Manuscripts and PDFs

Major syntheses, technical notes, working drafts and historical snapshots have different roles. These are research manuscripts; a PDF does not imply peer-reviewed publication.

Source notes govern current assumptions and corrections. Version and orphan audit · Complete note archive

Current synthesis

• Classical action scales: obstructions, conditional bounds and quantum premises TeX source. Current source note.

  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.

• Corpus audit of 1 October 2026: two-axis scores for all 175 notes, the uncorrected errors they expose, and where the drift sits TeX source. Current source note.

  Audit of 1 October 2026: Claude Opus 5.5 scored every note for interest and correctness and Claude Fable 5.1 reviewed the 26 selected digests; lists the 24 notes whose current text still states withdrawn or inconsistent claims, with three errors verified. Use it to find which notes need dated correction boxes; the condensed digests are published on the director's personal site, not here.

• Atlas of halving: what one refinement step does, and what emerges, in each dimension TeX source. Current source note.

  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.

• The mass gap as a finite list of theorems: the Hamiltonian lattice route TeX source. Current source note.

  Canonical operator/limit decomposition; finite-lattice and strong-coupling statements are distinct from continuum construction and a physical gap. T2′ at all bare couplings is stronger than the scaling-trajectory requirement.

• The mass gap: current position of this programme TeX source. Current source note.

  Maintained SU(3) map separates finite-lattice positivity, existential KS volume uniformity, rigorous Wilson strong-coupling bounds, weak scaling and continuum reconstruction. The October 2 precision distinguishes the exact sufficient KP condition, the integer 1059, the positivity condition for the 176 rate, and the linearization 1056. Scope: H1/H2 physical blocking and mixing, continuum construction and nontriviality remain open; ultraviolet stability does not complete OS reconstruction. KS 388 and adjacent 79 thresholds are withdrawn. Wilson and KS operators and finite-volume versus volume-uniform statements remain separate. Seven routes close only their stated norm/counting families.

• The cost of a mark: Newton's vanishing sagitta and a floor of order ℏ TeX source. Current source note.

  Newton/ancient-source synthesis combines conditional Gaussian recording bounds, general disturbance costs and polygon phases. The October 2 opening states supplied quantum action, coherence and domain-selection premises; section 10a integrates fixed-interval Kepler localization and centroids, the incoherent classical countercase, and reduced-phase Coulomb thresholds. Scope: Quantum states, instruments and positive hbar are premises. The full Planck constant is h=2 pi hbar; Coulomb selections use reduced hbar. Coherent-stream contrast requires coherence and a record topology; the incoherent-mixture countercase is exact for quadratic forces. Dirac finite-potential and critical-form selections have different endpoints; the KG radial endpoint does not prove full critical dynamics. The critical Dirac domain uses the original form closure and excludes the problematic compact source rewrite. Classical completions are not excluded. Independent necessity, universal calibration and historical submission obligations remain open. New additions are unrefereed; earlier reviews do not certify them, and literature priority is unestablished.

• What survives refinement: results on Newton's vanishing sagitta and the halving of lattice gauge fields TeX source. Current source note.

  Compact theorem compendium and proof links. For subsequent all-order weak-strip dressing, moving sources and nonlinear recording, read the later maintained notebook sections.

• What survives refinement: the pion, the Yang–Mills gap and Newton's action TeX source. Current source note.

  Joint Newton/SU(3)/pion synthesis separates supplied quantum action, selected Coulomb domains, coherence-dependent record contrast and physical spectral gaps. October 2 integration adds fixed-interval Kepler localization/centroids, Dirac/KG endpoints and corrected Wilson/KS inputs. Two elementary spectral-measure criteria require surviving observable weight. 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.

Current technical manuscript

• Strong coupling for abelian and non-abelian gauge theories: a shared existential gap TeX source. Current source note.

  Both compact U(1) and SU(3) have volume-uniform strong-coupling KS gaps by the group-general Yarotsky application with an existential threshold. The October 2 correction withdraws inherited explicit continuous-time rates and box location. A written U(1) finite-volume bound is 2g²−4P/g² in units ℏc/a on tori with sides at least four. Scope: The finite-volume bound depends on P=3N_s³ and gives no uniform threshold. No certified 4 lambda rate or box at g² about 400 survives. Perturbative absence of abelian running does not prove an exact compact-lattice flow or a single gapped interval; constructive arrival and continuum reconstruction remain open. Use strong-coupling-uniform-gap, the corrected continuous-time note and target-box note.

• A universal action floor needs a fixed action unit and no admitted similarity TeX source. Current source note.

  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.

• Every mark with body-independent noise is a momentum-transfer mark TeX source. Current source note.

  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.

• The Agmon bound controls the global excess only: the local large-field estimate does not follow TeX source. Current source note.

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

• The blocking criterion is monotonically worse in the block size: blocking gains nothing without a change of coupling TeX source. Current source note.

  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. 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.

• A sharp mechanical cost for a finite-duration reversal TeX source. Current source note.

  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.

• One action scale, two rules for composing paths TeX source. Current source note.

  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.

• A conservative receiver and the origin of a correlation scale TeX source. Current source note.

  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.

• Newton's cell action as a measure on cuts TeX source. Current source note.

  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.

• A surviving action defect and consistency under inserting cuts TeX source. Current source note.

  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.

• The dimension ladder: from 0+0 to 1+3, and what the action constant does on each rung TeX source. Current source note.

  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.

• Dobrushin's Wilson criterion: an explicit strong-coupling gap at g² > 444 and an open block test TeX source. Current source note.

  The single-link upper coefficient alpha≤18(e^(4 beta_W)−1) gives a volume-uniform SU(3) Wilson-transfer gap with exact lower rate log[1/(18(e^(24/g²)−1))] in units ℏc/a. The exact sufficient condition is g²>24/log(19/18); g²>444 suffices. The comparison table now distinguishes exact rates from leading logarithms. Scope: The leading estimates log(g²/432) and 4 log(g²/1056) are not rigorous lower rates. The polymer integer bound is 1059; its rate dominates only at sufficiently large coupling. Section 4’s single-link block criterion is vacuous in the interior by Elitzur; use confinement-scale-bands for invariant sub-block mixing and reasons-to-stop-as-research for feasibility. References remain metadata-level.

• Nonlinear feedback selects an action only while its energy source supplies power TeX source. Current source note.

  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.

• Local entangling gates preserve an energy gap uniformly in chain length TeX source. Current source note.

  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.

• Flowing before decimating leaves every norm-based criterion unchanged: what is needed is a statement about the state TeX source. Current source note.

  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.

• Conjugation by the flow preserves the spectrum, so the renormalization step is a truncation TeX source. Current source note.

  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.

• The flowed bound in free field theory: the coupling cancels, and the bound is informative only at the confinement scale TeX source. Current source note.

  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.

• Continuous spin in Galilei mechanics: a transverse boost moment TeX source. Current source note.

  The continuous-spin analogue at zero Galilei central mass measures a transverse boost moment, realized by a signed-mass dipole with unbounded energy. The elementary spacetime orbit has fixed-time curves. A supplied Poincaré spectral gap gives a boost–momentum uncertainty floor proportional to ℏΔ/c²; a positive continuous-spin label alone leaves energy arbitrarily close to zero. No positive-action selection or Yang–Mills construction is claimed.

• Galileo's comparison as two-path interference TeX source. Current source note.

  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.

• T2' is the absence of a zero-temperature phase transition, and the missing input is closedness of the gapped set TeX source. Current source note.

  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.

• Gaussian product blocking: the tree-level coupling never moves TeX source. Current source note.

  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.

• A uniform relaxation gap with nearest-neighbour interactions TeX source. Current source note.

  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.

• The Ising gap gives a conditional local parent Hamiltonian TeX source. Current source note.

  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.

• Continuous-time Kogut–Susskind expansion: corrected overlap bounds and an open explicit uniform threshold TeX source. Current source note.

  A continuous-time electric-basis polymer expansion for SU(3) Kogut–Susskind; its former volume-uniform gap (4/3)g² ℏc/a at g²≥388 (79 adjacent) was withdrawn on 2026-10-02: the anchored adjacent count misses later histories, the overlap shift measure lacked its multiplicity, and true-vacuum decay was unproved. Surviving: corrected overlap bounds, a finite-volume gap bound (8/3)g²−12|P|/g², a proved spectral lemma; the explicit uniform threshold is open. Scope: Strong-coupling lattice results only: the corrected finite-volume physical bound is (8/3)g²−12P/g² for tori with sides at least four; no explicit volume-uniform KS threshold stands. Yarotsky gives an existential uniform theorem. Compare the distinct Wilson operator with sufficient integer bound 1059; 1056 is linearized and 176 is leading activity only.

• The large-field action lower bound is immediate in the natural variable, and the constant field saturates it TeX source. Current source note.

  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.

• The large-field region in the Hamiltonian route needs an operator inequality, and the measure statement is weaker than that TeX source. Current source note.

  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.

• Upper bounds on the lattice gap: the Feynman–Bijl inequality, the abelian structure factor and the non-abelian dressing obstruction TeX source. Current source note.

  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.

• On the lattice one flow step has a uniformly bounded truncation error: the obstruction is decimation TeX source. Current source note.

  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.

• A Lieb–Robinson bound for the Kogut–Susskind Hamiltonian, with velocity ∝ Nc/g² TeX source. Current source note.

  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.

• Independent monotone receivers cannot suppress coincidences TeX source. Current source note.

  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.

• A gap is a scale the classical theory lacks: solved low-dimensional mass gaps and the positive-action question TeX source. Current source note.

  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.

• Two exact identities for the magnetic energy, a ground-state sum rule, and why pointwise Gibbs domination fails as Agmon does TeX source. Current source note.

  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.

• The mark cost is exactly ℏ/2, and the worst-case theorem survives as a statistical one TeX source. Current source note.

  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.

• Mechanical interference measures phase without fixing an action unit TeX source. Current source note.

  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.

• A monotone hierarchy of upper bounds from one flowed correlator, and why no bound of this kind can give m > 0 TeX source. Current source note.

  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.

• What a necessity argument for h > 0 must supply: a unit and an indeterminacy TeX source. Current source note.

  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.

• Galileo's falling parabola, Newton's refinement and the Planck-scale question TeX source. Current source note.

  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.

• Newton's record as a parallel move: what forgetting a record leaves TeX source. Current source note.

  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.

• A passive threshold creates events, but its scale belongs to the receiver TeX source. Current source note.

  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.

• A physical cut: elastic reversal, finite speed and memory TeX source. Current source note.

  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.

• The probabilistic Planck gap: Fτ(L+τ P/2m) ≥ ℏarcsin(1-2ε), and why it is resource-relative TeX source. Current source note.

  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.

• The gap is bounded by the variance of the averaged spatial Polyakov loop, with the small-volume scaling built in TeX source. Current source note.

  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.

• Newton's inscribed polygon differs from the parabola by the phase of its segments TeX source. Current source note.

  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.

• The two reasons to stop, as research problems: what a correlation inequality would buy, what a certified verification cannot, and where the research is TeX source. Current source note.

  Strong-side monotonicity and available finite certification leave weak-side RG as the decisive mass-gap target. Scope: The target becomes weak-side RG control, with explicit small-field constants; small-field-step-decay-and-threshold later tests that constants programme.

• The record costs disturbance: a floor for every instrument TeX source. Current source note.

  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.

• The record costs recoil: a floor for every probe state TeX source. Current source note.

  Arbitrary momentum-transfer probes retain a recoil bound, while grid probes show why Gaussian action-error bounds are not universal. Scope: Adaptivity requires resource suprema; Gaussian three-mark comparison uses infinite endpoint recoil. Record-costs-disturbance extends beyond this coupling class.

• The record's distance is a path length TeX source. Current source note.

  Bures-angle paths sharpen protocol constants and combine aperture and disturbance into one force-split accounting. Scope: Hybrid resource bounds and hypothesis preparation quantifiers matter; sharp single-shot constants do not prove protocol attainment or independently select ℏ.

• Inserting a point, subdividing a cell: how the limit is built TeX source. Current source note.

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

• Rivero's consistency conjecture: quartic paths and central-force angles TeX source. Current source note.

  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.

• One action constant from composition and spatial rotations TeX source. Current source note.

  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.

• The Schur error of the small-volume reduction is an ultraviolet problem: a fixed-lattice theorem is available, the renormalized regime is not TeX source. Current source note.

  Refined gap-transfer lemma and formal UV diagnosis. The proposed fixed-lattice gap proof with rigorous constants is explicitly not completed; renormalized/fibered Schur control remains open.

• Score-constrained ensembles and population transport TeX source. Current source note.

  Exact constrained action, reservoir-aware population momentum and fixed compact-shape obstruction. A width/phase mode has an exact positive non-Gaussian forward completion; nonlinear copies generate quartic exponential tails outside finite modes, with exact unread recoil energy. Written, internally checked. Physical constraint enforcement, closed apparatus and positive action necessity remain open.

• The self-sufficiency contrast: where Newton's axioms leave motion undefined, the theory with h > 0 defines it TeX source. Current source note.

  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).

• Series and parallel: where the gauge insertion law stops closing TeX source. Current source note.

  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.

• Independent settings test shared readiness, while leaving action units free TeX source. Current source note.

  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.

• One small-field blocking step, part 1b: the decay rate and the shape of the threshold, and why the weak side's deficit is ten orders of magnitude TeX source. Current source note.

  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.

• One small-field blocking step for SU(3), part 1: the Gaussian fluctuation integral with block averaging, with explicit constants TeX source. Current source note.

  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.

• The strong-coupling threshold made explicit: Yarotsky's proof gives g₀²~10¹⁰⁰, and only a direct expansion can give g₀²~10² TeX source. Current source note.

  Crude constant extraction exposes an astronomically restrictive Yarotsky threshold and motivates direct strong-coupling expansions. 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.

• The strong-coupling gap of the Kogut–Susskind Hamiltonian is uniform in the volume TeX source. Current source note.

  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.

• The SU(2) bridge midpoint in closed form: the curvature term is exact TeX source. Current source note.

  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.

• SU(3) in four dimensions: every constant of this programme, evaluated TeX source. Current source note.

  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.

• The bridge midpoint on a compact group at any cut: windings and the centre TeX source. Current source note.

  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.

• When an action plateau controls a spectral gap TeX source. Current source note.

  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.

• A finite-speed return bridge: cuts and midpoint crossover TeX source. Current source note.

  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.

• Thermodynamics of records gives a trade-off and no floor TeX source. Current source note.

  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.

• A topological energy floor survives shrinking while the action cost vanishes TeX source. Current source note.

  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.

• Compact U(1) under refinement: an explicit total-variation bound in three dimensions, and why it fails in four TeX source. Current source note.

  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.

• An explicit strong-coupling gap for the Wilson transfer matrix: SU(3) is gapped for g² ≥ 1059, with Δ ≥ (ℏ c/a) 4log[1/(176(e6/g2-1))] TeX source. Current source note.

  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.

• Which limits of Newton's construction reach the zero branch TeX source. Current source note.

  Locates the gap thesis of 2026-10-01: Galileo's comparison with Gaussian preparations and marks has a record family constant in h (Thm 1); on any finite partition with several nondegenerate paths the complete record has no h->0 limit while smeared records converge to the classical branch sum (Thm 2); exact for inertial motion with a graded velocity after its fold time (Thm 3). Written by Claude Fable, unrefereed.

• The zero-spacing limit for any plaquette action and any dimension TeX source. Current source note.

  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.

Working manuscript

• An action floor on transverse phase-space area produces the Yang–Mills quantum-mechanical gap, and a gap forces an action unit TeX source. Current source note.

  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.

• A shared action coefficient from classical composition TeX source. Current source note.

  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.

• The finiteness half of the mass gap is the nonvanishing of one smeared susceptibility TeX source. Current source note.

  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.

• The flow Jacobian's growth factor is sharp: it is the Nielsen–Olesen mode, so the large-field region cannot be flowed TeX source. Current source note.

  Qualitative unstable-mode amplification is useful. The source’s own displayed Landau spectrum does not justify its exact curvature-norm saturation headline; retain that factor caveat and the lattice-step compactness correction.

• Four-dimensional composition: exact Gaussian blocking and one-loop matching TeX source. Current source note.

  Exact Gaussian composition, generated vertices and finite one-step subtracted estimates. Depth-uniform nonlinear contraction and remainder control remain open.

• Four-dimensional parallel insertions: one-loop shifts and the logarithm TeX source. Current source note.

  Formal one-loop running under explicit generated-action/background matching hypotheses; a coefficient calculation does not provide stable iteration.

• The ground-state measure is the time-slice marginal of the Euclidean measure, and that imports the local large-field estimate TeX source. Current source note.

  Exact finite-lattice ground-state/slice-marginal identification transfers probability estimates actually established for that marginal. The action-cost calculation alone gives no normalized local rarity estimate.

• The large-field region costs more action than it has unstable directions, by the factor 2π²/g² TeX source. Current source note.

  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.

• Leibniz's law of continuity, read on records, selects a positive action floor for motions TeX source. Current source note.

  Continuity iff positive mark cost holds for the declared independent Gaussian, non-adaptive, preparation-invariant optimized family. It does not establish uniform complete-record continuity for arbitrary physical apparatus.

• The map as one conditional theorem: two hypotheses, both finite in kind, and an explicit lower bound m ≥ ℏ c γ'/a* TeX source. Current source note.

  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.

• Openings for the mass gap, and why it is not special to SU(3) TeX source. Current source note.

  Four scoped openings compare curvature, centre deformation, lower dimensions and spectral valley lifting. Continuum scaling needs positivity near g=0, not every bare coupling. The October 2 comparison uses Wilson sufficient bounds 1059 and 444, records the withdrawn KS 388 threshold, and preserves the unchecked curvature conversion. Scope: The curvature conversion to g²>32 and physical transfer remain unchecked; deformed-theory continuity and SU(3) large-N transfer remain hypotheses. KS numerical volume uniformity is withdrawn; finite-volume and existential uniform statements are distinct. The September 23 pause is historical; use current STATE and halving-atlas.

• Newton indeterminacy routes: the physical premise a record floor needs TeX source. Current source note.

  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.

• A fifth postulate for the Principia: joint determinacy of place and motion TeX source. Current source note.

  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.

• Reciprocal exchange fixes relative mechanical scales TeX source. Current source note.

  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.

• Shared radiation and closure under recording TeX source. Current source note.

  Live recording notebook: §8.22 realizes score correction with correlated positive classical preparation but fails ideal full pointer readout and free transport. Section 8.23 matches the Gaussian Fisher target with stationary switching and balanced mean energy; a preparation-independent microscopic force cannot implement all widths/means. Written, internally checked. Apparatus energy closure, phase reunion and independent positivity remain open; earlier results keep their qualifications.

• A Lorentz-invariant background and the state-route constant: a conditional link to the radiation unit TeX source. Current source note.

  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.

• A shared release produces exclusive fringe-weighted events TeX source. Current source note.

  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.

• Global spin phases quantize a ratio, not an absolute action scale TeX source. Current source note.

  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.

• Stabilizing a topological radius exposes an action coefficient TeX source. Current source note.

  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.

• Brownian free motion: the velocitas ultima denied, one action constant, and the floor TeX source. Current source note.

  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.

• The target box: the strong-coupling gap survives local perturbations, so the renormalization group has an explicit finish line TeX source. Current source note.

  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.

• SU(2) on the full mid-plane: the relative-order-t halving defect TeX source. Current source note.

  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.

• The SU(2) mid-plane on its small-field set: bounds and the missing comparison TeX source. Current source note.

  Live working notebook: all-order connected bad sets and dressed activities are proved only in the auxiliary weak strip; §37 is a centered quadratic moving-source test for the subtracted barrier part of (113). Full nonlinear comparison, physical recoupling and iteration remain open.

• Superdeterminism lives below every resolution; a floor gives it a lifetime TeX source. Current source note.

  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.

• The tangent groupoid, its zoom, and whole trajectories TeX source. Current source note.

  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.

• Thermal reliability selects a detector cost, not a universal action TeX source. Current source note.

  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.

• In three dimensions the mass gap is the positivity of one function at infinity TeX source. Current source note.

  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.

• The one-loop potential along the abelian valley of the torus is periodic, reproduces C133's linear term at the origin, and saturates at order 1/L TeX source. Current source note.

  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.

• Halving goes to the ultraviolet; confinement lives in the infrared TeX source. Current source note.

  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. 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.

• The weak-coupling gap in a small box reduces to three inequalities through the Schur complement TeX source. Current source note.

  Proved transfer lemma with corrected relative/fibered projection requirements. H1 nonzero-mode separation, H2 relative Schur control and H3 effective zero-mode gap remain open physical inputs. The displayed cutoff calculation uses SU(2); transfer to SU(3) also requires group-dependent estimates.

Older version

• An Agmon bound for the Kogut–Susskind ground state: large fields are suppressed at rate 1/g² per plaquette TeX source. Current source note. Later treatment.

  Retains an earlier route and its qualifications; use agmon-global-not-local for the current correction. Establishes the exact weighted ground-state Agmon identity and forbidden-region estimate on compact gauge configuration space. 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.

• One blocking step: the obstruction is the variation, not the size, of the inter-block coupling TeX source. Current source note. Later treatment.

  Retains an earlier route and its qualifications; use blocking-criterion-monotone for the current correction. Isolates scalar boundary-energy shifts from state-dependent Schur variation and counts tentative RG doublings to a strong-coupling threshold. 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.

• The confinement scale in three bands, with published numbers, and the verification restated gauge-invariantly TeX source. Current source note. Later treatment.

  Retains an earlier route and its qualifications; use reasons-to-stop-as-research for the current correction. 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. 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.

• The truncation error of a flow step is Gaussian times e2t\|G\|∞, so the renormalization step is small exactly on small-field configurations TeX source. Current source note. Later treatment.

  Retains an earlier route and its qualifications; use lattice-truncation-uniform for the current correction. 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. 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.

• The intermediate region as a finite verification: complete analyticity, the transfer matrix, and the two numbers whose meeting closes the proof TeX source. Current source note. Later treatment.

  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. The October 2 correction separates a required Wilson mixing margin from the existential Hamiltonian target-box tolerance. Scope: No numerical openness radius transfers between Wilson and KS operators. 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-as-research.

• The cost of a mark: a Newton-age floor for the Galileo comparison TeX source. Current source note. Later treatment.

  Retains an earlier route and its qualifications; use mark-cost-and-statistical-floor for the current correction. 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. 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.

• The Planck gap as a theorem: every marking protocol needs F²τ³ > 9mκ TeX source. Current source note. Later treatment.

  Retains an earlier route and its qualifications; use planck-gap-probabilistic for the current correction. Conditional classical bounded-error/bounded-recoil protocols require F²τ³>9mκ; a balanced two-mark construction succeeds above 36mκ. 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.

• On typical configurations the flow truncation is controlled at any coupling; the whole obstruction is the large-field tail TeX source. Current source note. Later treatment.

  Retains an earlier route and its qualifications; use lattice-truncation-uniform for the current correction. 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. Scope: Interacting use is perturbative. Its lattice-step pessimism is corrected by compactness; this note does not prove an interacting normalized rarity bound.

• What would unblock the confinement-scale region, and what was checked and fails TeX source. Current source note. Later treatment.

  Retains an earlier route and its qualifications; use reasons-to-stop-as-research for the current correction. Earlier confinement-route survey records where correlation cones, finite-size gap criteria and exact transformations fail to supply the weak-to-intermediate gauge bridge. 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.

Historical research snapshot

• Why the ancients argued about the arrow and not the sling TeX source. Current source note.

  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.

• From the continuum to discrete substrates: 't Hooft and other turns TeX source. Current source note.

  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.

• Holography in the lowest dimensions: at d=0 it is the wave-function identity, at d=1 it is the Schwarzian TeX source. Current source note.

  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.

• Action, Trajectories and Gaps: A Research Programme TeX source.

  This is a generated historical research source rather than an orphan result manuscript: its TeX identifies research/PROGRAMME.md as its maintained source, and the PDF labels its 2026-09-16 content historical and superseded by AGENTS.md. Preserve it for provenance only; its former priorities and process instructions must not be treated as live state, which is now carried by research/STATE.md and the governing repository instructions.

Orphan manuscript

• Action Differences, Fluctuations and Operational Bounds: Control results for a research programme on mechanical gaps TeX source. Later treatment.

  This early synthesis preserves useful accepted calculations and their premise boundaries, but it predates the later action-scale, record, and Newton insertion work; its central status map is superseded by action-scale-obstructions, while the detailed propositions and quantitative two-arm calculation remain useful as an archival technical source. The quantum discrimination result is conditional and does not establish classical necessity.

• A Positive Action Plateau from Classical Velocity Memory TeX source. Later treatment.

  The later composition-universality note uses the C019–C020 plateau as input and proves a conditional common-coefficient result, but does not reproduce the finite-chain spectral derivation or the short-window and noncommuting-limit calculations. The manuscript therefore retains the detailed proof source for C018–C021; its physical universality question remains explicitly open, and the plateau is not a quantum phase constant.

• Classical spins and the dynamical meaning of an operational state TeX source. Later treatment.

  The three principal results have since been separated into maintained notes: classical-orientation-closure (C125), hamiltonian-moment-descent (C127), and hamiltonian-finite-closure (C128). The manuscript remains a coherent reader-facing account of how local operational closure, physical interaction, and finite repair differ; its spin magnitudes supply action units but no universal positive action scale.

• Physical-Time Action Relaxation in an Elastic Collision Bath TeX source. Later treatment.

  The ordered-beam-preparation note routes directly to the maintained proof of C051, while composition-universality later uses the bath plateau as a preparation-equivalence countertest. Neither note replaces the full bath calculation or the paired C050/C051 construction, so this manuscript still serves as their technical source. The examples show preparation and rate dependence, not a universal action floor.

• Classical Paths with a Finite Action Defect: An exact two-regulator experiment TeX source. Later treatment.

  The two-regulator-audit note accurately routes readers to this manuscript and restates the distinct path, action, and scale limits, but the arbitrary-partition determinant, normalized Gaussian calculation, scalar stationary-phase limit, and parameter map remain detailed here. Its contribution is a regime-dependent finite defect, not a positive lower bound on all admissible paths.

• Same collisions, different transport: an exactly soluble two-speed gas TeX source. Later treatment.

  This manuscript adds no new claim beyond C050–C051 and is not the maintained proof source; it is a reader-oriented standalone exposition of the exact matched-parameter counterexample, with the maintained C051 proof linked through ordered-beam-preparation. Retain it if the teaching/publication use is intended, since its comparison and Palm-clock explanation are easier to teach than the technical collision paper.

• Time Refinement and an Action Scale TeX source. Later treatment.

  The later refinement-composition-and-limit note incorporates the exact Gaussian insertion/composition law and places it inside the broader Newton and gauge refinement construction. This earlier paper remains a compact, detailed source for the free-kernel family, bridge concentration, and unitary generator; its central conclusion is that refinement alone does not select a positive action scale, and positivity is an additional premise.