, not absence of a mass unit. No result derives positive action structure from consistency alone.

Priority 3 · conditional · conditional

Packages an RG fluctuation-decay hypothesis and robust finite-box mixing certificate into a proposed lattice decay/gap transfer, with physical lower scale ℏc min(γ*,γ′)/a*.

Scope: both hypotheses remain unproved; continuum existence, nontriviality and observable/limit control are separate obligations despite the note’s endpoint rhetoric. Physical-coupling H2 certification is later rejected in reasons to stop; current progress follows the refinement spine.

Priority 3 · open · mixed

Canonical map distinguishes finite-lattice positivity, strong-coupling volume uniformity, weak-trajectory scaling, continuum axioms and the small-volume corner. T1 follows from compact resolvent and positivity improvement; δ(g) must scale with aΛ for a finite physical gap.

Scope: ultraviolet stability is not a completed OS construction, and all-coupling T2′ is stronger than required. Openings supplies the scaling-region correction.

Priority 3 · open · mixed

Earlier major synthesis collects finite-lattice/strong-coupling results, upper-bound tools, exact identities and seven failed approaches, with explicit weak/strong threshold asymmetry. Later sections reject correlation monotonicity and certified physical-coupling H2 as weak-side solutions.

Scope: method closures have named scopes; claimed large-field measure closure exceeds action-cost/model-entropy ingredients. Follow reasons to stop and the current refinement construction.

Priority 3 · obstruction · mixed

Strong-side monotonicity and available finite certification leave weak-side RG as the decisive mass-gap target. corrects: what-would-unblock.

Scope: The target becomes weak-side RG control, with explicit small-field constants; small-field-step-decay-and-threshold later tests that constants programme.

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 · obstruction · mixed

Explicit Gaussian decay estimates lead to a heuristic nonlinear threshold far below the desired physical weak-coupling regime.

Scope: No nonlinear polymer expansion or rigorous impossibility theorem is proved; this diagnoses the limitation of the proposed constant-chasing method after the Gaussian step.

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 · mixed

Normalized barriered one-step bounds culminate in all-order weak-strip correlations and exact dressed activities, with moving-background source control proved only for a quadratic test model.

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 refereed 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 · conditional · formal

The regularized SU(2) one-loop torus valley potential is an explicit positive periodic Fourier series, with linear zero-mode term 2ℏc|a| and quadratic nonzero-mode correction; symmetry fixes that quadratic coefficient for general constant backgrounds.

Scope: This is a one-loop effective potential, not an interacting excitation-gap proof or UV-uniform Schur estimate. Supports Feshbach H3; the cubic-lattice correction sign is borrowed.

Priority 3 · conditional · mixed

Explains how one-loop exponential terms scale under halving and why fixed infrared observables require uniform ultraviolet transport, separating dimensional transmutation from a positive gap coefficient. is superseded by: reasons-to-stop-as-research.

Scope: The ladder is conditional on matching; coarse observations may miss fine spectral channels. Its finite-crossover-certification wording needs the later weak-side diagnosis, and some passages use g² for the classical coupling.

Priority 3 · result · proved-refereed

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

A proved relative Schur-transfer lemma reduces the small-volume gap lower bound to nonzero-mode separation, a relative coupling error and an effective zero-mode gap.

Scope: All three physical inequalities remain open; the bare sharp-cutoff vacuum fails UV control. A fibered projection and gauge-compatible cutoff identify the corrected estimates, not a completed fixed-lattice gap theorem. The displayed cutoff calculation uses SU(2); transfer to SU(3) also requires group-dependent estimates.

Priority 3 · result · proved-unrefereed

A closed-surface polymer criterion 20e²ρ≤0.84 yields volume-uniform local-observable decay and a Wilson-transfer gap lower bound in units ℏc/a.

Scope: ρ=1/g² is a leading activity estimate; the conservative all-representation bound uses ρ=e^(6/g²)−1. Quoted 176/1056 thresholds use these different readings; positive gap needs a strict margin. Wilson and KS are distinct operators.

Priority 2 · result · mixed

Both compact U(1) and SU(3) have volume-uniform strong-coupling Kogut–Susskind gaps; their weak-side flows distinguish them.

Scope: Nonabelian arrival in the target box with generated interactions controlled remains conjectural; perturbative absence of abelian running does not itself prove every global phase claim. Use strong-coupling-target-box and the mass-gap position.

Priority 2 · obstruction · proved-unrefereed

Explicit Kogut–Susskind Agmon constants yield extensive global magnetic-energy suppression, while a fixed local excess estimate degrades as inverse square-root volume. corrects: agmon-ground-state-suppression.

Scope: Corrects agmon-ground-state-suppression; total energy cannot supply the local RG estimate. The proposed pointwise Gibbs-comparison repair also fails by magnetic-energy-identities, Proposition 4.

Priority 2 · obstruction · proved-unrefereed

Connected estimates are present in Yarotsky's expansion, but the block-boundary norm/block-gap criterion worsens with block size: M² at strong coupling and M³ in the small-volume regime. corrects: blocking-step-obstruction.

Scope: Corrects blocking-step-obstruction; improvement requires RG coupling/effective-action change. The general monotonic statement explicitly assumes δ(g;M)≤M²δ(g;1); displayed asymptotic regimes supply its concrete examples.

Priority 2 · obstruction · mixed

Gauge symmetry makes single-link block marginals Haar; useful finite-block certification must control gauge-invariant sub-block mixing. Published lattice scales orient the gap between rigorous strong coupling and physical crossover. is superseded by: reasons-to-stop-as-research.

Scope: Continuum-scaled glueball estimates near β≈5.7 need the explicit lattice-softening caveat; proposed small-box certification is superseded strategically by reasons-to-stop-as-research, which prioritizes weak-side RG.

Priority 2 · obstruction · mixed

The single-link coefficient α≤18(e^(4β_W)−1) gives a volume-uniform SU(3) Wilson transfer gap for g²>444 in physical units ℏc/a.

Scope: Section 4's single-link block criterion is explicitly corrected as vacuous in the interior by Elitzur; retrieve confinement-scale-bands for gauge-invariant sub-block mixing and reasons-to-stop-as-research for feasibility.

Priority 2 · conditional · conditional

The zero-momentum Feynman–Bijl bound reduces finite excitation energy to one finite derivative expectation and positive flowed susceptibility, with volume cancellation.

Scope: Continuum Hamiltonian equal-time flow limits and nontrivial susceptibility are hypotheses; Lüscher's 4D Euclidean theorem does not discharge them. flowed-bound-free-field corrects the extra g² factor. This proves no lower gap bound.

Priority 2 · obstruction · proved-unrefereed

Flow conjugation preserves spectra and multiplication sup norms, so it leaves norm/gap blocking criteria unchanged. Useful smoothing requires measure/state-weighted or relative-form control.

Scope: The addendum's proposed local Agmon repair is superseded by agmon-global-not-local; magnetic-energy-identities also rejects the simple local Gibbs-comparison route. Keep the invariant-criterion theorem separate from those obsolete next steps.

Priority 2 · obstruction · mixed

Finite-lattice flow is a diffeomorphism with unitary implementation; its exact Hamiltonian conjugation preserves the gap. Free magnetic damping cancels electric stiffening. Any locality-criterion gain needs a separate controlled truncation of generated terms.

Scope: lattice-truncation-uniform corrects one-step pessimism; exact changes of variables themselves never establish new spectral separation.

Priority 2 · obstruction · mixed

Symmetry of the curvature operator and the Nielsen–Olesen unstable chromomagnetic mode show that gradient descent can amplify perturbations near large-field saddles while lowering action. is superseded by: lattice-truncation-uniform.

Scope: The note's exact saturation claim needs care: its displayed Landau spectrum includes the positive covariant-Laplacian contribution, so it does not by itself establish growth equal to the curvature-only norm exponent. Lattice-step uniformity is separately corrected in lattice-truncation-uniform.

Priority 2 · result · mixed

Separates finite-lattice continuity from infinite-volume gap closure and organizes possible obstructions using the gapped coupling set. Strong coupling supplies a nonempty region; global openness and closedness remain unproved.

Scope: appended correction recognizes bulk transitions compatible with a continuum gap: only the weak-coupling trajectory and T3 are needed. Openings supersedes the all-coupling requirement.

Priority 2 · conditional · mixed

Standard transfer-matrix spectral projection identifies the finite-lattice ground-state density with the infinite-time Euclidean slice marginal, transferring estimates actually proved for that marginal. It explains the scale-invariant four-dimensional large-field action cost.

Scope: action cost alone supplies no normalized nonperturbative probability estimate or Hamiltonian operator inequality. The truncation-exponent competition is partly superseded by operator inequality.

Priority 2 · result · proved-unrefereed

A continuous-time electric-basis polymer expansion gives an explicit SU(3) volume-uniform KS lower gap (4/3)g² ℏc/a at g²≥388, retaining exact excited-link decay. The general connected-history tree count distinguishes this threshold from the adjacent-growth estimate g²≥79.

Scope: this is a strong-coupling lattice finish line, not weak-side RG or continuum construction; compare Wilson expansion.

Priority 2 · result · mixed

Flow monotonicity and positivity give the Euclidean action lower bound η²/(4g²) when the flowed block L² field exceeds η². The dimensional cancellation is independent of block scale.

Scope: this is an action cost; claimed sharpness needs admissible boundary/flow control, and a normalized probability bound still needs partition-function estimates. See entropy model and operator obstruction.

Priority 2 · obstruction · mixed

Exact min-max/trial-state inequalities distinguish harmless positive rare perturbations from negative wells that can reorganize low energy despite small measure. The attempted Hamiltonian large-field comparison loses cutoff-scale surface energy.

Scope: rarity cannot replace the missing signed operator bound; Euclidean polymer estimates consume measure control directly. Growing truncation range removes the older fixed-range exponent competition in measure transfer.

Priority 2 · result · mixed

Derives KS Feynman–Bijl upper bounds, fixes strong-coupling gap order, and exposes an abelian structure-factor gapless channel. For semisimple gauge groups, no nonzero c-number linear electric observable obeys Gauss law.

Scope: the dressing discussion blocks this trial route, not all gapless states, and proves no lower bound. Ultraviolet plaquettes cannot control continuum finiteness; use flowed susceptibility.

Priority 2 · obstruction · proved-unrefereed

Compact lattice variables and coupling-independent Wilson flow bound one doubling’s Jacobian/locality constants, giving an exponentially small truncation tail with increasing lattice range. This corrects the claim that raw field supremum obstructs one lattice step.

Scope: neither conjugation nor truncation removes variables; generated interactions and controlled decimation/iteration remain. Flow before decimation rejects improvement of the projection criterion by smoothing alone.

Priority 2 · result · proved-unrefereed

Treating commuting unbounded single-link electric terms exactly leaves bounded plaquette interactions with unchanged supports, permitting a volume-uniform Lieb–Robinson estimate with velocity proportional to Nc/g².

Scope: the weak-coupling bound degrades and establishes no physical continuum light cone. Locality alone does not prove openness of the gapped set; the cited stability route still needs hypotheses absent from finite-coupling KS.

Priority 2 · obstruction · exact

Exact magnetic-energy Laplacian/gradient identities give a ground-state variance/kinetic sum rule. Exponential comparison functions reproduce the extensive threshold that prevents this argument from suppressing a local excess uniformly in volume.

Scope: the failure concerns this eigenvalue-equation comparison, not a general impossibility of local estimates. Ground-state measure transfer imports Euclidean estimates once those estimates are proved.

Priority 2 · obstruction · mixed

For a supplied positive spectral measure with finite nonzero moments, consecutive ratios upper-bound its spectral edge and improve toward negative indices. A gapless density suppressed faster than every power at zero has finite inverse moments, closing that positivity argument.

Scope: correlator reconstruction, temporal-flow admissibility and infrared integrability need their own hypotheses; finite moment ratios prove no positive global gap. See finiteness.

Priority 2 · result · mixed

The averaged fundamental spatial Polyakov loop gives Δphys≤ℏcg²N/[4L Var(averaged loop)] for the SU(N) Kogut–Susskind Hamiltonian. A positive volume-uniform physical gap therefore requires suitable loop self-averaging.

Scope: This is an upper bound, never a positivity proof; proposed small-box scaling uses additional zero-mode estimates, and strong-coupling Haar variance yields a weak bound.

Priority 2 · obstruction · mixed

Bare-fibre second-order estimates expose a UV-sensitive Schur slope, and the advertised fixed-lattice gap still lacks its rigorous-constant proof.

Scope: The proposed fixed-lattice gap theorem explicitly lacks its rigorous-constant proof here. Weak-coupling Feshbach reduction therefore still needs dressed-vacuum and uniform estimates, not merely subtraction of vacuum energy.

Priority 2 · result · mixed

The linearized SU(3) four-dimensional block step has exact covariance and an explicit fluctuation Poincare bound.

Scope: This describes the Gaussian reference; nonlinear normalized estimates and iteration are separate. The decay/threshold continuation diagnoses why crude propagation constants do not close the weak side.

Priority 2 · conditional · conditional

Strong-coupling expansion tolerances define a quantitative RG target class robust to bounded local corrections.

Scope: Tables separate adjacent-growth estimates from conservative general counts and retain the quoted expansion premises. Reaching the box by exact low-energy RG with controlled generated terms remains open; uses KS expansion.

Priority 2 · obstruction · formal

Crude constant extraction exposes an astronomically restrictive Yarotsky threshold and motivates direct strong-coupling expansions. is superseded by: kogut-susskind-strong-coupling-explicit.

Scope: Expected direct-expansion thresholds 40–70 were estimates, not derived results; the later KS calculation supplies its actual threshold. The weak-side boundary still controls the crossover width.

Priority 2 · result · known

Yarotsky's established theorem yields a volume-uniform strong-coupling KS gap and thermodynamic ground state.

Scope: The threshold is existential here, with explicit improvements linked separately. This controls fixed-cutoff strong coupling, not the weak-bare-coupling continuum trajectory; see lattice obligations.

Priority 2 · conditional · conditional

Conditional on physical construction, 3D Yang–Mills gaps have the one-variable form g²ℏ²c f(g²ℏL); a positive finite infinite-volume limit is the remaining spectral target. Constant modes identify the small-box scale.

Scope: Strong lattice coupling and small-volume expansions are separate regimes; local stochastic dynamics supplies neither invariant measure nor OS theory. All-coupling positivity is sufficient, not necessary for the continuum target.

Priority 2 · obstruction · reading

Earlier confinement-route survey records where correlation cones, finite-size gap criteria and exact transformations fail to supply the weak-to-intermediate gauge bridge. is superseded by: reasons-to-stop-as-research.

Scope: Its claim of exactly two unblocking routes is corrected later: monotonicity would help only the strong side, and current physical-crossover certification is infeasible. Keep scoped ingredients, not the historical next-task queue.

Priority 1 · obstruction · mixed

Establishes the exact weighted ground-state Agmon identity and forbidden-region estimate on compact gauge configuration space. is superseded by: agmon-global-not-local.

Scope: Its original n-plaquette/local-suppression interpretation is corrected by agmon-global-not-local; only global magnetic-energy excess control survives. Sections proposing local decimation control must be read through that correction.

Priority 1 · obstruction · mixed

Derives a covariant-flow Jacobian bound with Gaussian spatial decay multiplied by a curvature-growth factor, motivating small/large-field splitting and a range/flow-radius truncation budget. is superseded by: lattice-truncation-uniform.

Scope: Its inference that one lattice step lacks uniform control is explicitly corrected by lattice-truncation-uniform. Kernel-tail estimates require further operator/relative-form justification; the linked instability note's exact sharpness claim needs caution.

Priority 1 · result · exact

Free flowed magnetic-energy susceptibility and derivative expectation give gap upper bound 8ℏc/(3√(πt)), independently matched by the two-photon spectral mean; g² cancels exactly.

Scope: Corrects finiteness-half-flowed-susceptibility; the bound vanishes with increasing smearing in Maxwell theory. In interacting YM its coefficient becomes nonperturbative near confinement, and an upper bound cannot prove positivity.

Priority 1 · exploratory · formal

Constant-background Landau counting estimates η²/(8π²) unstable modes per four-dimensional block, giving action-cost/count ratio 2π²/g².

Scope: this is a model estimate, not a general entropy or convergent polymer theorem; bounded weight per mode and normalized measure control are additional premises. Compactness gives a constrained minimizer but does not itself supply a Gaussian expansion. The action bound is in lower bound.

Priority 1 · exploratory · mixed

Records four parked approaches—multiscale curvature, centre-stabilized compactification, three dimensions and spectral valley lifting—and clarifies group/action dependence of lattice bulk transitions. Continuum scaling needs positivity near g=0, not every bare coupling.

Scope: curvature-threshold conversion is explicitly unchecked; deformed-theory continuity and SU(3) large-N transfer remain hypotheses. Its pause/next-task language is historical; use current STATE and atlas.

Priority 1 · obstruction · mixed

Free flowed fields satisfy t√⟨|b_t|²⟩=g/(4π); the typical-field truncation estimate can be reduced by increasing range, while a volume supremum needs tail control. is superseded by: lattice-truncation-uniform.

Scope: Interacting use is perturbative. Its lattice-step pessimism is corrected by compactness; this note does not prove an interacting normalized rarity bound.

Priority 0 · obstruction · mixed

Isolates scalar boundary-energy shifts from state-dependent Schur variation and counts tentative RG doublings to a strong-coupling threshold. is superseded by: blocking-criterion-monotone.

Scope: Its proposed connected-estimate induction is superseded by blocking-criterion-monotone; the available connected estimate does not improve the criterion. One-loop doubling counts are orientation, conditional on controlled effective couplings, not continuum-gap bounds.

Priority 2 · result · proved-refereed

Bounded-force reachable sets compose exactly through the joint phase state, and observed cuts have explicit residual fibres and areas.

Scope: Pointwise force admissibility and non-disturbing ideal observations are premises; refining the latest record collapses area without necessarily recovering momentum. Finite-precision-cut tests realistic record errors.

Priority 2 · result · proved-refereed

Eliminating the receiver of an equal-mass three-body spring chain yields an exact cosine memory kernel and inherited quadratures; tagged position/momentum alone fail to compose. Two nearby exact phase observations recover the missing state.

Scope: Recovery is ill-conditioned as sampling shrinks, and all prepared modal actions contract while the memory law remains fixed; extends classical cut state.

Priority 1 · result · proved-refereed

One finite mechanical clock and four probes reconstruct a known four-dimensional spring receiver at fixed latency with error/disturbance O(λ²), using preparation width b=λ³ and exact final momenta.

Scope: Forces, masses, duration and clock energy stay bounded; preparation/record precision shrink. This is delayed reconstruction, not causal fresh measurements at each cut.

Priority 1 · result · proved-refereed

Exact bounded-force centre risks add across independently budgeted record blocks; at r=3/2, k identical independent blocks give k times the saturated canonical risk product.

Scope: A fixed global budget instead gives copy-count invariance. Regrouping preserves the original Cartesian constraints; replacing them with a global norm ball changes the apparatus class. Precision remains supplied.

Priority 1 · result · proved-refereed

A fixed smooth pulse design admits an exact common-record receiver curve varying both x and P despite known nonzero first-probe displacement, receiver/apparatus energies and all ten final apparatus coordinates.

Scope: Positive canonical risk is preparation/design-relative and uniform only over weak coupling in that fixed class; a curve does not establish projected area. Later calibration/recovery notes refine the boundary.

Priority 1 · result · proved-refereed

A constrained eleven-coordinate preparation chart and Borsuk–Ulam yield distinct receiver states with identical ten-component final apparatus records and fixed energies/first displacement.

Scope: The uniform full-receiver norm risk need not lie in x or P; it proves neither canonical action-product nor area floor. calibrated-canonical-ambiguity supplies a separate pulse-design construction for both canonical coordinates.

Priority 1 · result · proved-unrefereed

On the fixed R30 apparatus box, receiver recovery error is O((record error+calibration error)/λ); exact final records give matching canonical-risk product order min{1,(ε/λ)²}. At sufficiently weak coupling one record hides the entire bounded receiver domain.

Scope: Four calibration tolerances are supplied resources; exact calibration recovers the receiver by three-calibration-global-recovery.

Priority 1 · result · proved-refereed

Two classical impulsive canonical probes per cut, prepared with phase widths O(mesh⁴), recover the hidden spring receiver uniformly with O(mesh) error and O(mesh³) accumulated disturbance.

Scope: Probe count/mass grows, switching is external, final records are exact, and preparation precision improves. autonomous-finite-readout supplies the later finite-mass fixed-latency construction.

Priority 1 · result · proved-unrefereed

Initial clock position, exact receiver energy and eight final pointer records recover receiver, probe positions and unknown clock momentum on one fixed transverse preparation patch, with record-error/coupling stability.

Scope: Local-patch knowledge and known incoming probe momenta are premises; full-shell global recovery fails in global-clock-speed-ambiguity. The canonical risk product closes as record errors vanish.

Priority 1 · result · mixed

Finite conservative spring receivers have zero long-window response; an increasing equal-mode-energy chain yields Θ√(m/k) only with size-first limits, and a common hard speed ceiling closes that plateau. Its 39 sections also consolidate cut memory and the full readout/calibration sequence.

Scope: Specialized notes hold individual proofs; the final correlated-fibre curvature nonvanishing test remains open and parked.

Priority 1 · result · proved-unrefereed

Terminal zero probe phase, backward Hamiltonian flow and an energy adjustment produce an exact receiver-sign pair with identical full final apparatus records and both initial energies fixed. Canonical risks stay positive as coupling decreases.

Scope: Fixed preparation width and receiver excitation supply the scale; two points establish neither area nor exact minimax value. calibrated-displacement-ambiguity removes reliance on sign symmetry.

Priority 1 · result · proved-refereed

Exact nonlinear calibration of the finite four-record apparatus permits arbitrarily precise receiver reconstruction at one fixed nonzero coupling as preparation and final-record errors shrink. Accuracy times bounded disturbance also tends to zero.

Scope: Dynamics/calibration knowledge, improved precision and fixed latency are resources; fixed coupling alone creates no floor. fixed-preparation-ambiguity holds incoming width positive instead.

Priority 1 · result · proved-refereed

Unknown incoming probe momenta exactly compensate a two-dimensional receiver-shell patch at one common four-momentum record, giving positive position/momentum risk and projected canonical area bounds.

Scope: Fixed-width Cartesian preparation support and four-record access supply the obstruction; this is reconstruction risk, not an accuracy–disturbance lower bound. Extra calibration/records change the compatible fibre.

Priority 1 · result · proved-refereed

Unknown incoming apparatus coordinates compensate receiver changes while preserving every final apparatus record; sufficiently weak positive coupling can hide the entire fixed-energy shell. Its projected ellipse gives exact coordinate minimax risks.

Scope: the floor depends on preparation widths and receiver energy; supplying apparatus preparation restores recovery. Additional total-energy constraints are treated in energy-constrained ambiguity.

Priority 1 · conditional · reading

A source theorem excludes genuine reversible interaction on the minimal separable orientation composite under explicit operational premises.

Scope: Mechanical interaction need not descend reversibly to retained moments. This source/hypothesis audit does not verify the full reconstruction proof or supply an action scale; generator constraints checks only its initial technical step.

Priority 1 · result · proved-refereed

Sharing an ℓr record budget across bodies changes the minimax composition exponent through resource accounting.

Scope: The anomalous exponent reflects apparatus-resource sharing; independent budgets in block composition restore the appropriate extensive composition.

Priority 1 · conditional · conditional

A capped activated-rate detector with fixed attempt frequency has exact feasibility and optimal signal barrier reduction kBT[log(cτ/(atg))]+.

Scope: Incident-work bounds require a supplied transduction coefficient; variable prefactors evade barrier reduction. Temperature, deadlines and errors remain in the cost, which vanishes in admitted parameter limits and selects no universal action.

Priority 1 · result · proved-refereed

Two bounded-error position records give simultaneous sharp errors for the stated causal position/momentum estimator, and both errors vanish when precision improves faster than sample separation and delay vanishes.

Scope: Known initial state, non-disturbing records and a force ceiling are inputs; this is not optimal minimax recovery. The equal-error optimum is δ*=2√(mε/F), with momentum error 2√(mFε).

Priority 0 · result · mixed

A selected correlated calibration line suppresses canonical ranges on one common-record fibre. Signed-coupling symmetry cancels the proposed tangent coefficient exactly; moving-reference terminal quadratures give homogeneous quadratic response and a conditional curvature lower bound.

Scope: This is local-fibre support, not global minimax recovery; both projected curvature coefficients remain unproved nonzero. action-scale-obstructions parks further apparatus continuation.

Priority 0 · result · proved-unrefereed

Eight final pointer coordinates plus final clock momentum recover receiver, incoming probe positions and unknown clock speed globally on a fixed bounded domain at small positive coupling. Reaction is included; exact receiver energy is unnecessary.

Scope: Initial clock position and incoming probe momenta remain supplied. Record precision controls the vanishing canonical risk product; full-clock-phase-recovery hides clock offset too.

Priority 0 · result · proved-unrefereed

Joint final pointer and clock phase-space data globally recover the receiver and unknown incoming positions/clock phase on a bounded domain, with reaction retained. At fixed nonzero coupling, record errors tending to zero remove reconstruction risk.

Scope: incoming probe momenta remain supplied; unknown full apparatus preparation restores ambiguity in R25.

Priority 0 · result · proved-unrefereed

Eight final pointer coordinates globally recover a bounded receiver state and unknown probe positions at sufficiently small positive coupling, including clock reaction. Explicit inverse bounds make coordinate-risk product vanish with record precision.

Scope: initial clock phase and probe momenta are supplied, and joint classical record access is assumed. Hidden clock removes that calibration premise.

Priority 0 · result · proved-unrefereed

An early-pulse apparatus admits two distinct clock speeds and equal-energy receiver states with identical full pointer records, persisting with positive-coupling reaction and separating both receiver coordinates.

Scope: this is a design-existence, preparation-dependent coordinate-risk bound, not canonical area. It preserves local calibrated recovery; final clock momentum repairs the missing record.

Priority 0 · result · proved-unrefereed

Unknown clock position/speed allows a smooth equal-record receiver family on an exact energy shell, with both canonical coordinates varying despite eight exact pointer records and known probe momenta.

Scope: fixed pulse design and positive preparation widths give coordinate risks, not enclosed canonical area or a universal floor. Revealed clock information is progressively tested in local recovery.

Priority 0 · result · proved-unrefereed

In a fixed admissible pulse design, unknown probe positions compensate a receiver-shell patch despite four initial and four final momenta being recorded; clock reaction supplies an invertible λ² response.

Scope: The reconstruction lower-bound product scales as (λb)² and is preparation-relative, not universal or a risk-vanishing theorem. Full-pointer-recovery tests the richer-record endpoint.

Priority 0 · exploratory · proved-unrefereed

A prescribed classical shared-release receiver reproduces exclusive fringe probabilities with explicit finite-window and delay costs.

Scope: Shared control, linear hazards and readiness are supplied architecture. At vanishing energy, finite-window detection probability vanishes; this supplies neither local independent detection nor a universal action floor.

Priority 0 · result · proved-refereed

One uncertain apparatus calibration leaves an exact equal-record receiver curve with design-dependent canonical ambiguity.

Scope: Projected curve area remains zero, arbitrary designs need nonzero cofactors, and ambiguity shrinks with calibration tolerance at fixed coupling. Uses tolerance recovery.

Priority 0 · result · proved-refereed

Three exact initial probe-position calibrations, apparatus energy and full final apparatus data give uniform global receiver recovery on a bounded convex domain, for a fixed pulse design and suitably reduced fixed preparation box.

Scope: Record-error amplification is O(1/λ); at fixed coupling its canonical product tends to zero. Calibration errors need the tolerance extension.

Priority 0 · result · proved-refereed

Two calibrated probe positions and both exact energies remove local degeneracy in a fixed apparatus design but leave two globally distinct receiver preparations with identical complete records and positive canonical separation.

Scope: The preparation-relative bound is neither positive area nor sharp global minimax risk. A third calibration gives uniform global recovery on its stated reduced domain.

Priority 1 · historical · reading

Separates Aristotle's individually possible versus simultaneous divisions from the project's modern claim that re-solving a discretized trajectory can readjust vertices.

Scope: Sampling, variational re-solving and exact marginal composition are distinct; no ancient path-integral or Newton-transmission claim is supported. cut-point-consistency treats fixed surrounding endpoints only.

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 1 · historical · reading

A source-premise map identifies Hardy connectivity and CDP purification without deriving a mechanical action scale.

Scope: These elementary exclusions neither reject every classical phase-space model nor supply a mechanical action scale; full reconstruction proofs were not read. Hamiltonian-finite-closure tests whether an interacting mechanical description can satisfy the required finite closure.

Priority 0 · historical · reading

Reads ancient arrow arguments as first-order motion questions, Newton's sling as second-order deviation under inertia, and the firebrand circle as sampling/appearance.

Scope: The corrected note explicitly retains Plutarch's sling–moon witness; any claim that ancient philosophical sling arguments are absent is false. The force-rung analogy uses planck-gap-paper, not a historical transmission proof.

Priority 0 · historical · reading

Conversation provenance connects discrete/deterministic programmes to ancient cuts, information loss and the Newton refinement thesis; a later Gaussian groupoid test admits both zero and positive action branches.

Scope: Most historical references/interpretations are explicitly from memory and unverified; three automaton DOIs are metadata-verified only. Information-loss equivalence classes lack a specified record-floor implication; no live queue change follows.

Priority 0 · historical · reading

Ancient contact and assembly arguments supply qualified historical counterparts to interface-state composition.

Scope: Vasubandhu and Ibn Sina are differentiated from modern interface-state arguments; Chinese restorations and the Liu Hui witness need edition/facsimile qualification. Resemblance proves neither transmission, quantum structure nor an action unit; compare classical cut state.

Priority 2 · result · reading

Claude Opus 5.5 scored every note from 1 to 10 for interest and for correctness under one rubric, and Claude Fable 5.1 reviewed the 26 selected digests against the notes. The Newton, refinement and gauge spine scores 8 or 9 for correctness; 24 notes keep uncorrected overclaims or inconsistent numbers, 20 of them closed mass-gap routes. Verified errors: planck-gap-derivation area forms off by a factor 2, strong-coupling-target-box q=4 column, small-field-step-decay abstract constants. corrects: planck-gap-derivation.

Scope: Scores are a model reading with reasons on file, not refereeing; §3b items are the scorer's reports, unverified; interest anchors measure what an outside expert would take, not novelty. The two digest PDFs and the score table are published on the director's personal site, not here.

Priority 2 · historical · reading

Source-backed digest states the Navier–Stokes alternatives and continuum infinite-volume Yang–Mills existence/gap target, including forcing distinctions, nontriviality, axiomatic strength and 0<m<∞.

Scope: official originals govern details; confinement and isolated particles are additional questions. Its attributed 2026 announcement is recorded source context, not independent proof verification. Use this definition before interpreting lattice or auxiliary-clock gaps.

Priority 2 · result · known

Evaluates group-dependent constants in the earlier SU(3) programme: Casimirs, strong-coupling normalization, locality velocity, magnetic identities, running coefficients and flux-sector counts.

Scope: Lookup conventions remain useful; extrapolated doubling counts, small-volume interpretation and older route status are not new proofs. Retrieve maintained dependency notes for current thresholds and corrected physical/continuum scope.

Priority 1 · result · proved-refereed

Two-sided product-test positivity restricts reversible generators to a tensor power of a seven-dimensional local matrix space.

Scope: This is a necessary ambient linear-space restriction, not a full admissible Lie algebra or reconstruction; later source steps and physical clock/action identification remain separate.

Priority 0 · result · mixed

Early working calculations cover independent composition, telegraph bridge crossover, auxiliary spectral bounds and analytic continuation. Hidden slow modes preserve a velocity plateau while closing the full relaxation gap.

Scope: Historical script/sampler evidence and uncompiled Lean are not active verification instructions. Retrieve maintained composition-universality, telegraph-return-bridge, checkerboard-dynamics and susceptibility-gap for durable statements.

Priority 0 · result · mixed

Routes to the older technical manuscript’s action contractions, normalized Hessian bound, short-time kernel expansion and quantum two-arm discrimination threshold.

Scope: the threshold assumes supplied phase/Born rules and independent-copy resources; increasing copies removes a fixed imperfect-confidence threshold. This note contains a location table rather than independent derivations. Later matched comparisons are in Galileo interference.

Priority 3 · open · mixed

Maintained map of directional and arbitrary-fraction refinement, grouped by dimension and gauge group, linking exact identities, formal running, corrections and open cells. Tracks the gauge weak-strip activities and Newton complete-record/nonlinear counterexamples through recent appended sections.

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

Priority 3 · result · mixed

Exact insertion laws and summable defects construct mechanical refinement limits and frame the unresolved higher-dimensional gauge construction. 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 · mixed

A compact theorem map collects Newton cut-record results and exact/barriered gauge-halving results with source proof locations.

Scope: Source notes carry current proofs and corrections; finite-order derivative bounds and rare-cluster estimates are not polymer convergence. Newton continuity remains model-relative; SU(N) transfer stated here is one-loop, not a full nonlinear continuum theorem.

Priority 3 · exploratory · mixed

Relates tangent-groupoid zoom and principal symbols to single-step refinement, and time-graded composition to whole trajectories; constant-force action/kernel laws provide the explicit example.

Scope: Geometric identifications are readings of established constructions. Diffusive quantum-path interpretation presupposes supplied positive ℏ and does not establish action necessity; uses refinement construction and fifth postulate.

Priority 3 · conditional · conditional

Sets the joint-paper frame: Newton action necessity and pure SU(3) continuum gap are the two proof goals; massless/massive pion physics tests symmetry and surviving spectral weight. Two elementary spectral-measure limit criteria distinguish physical thresholds from disappearing eigenvalues.

Scope: Neither goal is proved; September 29 updates older H3-first routing to UV halving, and STATE controls today’s 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-refereed

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 · proved-refereed

Fixed-energy spring position conditional kernels fail semigroup composition; independently refreshing hidden momentum at every cut freezes terminal position in L². Retaining canonical momentum restores exact composition on every partition.

Scope: Observational cuts retain correlations; resets change the experiment. Orbit action E/ω remains preparation-selected, and three-body-cut-memory shows reduced tagged phase state can still need receiver memory.

Priority 2 · result · proved-refereed

Constant-force sampled-chord defects vanish as F²T|π|²/(24m); Gaussian bridge restriction consistency fixes any supplied κ, including zero. Fixed positive κ makes raw kinetic excess diverge while its per-node estimator converges.

Scope: Changing κ with refinement changes preparations; a fresh inserted cut costs expected κ/2 only under the supplied Gaussian rule. Newton's impulsive polygon is a different scheme.

Priority 2 · result · mixed

Organizes supplied action scales, rotor/string energies and generated YM scales by dimension, with explicit ℏ exponents and the 4D exponential. Proposes universal wave-field thermal consistency as an extension of Theorem U.

Scope: That extension is conjectural; refinement error budgets alone establish no higher-dimensional construction. Paper-spine readings, gravity least-length route and renormalon comparisons require their stated premises.

Priority 2 · result · proved-refereed

Equal-mass elastic return bridges retain velocity memory and obey sharp bounded-speed midpoint variance bounds, while tracer action scales as mu²Δ/2 and vanishes under refinement. Scalar probability convolution semigroups with exact finite propagation reduce to deterministic drift.

Scope: Prepared partner separation changes with the interval; internal-state models such as telegraph-return-bridge evade the scalar closure premise.

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 · proved-refereed

Exact count/simplex return paths give consistent velocity-resolved cuts, a midpoint atom plus beta-mixture law, controlled vanishing polygon-action error and an explicit long-window crossover to a supplied plateau.

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 2 · result · proved-refereed

Gaussian bridge polygons can converge uniformly to a classical path while retaining excess action ℓ/2 when dimension times the vanishing action coefficient tends to ℓ.

Scope: The residual is regulator/preparation-selected, not a hard action-value floor; running bare mass to preserve a reference kernel is a different experiment. The full derivation is the regulator-limits manuscript.

Priority 1 · result · mixed

Derives fluid/gauge scaling and dissipation identities and distinguishes finite-box fluid decay, auxiliary Markov mixing and physical Hamiltonian spectral gaps. Lists invariant-law, continuum/volume, reconstruction and observable-completeness obligations for proposed bridges.

Scope: Historical orientation, with lower-dimensional cited SPDE results and conditional transfers; three-continuum-limits and mass-gap-obligations-lattice carry the current programme.

Priority 1 · result · mixed

Connects source-inspired spatial cutting to Gaussian time-kernel composition: convolution preserves any supplied variance coefficient κ/m, including the deterministic zero member.

Scope: Positivity and oscillatory analytic continuation are separate physical premises. Historical source tasks and old next-step instructions are context; refinement-composition-and-limit develops the maintained insertion framework.

Priority 1 · result · proved-unrefereed

Ballistic-support scalar convolution probability semigroups are deterministic drifts; scalar translation-invariant strongly continuous unitary groups with compact light-cone kernels are translations times a phase.

Scope: The unitary proposition is a working unrefereed derivation; nontrivial telegraph/Dirac dynamics retain internal state. Cone/arrow comparisons are modern heuristics, and the supplied crossover scale does not establish action positivity.

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 · proved-refereed

Derives exact terminal canonical area from a clipped bounded-force lens, with additive precision/delay and mixed extrusion terms; area vanishes uniformly when position error and residual delay vanish together.

Scope: A central record can retain finite momentum width while area closes. Non-disturbing record access is supplied; two-position-recovery addresses recovery of both coordinates.

Priority 1 · result · proved-refereed

A written coupling lower bound and magnetization eigenmode prove the known full relaxation gap a[1−tanh(2b)] for every periodic N≥3. Bounded coupling and a per-site clock floor give volume-uniformity.

Scope: growing coupling or a fixed total clock closes it; this auxiliary stochastic rate selects neither a quantum Hamiltonian nor action normalization. See Hermitian transfer.

Priority 1 · result · proved-refereed

The weighted-space unitary transform produces explicit positive three-site Ising terms, a unique square-root Gibbs ground state, and the exact inherited gap a[1−tanh(2b)]. Supplied action K converts this to energy.

Scope: physical real-time dynamics, state interpretation, clock normalization and infinite-volume construction remain additional premises. Established stochastic-parent correspondence, with a derived local specialization; see foundational value.

Priority 0 · result · known

Routes the former standalone midpoint-crossover derivation to maintained telegraph-return-bridge, sections 6–9, preserving C037–C038 and source links.

Scope: This file contains no independent calculation; retrieve the consolidated manuscript for preparation, beta law, moments and mass/window constraints.

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: references are metadata-level and holographic interpretations are orientation; large-N capped geometries prove no SU(3) theorem. The usable standard marginal identity is in measure transfer.

Priority 0 · result · known

Separates the known size-independent Ising relaxation gap, logarithmically growing global mixing time, and energy gap obtained by multiplying a stochastic parent by supplied action.

Scope: equilibrium weights fix neither generator nor clock normalization; further spectra add no action-selection mechanism. Return only for a physical-generator comparison or distinctive conceptual claim. Uses interacting gap and Hermitian parent.