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

• Gauge refinement: exact steps, scale dependence and the infrared obligations TeX source. Current source note.

  Exact series/parallel operations, arbitrary cuts and group image sectors organized by dimension and scientific dependency. Retains exact circle/rotor identity and one-loop scaling, separates depth/volume uniformities, fine-measure construction and spectral visibility. Corrects unsupported renormalon identification and stable-quartic non-Borel claim. No scale-to-gap or confinement implication is assumed. it adds no theorem; read linked derivations for hypotheses. Sharp Gaussian fixed-point assertions require the later composition correction.

• Gauge spectral limits: finite lattices, coupling sets and the continuum target TeX source. Current source note.

  One canonical lattice-to-continuum spectral treatment contains T1 finite compact-resolvent positivity, strong-coupling and Wilson/KS distinctions, finite-volume analytic coupling dependence, conditional limiting spectral thresholds and lost-weight counterexample. Integrates former position map and exact Wilson sufficient constants. All-coupling positivity/phase equivalences, large-field rarity, relative flow truncation and exhaustive route closures remain unproved; continuum construction and physical spectral transfer stay separate. Finite-lattice existence, strong-coupling positivity, all-coupling stability, continuum scaling and reconstruction are separate. The small-volume field asymptotic and positive SU3 matrix gap are conditional; Lambda_lat is inverse length and m is an energy. No monotone crossover is proved. The small-volume scaling statement names a centre sector; it is not automatically the full finite-box gap.

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

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

Current technical manuscript

• Classical action selection: units, symmetries and orbital thresholds TeX source. Current source note.

  Dimensional and similarity tests distinguish action units from floors. Sharp excitation and singular-Coulomb bounds retain their premises; the integrated C130 radial-action calculation gives the supplied-quantization Sommerfeld energy correspondence. Scope: Each theorem states its observable, preparation and admitted class. A dimensional unit supplies no positivity; fixing a Coulomb coupling requires no charge quantization. Classical thresholds do not determine quantum operator domains or identify a universal floor with physical hbar.

• A bounded Alpoge–Sussman peek: what returns to Newton in 0+1 TeX source. Current source note.

  A pinned selected-source peek motivates an exact quartic connected-response test: m4-3m2^2=-(b/hbar)(m2 m6-m4^2)<0. The equal-cell cumulant is asymptotic to -3 lambda hbar^3 T^5/(256m^4), while A_hbar fourth derivative divided by w tends to 6 lambda. A full Gaussian exponential-observable reflection test admits the zero-action member. Scope: External RP1, QED3 and YM manuscripts remain unaudited; reading is passage-level or theorem statements as recorded. The scalar cumulant sign differs from the fermion determinant probe. A shrinking-cell cumulant is not a macroscopic continuum witness; positive hbar is supplied. No Yang-Mills result, quantum identification or independent Newton action necessity is established.

• Classical apparatus: construction, complete records and stable recovery TeX source. Current source note.

  One autonomous finite-mass Hamiltonian apparatus has vanishing-error/disturbance quiet-preparation limits; exact nonlinear calibration and complete pointer/clock records give bounded-domain recovery. Impulsive repeated-cut readout has separate growing-probe resources. Preparation precision, exact model, known incoming probe momenta and joint final record access are supplied. Delayed known-flow reconstruction differs from causal unknown-force prediction; energy-assisted recovery is local unless final clock records are included.

• Time blocking, Newton polygons and explicit quantum action TeX source. Current source note.

  For arbitrary unequal time grids, the exact Dirichlet Green row sum gives a contraction factor Lambda T squared over 8, polygon convergence and a uniform inverse-Hessian/source-response bound. Gaussian fluctuation widths retain an explicit hbar factor. Scope: Short two-endpoint intervals and stated global curvature bounds or invariant local tubes. Gaussian covariance is a quadratic reference, not a full nonlinear measure estimate. Harmonic boundary caustics do not obstruct initial-value uniqueness; positivity of hbar and physical weighting remain unproved.

• Constructing Newton motion: refinement, continuation and regularity TeX source. Current source note.

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

• Blocking and Schur bounds: the surface-to-gap obstruction TeX source. Current source note.

  Surface-to-gap criteria scale unfavourably in specified regimes. All-coupling monotonicity requires a comparison of block gaps. Connected polymers alone do not prove the proposed Schur-variation inequality; boundary counting and sectors require checking.

• 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.

• Preparation, calibration and exact-record fibres of a classical apparatus TeX source. Current source note.

  A common inverse apparatus chart proves shell saturation, energy/calibration ambiguities, rank-controlled global recovery, matched tolerance/coupling crossover and directional calibration fibres. Signed-coupling symmetry cancels the proposed canonical tangent; finite-width quadratic quadratures retain a conditional curvature test. The calibrated global inverse uses a reduced fixed box and nonzero displacement; shell, full-state, coordinate-risk and projected-area claims have distinct hypotheses. Correlated estimates concern one local fibre; both projected curvature coefficients remain unproved nonzero. No universal action is selected.

• Bounded-force records: reachability, minimax recovery and composition TeX source. Current source note.

  Exact force lenses and clipped-cut areas, a symmetric-fibre minimax theorem, all three finite-horizon phase-risk regimes, independent composition and shared/block budget laws are proved in one information model. Known initial state, arbitrary bounded force, nondisturbing records and supplied deterministic error bands are premises. No whole-class speed ceiling or back-reaction is included. Precision/delay refinement closes the action-risk products; a shared norm budget differs from independent apparatus budgets.

• Canonical averaging and the Gaussian sector of Newton preparations TeX source. Current source note.

  Conditional Gaussian-sector classification from canonical mixing and weak or W2 closure; a two-copy density-ceiling counterexample and entropy-floor model with actual cusp transport and W2 closure. A later Gaussian-meter result disproves selective record closure for this entropy witness; its scale remains supplied.

• 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.

• Operational spin states, reversible interaction and finite closure TeX source. Current source note.

  Canonical proof treatment with explicit hypotheses and incorporated corrections.

• Classical fluctuation laws and conditional action universality TeX source. Current source note.

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

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

  Finite conservative harmonic receivers have zero long-window response; the prepared increasing-chain plateau depends on limit order and closes under its hard speed-support condition. Exact two-body cut/reset and three-body elimination/observability proofs retain the missing classical memory. Receiver energy and modal preparation supply all action scales. Resets alter the experiment, exact two-time recovery becomes ill-conditioned, and the apparatus/calibration/minimax proofs now reside in their separate canonical treatments.

• Constructing a 0+1 field theory: the quartic oscillator and the Yang–Mills matrix model TeX source. Current source note.

  Assembles the known quartic oscillator into a full-time Euclidean process with vacuum, reflection positivity, a non-Gaussian marginal and a gap, and extends the architecture to the prior SU(2) matrix and singlet theorem. Supplied hbar>0 and bounded-cylinder reconstruction; no relativistic covariance, four-dimensional construction or action necessity.

• Quantum preparations select continuations at a bounded Newton cusp TeX source. Current source note.

  Explicit Gaussian families with the same position density and limiting incoming history select opposite outgoing C1-cusp branches; superpositions yield mixtures. The logarithmic preparation and local-time scope are explicit.

• Refinement action: cut energy, quantum phase and Gaussian information TeX source. Current source note.

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

• Finite propagation: scalar rigidity, internal states and record limits TeX source. Current source note.

  Scalar ballistic-support probability convolution laws are deterministic drift, and scalar translation-invariant unitary groups with light-cone kernels are translations up to phase. Internal-state identities, the matrix conjecture, hidden slow mode and exact additive-readout double limit retain separate scope. Written scalar proof imports Paley–Wiener–Schwartz and harmonic estimates; no independent peer-review or novelty claim. Matrix classification is unproved, positivity permits killing, and readout bias is not intrinsic action or physical hbar.

• Quantum wall dynamics: boundary choice, reflected packets and impulse TeX source. Current source note.

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

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

  Derives the explicit one-link Wilson sufficient criterion g²>444 for SU3 and its transfer-gap rate. The proposed single-link block-marginal test is empty in the gauge interior; a joint invariant certificate and weak-trajectory control remain additional obligations. A block marginal criterion based only on interior single-link distributions sees Haar marginals and certifies no required joint mixing. No box size or monotone gapped interval is established; operators and continuum limits remain distinct.

• 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.

• Spectral upper bounds: smeared susceptibilities and moment hierarchies TeX source. Current source note.

  One variational and positive-spectral-measure treatment gives intensive susceptibility bounds, conditional moment ratios and exact spatially flowed Maxwell constants. Temporal-flow admissibility and infrared moments remain assumptions. All finite inverse moments can coexist with a zero edge; complete moment data may determine the observed channel, which need not be the global gap. Four-dimensional flow UV finiteness supplies neither a Hamiltonian insertion nor infrared integrability. A single observed edge can miss lower sectors; finiteness of every inverse moment gives no positive lower edge. The conditional continuum construction and weighted derivative/susceptibility limits remain open.

• Gauge flow: exact conjugation, Jacobian locality and spectral truncation TeX source. Current source note.

  Finite-lattice flow preserves spectra and multiplication norms and has a local first-derivative path bound. Exact continuum variation distinguishes the true Jacobian from an auxiliary Gaussian propagator with an absolute tail budget. Landau growth is gB-k_parallel^2; inverse metrics and relative Hamiltonian truncation remain open. Auxiliary bounded-coefficient heat evolution is not identified with every gauge-flow Jacobian. Absolute kernel errors and free variances do not establish interacting suprema, inverse-metric control, relative spectral truncation or iteration.

• 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.

• 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 Gaussian meter breaks the classical entropy-floor witness TeX source. Current source note.

  A positive-probability Gaussian-meter record breaks the classical entropy-floor witness, while Gaussian inputs retain the floor. A W2-closed Gaussian-mixture repair survives meter and copy operations but fails an explicit nonlinear shear. Quantum state positivity and the scale are supplied; the selection counterexample uses unbounded preparation apertures.

• Tangent-groupoid scaling and the quantum construction of Newton motion TeX source. Current source note.

  Current canonical treatment of interacting tangent scaling, finite-scale insertion, state limits and normalized bridge consistency; the maintained note supplies the hypotheses and limits of each result.

• Ising relaxation, local parent Hamiltonians and clock normalization TeX source. Current source note.

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

• Large gauge fields: action cost, constrained minima and model entropy TeX source. Current source note.

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

• Rare gauge configurations and signed operator comparisons TeX source. Current source note.

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

• Variational lattice-gap bounds: plaquettes, electric trials and holonomies TeX source. Current source note.

  The KS Feynman-Bijl identity gives plaquette strong-coupling order, an abelian structure-factor gaplessness test and an exact averaged-Polyakov susceptibility bound. Constant-colour electric trials vanish for semisimple gauge algebras; a universal string cost for all field-dependent dressed trials is unproved. An exact SU2 adjoint-loop bound targets the neutral centre sector; the one-patch fundamental variance is not a global vacuum variance. Nonzero variances and stated domains are required. The Gauss-law exclusion is semisimple, not every nonabelian group; a nonvanishing upper bound proves no converse gap criterion. Small-volume expectation transport and any universal dressing cost remain unproved.

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

  The support-preserving interaction picture reduces commuting unbounded single-link electric terms to bounded plaquette locality, with volume-uniform coefficient of order Nc/g². Numerical prefactors, clustering lengths, physical propagation and stability need their separate hypotheses. The old 32e coefficient is not certified from the cited theorem conventions. The dynamical quotient hbar v/Delta alone is not a general clustering length; a diverging upper estimate proves no actual superluminal speed or gap stability.

• Lifting a Lipschitz Newton certificate to quantum mechanics TeX source. Current source note.

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

• Classical detector records: thresholds, reliability and coincidence bounds TeX source. Current source note.

  Fixed-split conditionally independent monotone receivers obey the sharp coincidence-product bound. Passive barriers, exact capped-Arrhenius retention/deadline optimization and shared-release countermodels retain their separate premises, energy accounting and causal timing. None selects a universal physical action. Scope: Fixed routing, conditional independence and all-gate sampling are essential to the coincidence bound. Thermal costs require fixed prefactor, temperature and transduction; the exclusive shared-release model changes the independence premise and supplies its hazards.

• Low-dimensional gauge gaps: matrix confinement, conditional floors and scales TeX source. Current source note.

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

• Marking Newton's comparison: classical widths and quantum probe costs TeX source. Current source note.

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

• Gauge crossover mechanisms: spectral scope, comparison tools and source evidence TeX source. Current source note.

  One gauge-mechanism comparison keeps curvature, centre deformation, three-dimensional scaling and valley lifting beside correlation inequalities and finite certification. A weak scaling trajectory needs less than all-coupling positivity; temporal operators, rank and physical-hbar limits remain distinct. Curvature normalization and physical spectral transfer are unchecked; deformed and lower-dimensional theories need their own continuation. Loop monotonicity is not vacuum-gap monotonicity. Large-N factorization is not the physical hbar-to-zero limit at fixed rank, and no exhaustive route closure is established.

• 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.

• Radiation, reservoirs and the action-scale question TeX source. Current source note.

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

• Competing time-cell branches: a conditional positive-action bound TeX source. Current source note.

  A double-well time-cell response ceiling gives a uniform positive lower bound on the supplied action multiplier using an exact variance identity. The small-action source transition has a tanh profile. A specified global-minimum zero extension is discontinuous, while classical branch memory and a thermal weight are explicit controls. Scope: Euclidean positive-weight family at fixed cell parameters with d>0. The response ceiling and physical branch law are independent premises; alternate zero-action extensions are not excluded. No universal quantum identification or Planck value is proved.

• The Newton cusp: classical histories, conditional floors and records TeX source. Current source note.

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

• Newton's comparison: areas, action defects and controlled reunion TeX source. Current source note.

  Canonical Galileo/Newton geometry, source paragraph map, matched-endpoint action and arbitrary-partition defect, receding-centre correction and exact controlled F,−F,F reunion. Conditional copy thresholds are kept separate from action quantization. Spatial areas are frame-dependent, endpoint choices change coefficients, and deterministic work differs from energy uncertainty. Quantum phase, coherent arm control, copy count and measurement rules are supplied; no universal action floor or established fluxional-time interpretation follows.

• A nonlinear Newton time cell: controlled fluctuations and induced stiffness TeX source. Current source note.

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

• From multiscale quantum mechanics to Newton's initial-value problem TeX source. Current source note.

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

• Quantum record bounds: aperture, disturbance and retained resources TeX source. Current source note.

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

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

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

• From quantum time slices to relativistic and Newtonian motion TeX source. Current source note.

  Positive-energy relativistic and Newtonian packet limits with O(h+mesh+c^-2) centered mean and smooth-observable bounds, fixed-action strong resolvent convergence, subordination and a spectral Dirichlet wall. The sharper rate assumes c bounded below; no interacting relativistic field theory or independent action necessity is asserted.

• 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.

• Composition and action normalization: reciprocal mechanics, rotations and spin phases TeX source. Current source note.

  Reciprocal spring responses fix compatible inertia ratios through a connected-graph cycle test. Newton derivation compatibility and mechanical rotation closure equate constituent algebra constants, admitting zero. A supplied spin phase imposes 2s/K integrality while common normalization and arbitrarily small local cap actions remain free. Scope: Reciprocal dynamics, canonical product structure, measured generator identification and U(1) phase transport are distinct premises. Equality or integral ratios do not identify a nonzero universal physical hbar; a reference calibration or positivity law is additional.

• Coulomb completion and its classical limit: domains, collisions and preparation 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.

• Gaussian gauge blocking: exact covariance, decay and the nonlinear threshold TeX source. Current source note.

  Exact linearized four-dimensional tube blocking gives the 4/9 constrained gap and Gaussian covariance. A distinct soft constraint has gap 2/9 and a coefficient-specific decay criterion. Correct two-link normalization and infinity-norm shell sum replace the old threshold table; nonlinear KP control remains conditional. Scope: Hard conditioning and the soft operator differ; no finite-remainder hard numerical rate, normalized interacting expansion, optimality barrier or physical continuum gap is proved.

• Kogut–Susskind strong coupling: stability, finite-volume bounds and history estimates TeX source. Current source note.

  Yarotsky stability gives an existential volume-uniform KS gap and a specified translation-invariant perturbed target region. Exact finite-volume electric-loop bounds, corrected all-root overlap obligations and restricted continuous-time counts share one proof structure; numerical uniform thresholds remain withdrawn. Scope: Existential constants are not numerical thresholds. Restricted anchored histories do not control late roots, general shapes or normalized true-vacuum insertions. Physical continuum transfer and thermodynamic consistency retain separate hypotheses.

• 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 internally checked spin-1 matrix entries retain axial anisotropy and explicit image bounds. Scope: Isolated-cube factorization uses independent bridges. Fixed-spin formulas do not control the full heat-kernel character sum or shared-edge mid-plane; image separation degenerates at the cut locus. Supplies curvature inputs to the full-plane calculation.

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

• Three-dimensional Yang–Mills: conditional scaling and the physical gap limit TeX source. Current source note.

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

• Topology, size stabilization and action normalization TeX source. Current source note.

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

• 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. 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.

• Coherence, resolution and the classical zero-branch limit TeX source. Current source note.

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

• 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

• Classical cuts: source evidence and information at an interface TeX source. Current source note.

  Unifies Aristotle and Plutarch cut evidence with Indian contact, Avicennian, Chinese dissection/statics and cohesion witnesses, preserving editions and reading limits. Separates sampling, re-solving, marginalization and measurement, and retains fixed-time compatibility premises without an action-selection claim. No historical transmission, ancient path-integral theorem or Newton quantum anticipation is established. Unproofread and incomplete witnesses retain their access limits; contextuality is a source lead, not an applied theorem, and compatibility patterns select no dimensional scale.

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

  Independent article exposition of classical spin operational descent and finite-observable obstruction. The canonical orientation treatment integrates the subsequent generator/source qualifications; this earlier article is retained as a working manuscript.

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

  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.

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

  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.

• Gauge vacuum estimates: global tails and conditional slice transfer TeX source. Current source note.

  Exact magnetic Laplacian/gradient identities and variance sum rule feed finite-volume global Agmon tails with explicit all-path constants. Perron slab and anisotropic temporal limits conditionally identify ground-state slice probabilities; normalized local rarity and operator comparison remain separate. Global thresholds and prefactors depend on volume; local large fields need not force a global excess. Temporal limits are iterated at fixed spatial cutoff; no uniform local probability, signed operator inequality, continuum construction or physical gap follows.

• Quantum-to-Newton results: literature and Lean verification targets TeX source. Current source note.

  Concrete verification targets and bounded prior-art assessment for the C1,1 lift, smooth states, wall impulse, cusp selection, relativistic extension and Euclidean tower. Includes mixed-order fixed-action convergence, exact constant-force joint-limit ray separation, and a separate variable-mass ordering example. None is a compiled Lean proof and publication priority is unestablished.

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

  In the non-adaptive Gaussian mark model with preparation-invariant tests, optimal force discrimination is continuous at zero force iff κ>0; finite schedules realize spent cut action/κ. Historical passages support an explicitly modern record-level reading of continuity. Scope: The optimized Gaussian-family iff requires saturation/dense-protocol availability and a balanced zero-noise family at zero floor. The historical record-level reading is an additional interpretation; all quantum aperture bounds require hybrid-state envelopes.

• Conditional gauge-gap transfer: multiscale control, mixing and certification TeX source. Current source note.

  H1 multiscale fluctuation/quasi-local Gibbs control and H2 neighborhood mixing imply a volume-uniform Wilson-transfer gap, with continuum transfer conditional on existence and dense observables. Finite certificates, gauge marginal reduction, interaction robustness and numerical bands retain their distinct burdens. No physical trajectory, applicable finite-size certificate or continuum theory is constructed. Compactness, a scalar coupling, gauge-invariant expectation or numerical meeting scale does not prove the full interaction/mixing hypotheses; KS and Wilson thresholds differ.

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

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

  Canonical source with assumptions and later corrections integrated; internal model checks confer no independent peer-review status.

• 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.

• Score-constrained ensembles: action, records and the controller TeX source. Current source note.

  Canonical source with assumptions and later corrections integrated; internal model checks confer no independent peer-review status.

• Classical recording: complete memories, refinement and nonlinear limits TeX source. Current source note.

  Canonical source with assumptions and later corrections integrated; internal model checks confer no independent peer-review status.

• 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.

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

• 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.

• One-loop torus valleys: local confinement, periodicity and spectral limits TeX source. Current source note.

  The one-loop SU2 valley has bounded periodic full potential and an unbounded nonperiodic subtracted correction. For the declared flat 4pi holonomy operator, centre-odd and cubic-invariant trials give an explicit g^(8/3) exp(-c/g) upper gap, so local g^(2/3) spacing is not its global gap. Flat measure, scalar kinetic operator, all centre characters and the specified one-loop potential are benchmark inputs. No identification with PHP, full field-theory splitting, matching tunnelling asymptotic, SU3 theorem or continuum gap is proved. Gauging centre translations or restricting their character changes the admissible trial; neutral local cyclic space differs from the full finite-box Gauss space.

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

  Proves abstract Schur gap-transfer lemmas, including a constant-shift/relative-error version. Physical H1–H3, UV-uniform domains and bounds remain open, and H3 must identify the centre sector and observable algebra before a local matrix spacing can represent its compressed gap. H1–H3, projection domains and uniform constants remain open. Bare sharp-cutoff control fails; linear valley confinement does not automatically control quadratic multiplication globally. Formal SU2 cubic slope and Berry terms are not certified SU3 or continuum bounds. The scalar centre-sector countertest is not a construction or identification of PHP.

Historical research snapshot

• Why the ancients argued about the arrow and not the sling TeX source. Later treatment.

  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.

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

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.

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

  The composition-universality note summarizes the paired C050/C051 preparation countertest. It does not replace the full bath calculation or the paired construction, so this manuscript remains 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 Newton synthesis 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 composition-universality. 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.