navstokgap

Claim ledger

Updated 2026-09-16. Current verification uses written derivations and source/proof review. Script checks in earlier entries are historical under AGENTS.md’s hard rule. Each entry names its assumptions and supporting artifact. IDs remain stable through revision.

The P04 synthesis consolidates the accepted action-selection and gap arguments through C123. It adds no claim IDs or literature assertions; detailed proofs and audit statuses below govern.

Q01 interaction dependency (B70)

The B70 map applies de la Torre et al.’s Theorems 1–2 to C125. A continuous reversible nonlocal extension cannot preserve its minimal composite; the full quantum operations conclusion includes identical-copy and ancilla/measurement/discard closure. Evidence: selected primary theorem/setup passages and written application review. Literature status: established source theorem and derived application, no novelty asserted. Proof status: hypothesis map and consequence reviewed; C129/B77 subsequently accepts the initial generator restriction; later Lie-algebra steps and imported universality remain unaudited. C125 remains accepted for its original local-operation class; no new theorem ID or positive action claim is added by this source milestone.

Mechanical and operational results

ID Statement Evidence
C001 Constant force, matched endpoints: chord action excess \(F^2T^3/(24m)=T\delta E/12\) Constant-force note and symbolic script
C002 Positive action differences in the fixed-endpoint constant-force class have infimum zero Foundations Proposition 1; exact family \(\eta=a t(T-t)\)
C003 Absolute action differences accumulate at zero along a continuous variation with nonzero second variation Foundations Proposition 2; restricted action is \(C^2\) in its parameter
C004 The constant-force Dirichlet Hessian has lowest eigenvalue \(m\pi^2/T^2\); scaling the variation scales its quadratic cost continuously Foundations §3; \(L^2\) normalization
C005 The free Schrödinger kernel tends to \(\delta\), with first short-time correction \(i\hbar\tau\delta''/(2m)\) Foundations §4; Schwartz-distribution Fourier proof
C006 Fixed-\(N\), equal-prior quantum two-arm discrimination has an explicit positive first-lobe action threshold for \(1/2<p\le1\) Foundations Proposition 3; specified pure states, phase rule and joint measurement
C007 For fixed \(1/2<p<1\), C006’s threshold tends to zero as \(N\to\infty\); at \(p=1\) it is \(\pi\hbar\) for finite \(N\) Explicit formula and endpoint checks
C008 The free relativistic particle admits arbitrarily small positive fixed-endpoint action differences within a strict speed margin Foundations §6; explicit small-amplitude family
C009 Free normalized Gaussian kernels on the line compose at every finite partition with unchanged action parameter \(\kappa>0\) Time-refinement Proposition 1; square completion and normalization checks; B06 assumption audit
C010 A weakly continuous centered Gaussian convolution semigroup has variance \(at\), \(a\ge0\); at fixed mass \(\kappa=ma\) remains free Time-refinement Proposition 2; characteristic-function proof, including the degenerate family
C011 Fixed-endpoint Gaussian bridges have linear mean and variance \(\kappa s(T-s)/(mT)\); fixed finite-dimensional laws concentrate on the free classical path as \(\kappa\to0\) Time-refinement §3; square completion, Chebyshev and union bound
C012 For every \(\kappa>0\), the free multiplier \(\exp[-i\kappa t k^2/(2m)]\) gives a strongly continuous unitary group on \(L^2(\mathbb R)\) with Hamiltonian domain \(H^2\) Time-refinement §4; Fourier proof and composition check; B06 assumption audit
C013 At any positive finite time partition, the free endpoint Hessian has determinant \(m^{N-1}T/\prod\tau_j\) and inverse \([\min(t_i,t_j)-t_it_j/T]/m\); these give the normalized bridge law Regulator-limits Proposition 1; B08 proof review; exact nonuniform-partition checks
C014 Under that bridge, \(2\Delta S/\kappa\sim\chi^2_{N-1}\); if \(\kappa_N\to0\) and \((N-1)\kappa_N\to\ell<\infty\), a common coupling gives uniform almost-sure path convergence and \(L^2\) action excess \(\ell/2\) Regulator-limits Proposition 2; B08 review of covariance coupling and moments
C015 Smooth fixed-endpoint sine perturbations with amplitude proportional to \(1/n\) retain excess Newtonian action \(mv_*^2T/4\) within a strict speed bound; acceleration grows with \(n\) Regulator-limits §4; off-shell paths, direct integration and B08 review
C016 At one SPD quadratic critical point, the normalized oscillatory amplitude has a scalar squared-modulus limit \(|O(q_*)|^2/\det A=\langle\delta^{(d)}(\nabla F),|O|^2\rangle\) for each Schwartz test function Regulator-limits §5; fixed dimension, Fourier proof; B08 constants review
C017 For a physical free Lagrangian, \(\epsilon=b\tau\) gives energy units to \(b\); choosing the bare coefficient \(m_B=\epsilon m_R/\kappa_R\) exactly preserves the normalized reference kernels Regulator-limits §6; explicit reference condition, fixed square-root branch; B08 algebra/units review
C018 For almost-sure absolutely continuous paths with speed at most \(u<c\), \(\mathsf h_\Delta=m\operatorname{Var}(X(t+\Delta)-X(t))/\Delta\) lies in \([0,mu^2\Delta]\) and tends to zero as \(\Delta\downarrow0\) Classical-action-field Proposition 1; elementary variance proof; B09 and coordinator review
C019 A stationary finite irreducible reversible velocity chain with nonzero centered velocity gives \(\mathsf h(\Delta)\uparrow H_*=2m\langle v,(-Q)^{-1}v\rangle_\pi>0\), with spectral bounds and convergence estimate Classical-action-field Proposition 2; fixed-model action plateau, mean-zero inverse; spectral proof and B09 review
C020 Symmetric velocities \(\pm u\), \(0<u<c\), reversed at rate \(\lambda>0\), give \(H_*=mu^2/\lambda\); universality across masses requires \(\lambda_m=mu_m^2/H_*\) Classical-action-field §§4, 6; exact correlation, variance and telegraph equations; 16-check suite and B09 review
C021 In the formal scaling \(u_\lambda^2=a\lambda\), \(a>0\), iterated second-moment action limits are \(ma\) (rate first) and zero (duration first) Classical-action-field §5; explicit formula; scaling leaves a fixed speed ceiling; B09 review
C022 A centered bounded refreshed bath, \(m\ge M>0\) and independent rate \(\nu>0\) give unique stationary tracer variance \(S_*=Ms^2/m\), covariance \(S_*e^{-\gamma r}\) and action plateau \((m+M)s^2/\nu\), \(\gamma=2\nu M/(m+M)\) Collision-action-relaxation Proposition 1; invariant series and contraction; B10 and coordinator review
C023 With centered bounded initial velocity, \(\mathsf a(t)=2m\int_0^\infty\operatorname{Cov}(V(t+r),V(t))dr\) obeys \(\dot{\mathsf a}=\beta(H_*-\mathsf a)\), \(\beta=4\nu mM/(m+M)^2\); displacement retains a separate window formula Collision-action-relaxation Proposition 2; physical preparation time and action units; B10 and 15 checks
C024 At fixed masses/clock and rest preparation, scaling incoming velocities by \(0<\epsilon\le1\) preserves the collision premises and scales \(H_*\) and \(\mathsf a(t)\) by \(\epsilon^2\) Collision-action-relaxation §4; class countermodel to a uniform positive plateau; B10 review
C027 For constant-force sampled chords on arbitrary positive partitions, \(D_\pi=F^2\sum\tau_j^3/(24m)\le F^2T|\pi|^2/(24m)\); equal steps minimize at fixed count, and every split reduces the error Cut-point note §1; matched endpoints; B12 and exact checks
C028 In the scalar Gaussian endpoint-bridge family at fixed mass/times, exact restriction consistency forces fixed \(\kappa\ge0\); for positive \(\kappa\), \(D_\pi\to\infty\) in probability but \(2D_\pi/(N-1)\to\kappa\) in mean square Cut-point note §§2–3; covariance proof, C014 finite chi-square law and Chebyshev; B12
C029 Inserting a node after lengths \(a,b>0\) in the fixed-\(\kappa\) free bridge adds conditional mean kinetic action \(\kappa/2\), using variance \(\kappa ab/[m(a+b)]\) Cut-point note §4; square completion; B12 and nine-check suite
C030 Equal-mass particles prepared with opposite velocities \(\pm u\) and separation \(u\Delta\) exchange velocities at the midpoint; the tracer returns with \(\kappa_{\rm mid}=mu^2\Delta\) and action \(\kappa_{\rm mid}/2\) Physical-cut note §1; total energy \(mu^2\), momentum zero, preparation changes with duration; B13 and twelve checks
C031 For absolutely continuous paths with fixed endpoints and speed at most \(u\), \(\kappa_{\rm mid}=4m\operatorname{Var}Y/\Delta\le m\Delta(u-|v|)^2\), \(v=(z-x)/\Delta\); the bound is sharp Physical-cut note §2; endpoint-conditioned laws, two-segment action with explicit bias term; B13
C032 A probability convolution semigroup on position with support in \([-ut,ut]\) for all \(t\ge0\) is \(\delta_{bt}\), \(|b|\le u\) Physical-cut note §3; variance additivity and ballistic bound; spatially homogeneous stationary independent increments only; B13
C033 The explicit density-disintegrated telegraph return bridge from \((0,+u)\) to \((0,-u)\) has odd-count weights \(w_k=(\lambda T/2)^{2k}/[(k!)^2I_0(\lambda T)]\) and midpoint atom \(1/I_0(\lambda T)\) at \(uT/2\) Return-bridge note §§1–3; fixed \(u,\lambda,T>0\), independent occupation simplexes, right-continuous velocity version; B14 and coordinator proof/source review
C034 This fixed bridge has consistent cut restrictions and \(0\le S[X]-S_\pi\le(mu^2/2)N_T|\pi|\), giving almost-sure and \(L^1\) polygon-action convergence to \(mu^2T/2\) Return-bridge note §§4–5; positional samples, kinetic functional, deterministic shrinking meshes; B14, 12 symbolic checks and 45 rational partition cases
C035 Independent composition gives the mass-weighted COM coefficient and complementary reduced-mass coefficient; if a finite nonnegative mass-only coefficient on all \(m>0\) equals its COM-composed value, it is constant Composition note §§1–2; common observable and preparation class, product-state memory retained; monotone additive-function proof, B16 and five checks
C036 Under C035 for Green–Kubo plateaus, one stationary two-state reference with \(m_0,u_0>0\) and \(0<\lambda_0<\infty\) forces the shared coefficient \(K=m_0u_0^2/\lambda_0>0\) Composition note §§3–4; C020 input; finite-product consistency family and preparation/correlation countertests; B16 and coordinator proof/source review
C037 In the C033 bridge, \((Q+1)/2\mid K=k\) is Beta\((k+1,k)\) for \(k\ge1\), with \(Q=1\) at \(k=0\); \(\kappa_{\rm mid}/H_*=zR(1-R)\), \(R=\int_0^zI_0(s)ds/[zI_0(z)]\), has cubic onset and limit one Combined return-bridge note §§6–8; same density-disintegrated return preparation, \(Q=2X_{T/2}/(uT)\); factorial convolution, beta moments, Bessel integral proof; B17 and 29 exact checks
C038 A prescribed positive increment or return-midpoint coefficient requires \(T\ge K_*/(mu^2)\); a common finite window and uniform speed ceiling force a mass-independent coefficient to zero if admissible masses approach zero Combined return-bridge note §9; C018/C031 necessary bounds, explicit small-mass limit; B17 and coordinator review
C039 The normalized coin \(C_\varepsilon=(I-i\omega\varepsilon\sigma_x)/\sqrt{1+\omega^2\varepsilon^2}\) with opposite translations gives \(U_{T/N}^N\to e^{-iTH_D/K}\) strongly on \(L^2\); the rest-subtracted positive branch tends to free Schrödinger evolution as \(c\to\infty\) Checkerboard note §§1–4; supplied \(m,c,K>0\), \(\omega=mc^2/K\), complex amplitudes; self-adjoint domain \(H^1\), Fourier/dominated-convergence proof; B18, 28 exact checks and three mode tests
C040 Ideal direction measurement after every coherent step gives flip probability \(q_\varepsilon=\omega^2\varepsilon^2/(1+\omega^2\varepsilon^2)\) and probability of any flip at most \(\omega^2T\varepsilon\to0\), a ballistic limit Checkerboard note §5; initial definite direction, Born rule and repeated projective measurement, fixed \(\omega,T\); B18 addendum and coordinator proof review
C041 The C019 product bounds give an upper bound on the full relaxation gap; an independent slow label keeps \(H=mu^2/\lambda\) fixed while \(\gamma_1=2\min(\lambda,\epsilon)\to0\) Spectral-control note §§1–3,5; finite reversible \(L^2(\pi)\), fixed positive \(m,u,\lambda\), supplied energy unit; B20, 20 finite checks
C042 If centered velocity-unit observables have frame lower bound \(\alpha>0\) and total susceptibility \(S\), then \(\gamma_1\ge\alpha/S\); uniform coverage and response bounds give a uniform family gap Spectral-control note §4; full centered-space frame, eigenbasis proof, varying-mass normalization; B20 and coordinator review
C043 The bounded-acceleration return is feasible iff \(T\ge2u/a\) and has unique minimum kinetic cost \(mu^3/(3a)\); longer durations allow positive excess costs tending to zero Bounded-turn note §§1–2; \(W^{2,\infty}\), prescribed \(\pm u\), return endpoints, \(m,u,a>0\); external-control kinetic functional; B21 and 27 checks
C044 Lipschitz velocity gives sharp chord kinetic error \(0\le S_K-S_\pi\le ma^2\sum d_i^3/24\le ma^2T|\pi|^2/24\) Bounded-turn note §3; arbitrary partition, pair-variance proof, affine equality case; uniform fixed-\(m,a,T\) convergence; B21
C045 Four distinct bounded velocities can have fixed positive \(H=mu^2/\lambda\) and closing gap \(2\lambda\delta^2\); recovering the slow sign requires gain at least \(\sqrt2/\delta\) Spectral-control note §6; independent signs, \(\epsilon=\lambda\delta^2\), \(0<\delta\le1/4\), fixed \(m,u,\lambda>0\); B22 and exact checks
C046 Independent reversible factors with local frame bounds satisfy \(\gamma_{\rm prod}=\min_i\gamma_i\ge\min_i\alpha_i/S_i\) despite an unobserved mixed sector Spectral-control note §7; tensor eigenbasis, unchanged constituent clocks, \(S_i=\sum_aH_i/(2m_i)\); B22 and product-matrix checks
C047 Finite connected harmonic receiver at fixed centre velocity and independent fixed-energy uniform phases has \(h_i(\Delta)=2m_i\sum_j w_{ij}(1-\cos\omega_j\Delta)/(\Delta\omega_j^2)\to0\); independent random centre velocity adds \(m_i\sigma_0^2\Delta\) Conservative receiver note; exact modal/two-body proof, bounded-energy speed margin, necessary joint-limit bound; B23 review and 15 checks
C048 Equal-mode-energy odd harmonic rings have iterated action limits \(2\Theta/\Omega\) and zero in opposite size/window orders; \(z\to\infty\), \(z^2/N\to0\) gives the positive joint limit Receiver note §5; \(\Omega=2\sqrt{k/m}\), fixed \(m,k,\Theta>0\), zero centre mode, density and Lipschitz proofs; B24
C049 A common hard speed ceiling in C048’s phase preparation forces \(\Theta_N=O(N^{-1})\) and \(\sup_\Delta h_N\to0\); bounded total energy also suffices for this closure Receiver note §6; exact site-zero support maximum, cosecant sum and uniform bound; B24 and checks
C050 Equal-mass Poisson hard-point gas at density \(\rho\) with iid velocities \(\pm u\) and Palm tag has independent reversal waits of rate \(\rho u\) and plateau \(mu/\rho\); fixed-density cooling scales it linearly Collision paper §5; ordered-gap proof, stationary velocity preparation, infinite gas first; B25 review
C051 Independent uniform phases of two opposite-speed lattices, each spacing \(2/\rho\), give a stationary square-wave tag and \(0\le\mathsf h(\Delta)\le m/(3\rho^2\Delta)\to0\) at C050’s mass, speed, density and mean collision rate Collision paper §6; Palm/intersection proof and folded cubic variance; B26 review; joint mark/position preparation changes
C052 | Fixed external relativistic Kepler potential has regular bound domain \(\lvert L\rvert>k/c\), \(E_{\min}(L)\le E<mc^2\) and excluded angular-action infimum \(k/c\) | M07 note; written effective-potential, continuation and endpoint-time proofs; B27/B28 and coordinator visual/source review |
C053 | For every fixed softened core \(a>0\) and \(m,k,c>0\), every \(\lvert L\rvert>0\) admits a regular bound circle, giving angular-action infimum zero; at each fixed \(0<\lvert L\rvert<k/c\) minimizing radii tend to zero as \(a\downarrow0\) | M07 note and gap laboratory Kepler section; compact-sublevel and scaling proofs; B28 and coordinator review; no computational verification |
C054 | Uniform circular phase gives transport response \(\ell(1-\cos z)/(\gamma z)\) and averaged position-conditioned coefficient \(\ell\sin^2z/(2\gamma z)\); fixed positive action fraction bounds window ratios | A15 note, \(z=\omega\Delta\); phase-average proofs and B29 |
C055 | Twice the projected canonical covariance area equals orbital action for uniform circular phase and is affine-symplectic invariant; other phase preparations conserve their own value, including zero; C052/C053 thresholds transfer within uniform preparation | A15 covariance/orbit-map proofs, gap laboratory, B29 and coordinator source review |
C056 | External-potential dilation preserves mass, velocity, energy and invariant pushforward preparation; normalized orbit action, covariance area and corresponding-window response scale by \(a\) | A16 proof; smooth transformed domain, finite moments; B30 and coordinator written review |
C057 | A class containing arbitrarily small contractions of one positive finite degree-one action observable has positive-value infimum zero; a common finite value must be zero under any nontrivial admitted contraction | A16 closure proposition; model/preparation class explicit; coupling and force countertests; B30 |
C058 | One fixed smooth confining potential with global force ceiling admits radially stable uniform-phase circles with \(\ell_R\sim\sqrt{mK}R^2\to0\), vanishing speed/acceleration and finite limiting period | A17 proof; exact force balance, effective energy and asymptotics; B31 |
C059 | Relativistic circular motion with \(f\le F_{\max}\) and independent speed floor \(v\ge v_*>0\) satisfies \(\ell\ge m^2v_*^3/[F_{\max}(1-v_*^2/c^2)]\) | A17 identity \(\ell=P^2v/f\) and monotonicity; uniform-phase covariance transfer C055; B31 and coordinator review |
C060 | Regular closed \(C^2\) trajectories with speed floor, stated momentum law and force ceiling satisfy \(T\ge P_*\mathcal K/F_{\max}\) and normalized orbit action \(J\ge P_*^2v_*\mathcal K/(2\pi F_{\max})\ge P_*^2v_*/F_{\max}\) | A18 proof; Fenchel–Borsuk input, momentum derivative and monotonicity; B32 and coordinator review |
C061 | C060’s lower constant is attained by a circle in a smooth globally force-bounded confining potential | A18 cutoff construction; equality over the admissible model class, not every fixed potential; B32 review |
C062 | Fixed-energy uniform-phase two-body spring receiver: stationary position-conditioned kernels have mean composition defect \(-q\sin\omega s\sin\omega t\); phase state restores exact composition, and two positions recover momentum off half-period sampling | R03 §§1–2; written derivation, B33 and coordinator review |
C063 | At fixed duration and initial position, independent equal-step conditional direction resets converge terminally in \(L^2\) to that position; stationary one-time law is preserved but two-time correlation freezes. Canonical \(J=E/\omega\) tends to zero across cooled preparations at fixed Hamiltonian | R03 §§3–4; explicit moment recurrences and ellipse integral; B33 review |
C064 | Centre-reduced equal-mass three-body chain with tagged phase retained has exact cosine memory \(\Gamma(t)=(9k/20)\cos(\sqrt{5k/(2m)}t)\) and inherited receiver quadratures; identical tagged phase can have different futures at fixed energy | R04 §§1–3; canonical reduction, elimination and fixed-energy pair; B34 review |
C065 | Two sufficiently close exact tagged-phase samples recover the receiver pair through \(\det B_\delta=g^2\delta^4/(12\mu\nu)+O(\delta^6)\); inverse sensitivity grows while cooling closes modal actions without changing the memory kernel | R04 §§4–5; Taylor coefficients, analytic determinant and normal-mode scaling; B34 review |
C066 | Two ordered canonical probe shears give exact pre-cut phase errors and back-reaction; intrinsic rectangular-support products are \(s_qt_q\) and \(s_rt_r\), with zero infimum across shrinking positive-volume probe preparations | R05 §§1–2; fixed gains, ideal impulses and records; B35 and coordinator written review |
C067 | At fixed receiver, system preparation and horizon, fresh probe widths \(O(\eta^4)\) give uniform hidden-state reconstruction error \(O(\eta)\) and trajectory disturbance \(O(\eta^3)\), while probe count and fixed-mass apparatus mass grow as \(O(\eta^{-1})\) | R05 §§3–4; exact kick-inclusive error identity and bounded-flow sum; B35 review; scalable impulsive instrument |
C068 | Fixed finite-mass clock and four probes with smooth coordinate coupling \(\lambda K\sum f_j(s)X(x)R(q_j)\) give persistent momentum records \(\boldsymbol\pi(T)=\boldsymbol\pi(0)-\lambda\mathcal A z_0+O(\lambda b+\lambda^3)\) with invertible fixed-width pulse matrix; receiver disturbance is \(O(\lambda b+\lambda^2)\) at fixed horizon | R06 §§1–3; positive kinetic energy, compact receiver preparation, bounded forces, clock reaction included; B37 and coordinator written review |
C069 | For C068 with known dynamics and exact final momentum records, incoming apparatus width \(b=\lambda^3\) gives delayed initial-state reconstruction and trajectory disturbance \(O(\lambda^2)\); canonical initial-position error times finite-horizon momentum disturbance tends to zero as \(O(\lambda^4)\) | R06 §4; fixed masses, pulse widths and clock mean energy; increasingly precise preparation and weak coupling; reporting refinement is sampling after fixed latency; B37 review |
C070 | Exact delayed past records with unresolved force \(f\in L^\infty\), \(|f|\le F\), give coordinate minimax prediction errors \(F\ell^2/(2m)\) and \(F\ell\), simultaneously attained by inertial prediction; risk product is \(F^2\ell^3/(2m)\) | R07 §§1–2; identical-record force pair and endpoint-integral upper bounds; B38 review |
C071 | R07’s exact joint reachable region satisfies \(|z-d/2|\le(1-d^2)/4\), \(|d|\le1\), and has canonical area \(2F^2\ell^3/(3m)\); force-budget or delay closure sends the area to zero | R07 §§3–4; fixed-impulse extremizers, convex interpolation and area integral; B38 review |
C072 | For arbitrary measurable bounded force, \(K_T=S_bK_a+K_b\) at every interior cut; all unobserved refinements preserve the endpoint set, while replacing the intermediate lens by its marginal rectangle strictly enlarges it | R08 §§1–2; split/concatenate proof and saturated-impulse counterexample; B39 review |
C073 | Exact cut position leaves momentum interval of width \(D(\zeta)=Fa[\sqrt{2-4\zeta}+\sqrt{2+4\zeta}-2]\) and terminal canonical area \(2F^2b^3/(3m)+Fb^2D(\zeta)/m\); area tends uniformly to zero as \(b\to0\) although the central momentum width stays positive | R08 §§3–4; ideal available records, lens inversion and determinant-one shear; B39 written review |
C074 | A bounded-error position strip clipped to the force lens has terminal area \(A_C+2F^2b^3/(3m)+2Fbw+Fb^2D_*/m\), with strip width \(w\) and extreme compatible momentum span \(D_*\) | R09 §§1–2; segment extrusion, monotone fibres and B40 review |
C075 | At fixed \(F,m,T\), R09’s area is at most \(4FT\varepsilon+2F^2ab^2/m+2F^2b^3/(3m)\) and closes uniformly as precision error and delay vanish at any relative rates; central momentum uncertainty persists | R09 §3; written bound and exact/full/no-delay endpoint checks; B40 review |
C076 | Two bounded-error positions give exact compatible-set propagation and estimator-specific worst-case terminal errors \(R_p=B+Fb\), \(R_q=\varepsilon_2+bB/m+Fb^2/(2m)\), where \(B=m(\varepsilon_1+\varepsilon_2)/\delta+F\delta/2\) | R10 §§1–2; weighted force integral and simultaneous saturation; B41 review |
C077 | If \(\delta,b\to0\) and \((\varepsilon_1+\varepsilon_2)/\delta\to0\), both coordinate errors close uniformly; equal errors and \(b=\delta_*=2\sqrt{m\varepsilon/F}\) give action-error product \(28\sqrt{mF}\varepsilon^{3/2}\) at fixed horizon | R10 §3; timing feasibility and written bounds; established equal-error differentiation optimum, derived mechanical product |
C078 | For compact centrally symmetric convex inputs, linear records with bounded errors and a scalar linear target, global minimax error equals the maximal target magnitude in the central compatible fibre | R11 §1; half-difference and midpoint proof; B42 review |
C079 | At \(d=2\sqrt{m\varepsilon/F}\), \(T-b\ge2d\), two position records and the entire bounded-error position history have identical coordinate minimax risks \(P_*=Fd+Fb\) and \(Q_*=\varepsilon+Fdb/m+Fb^2/(2m)\); at \(b=0\) their product is \(2\sqrt{mF}\varepsilon^{3/2}\) | R11 §§2–3; established prepared opposite-motion pair, R10 upper bound, blind-delay extension; B42 review |
C080 | Product symmetric convex input/error classes with all constituent records give exact scalar minimax radius \(\sum_i \lvert a_i\rvert r_i\); canonical centre/relative radii follow, and n identical copies have \(H_R=nH_1\). Mass-only nonnegative coordinate-radius closure on all positive masses forces \(Q(m)=q_*\), \(P(m)=p_*m\) | R12 §§1,4; product-fibre extremizers and nonnegative additivity; B43 written review |
C081 | For product bounded forces/errors, complete records, known initial states and \(T\ge\max_i4\sqrt{m_i\varepsilon_i/F_i}\), centre risks are \(Q_A=E\), \(P_A=2\sum_i\sqrt{m_iF_i\varepsilon_i}\); retaining only the centre record gives \(Q_B=E\), \(P_B=2\sqrt{MF_\Sigma E}\), strictly larger momentum risk unless \(F_i/(m_i\varepsilon_i)\) is common | R12 §§2–3; R11 sharp radii, exact force/error image, Cauchy–Schwarz; B43 written review |
C082 | Known initial phase, complete position records of error epsilon and arbitrary force bound F give exact finite-horizon radii \(Q=\varepsilon\min(s^2/2,1)\) and \(P=\sqrt{mF\varepsilon}V(s)\), with \(s=T\sqrt{F/(m\varepsilon)}\) and \(V=s,\sqrt{2s^2+4}-s,2\) on intervals split at sqrt(2),4; blind-delay radii are \(P+Fb\), \(Q+bP/m+Fb^2/(2m)\) | R13 §§1–2; rearrangement and path-strip extremizers; B44 written review |
C083 | R12 product and aggregate radius formulas extend to every horizon using C082; aggregate position risk strictly increases exactly when some \(F_iT^2/2<m_i\varepsilon_i\) and another is strictly greater. The F,16F example loses both phase coordinates; identical-copy risk products remain extensive at every horizon | R13 §3; product fibres, exact image and minimum-of-sums identity; B44 written review |
C084 | Two identical constituents with pointwise quadratic shared record-error budget have exactly the one-body centre risks at mass 2m, force bound 2F and precision epsilon/sqrt(2), for full or averaged records, at every horizon; blind-delay extension and saturated factor 2^(1/4) follow | R14 §§1–2; exact projection/synchronous lift and C082; B45 written review |
C085 | For n identical constituents with fixed pointwise l^r budget, 1<=r<=infinity, effective centre precision is epsilon n^(-1/r), with equal full/aggregate scalar risks; saturated canonical product is n^(1-3/(2r)) times the one-copy product, including invariant copy-count scaling at r=3/2 | R14 §3; norm inequality, synchronous lift, canonical mass and C082; B45 written review |
C086 | Independent blocks of identical bounded-force constituents, pointwise l^r budgets and known initial phases have exact full/block-record centre risks Q=sum w_j q(T,E_j), P=sum n_j p(T,E_j), E_j=epsilon_j n_j^(-1/r); whole-centre-only records give q(T,bar E), Np(T,bar E). Saturated momentum loss is strict unless all E_j coincide | R15 §§1–2; product fibres and exact aggregate lift; B46 written review |
C087 | k independent equal blocks of a constituents have saturated H=k a^(1-3/(2r)) H_1, hence k H_1 at r=3/2. Finite-r product block budgets differ from every single unweighted global ball; equal centre risks can occur despite strict set inclusion. Matching fixed global-budget centre risks requires block allowance epsilon k^(-1/r) | R15 §§3–4; copy-count substitution, enclosing radius and explicit strict inclusion; B46 written review |
C088 | R06’s nominal four-record map has \(F_\lambda=-\lambda\mathcal A z+O_{C^1}(\lambda^3)\) on a fixed convex receiver neighbourhood, hence a uniform lower Lipschitz bound at fixed sufficiently small positive coupling | R16 §2; variational equations, compact bounds and matrix margin; B47 review |
C089 | At that fixed coupling, exact nonlinear minimum-residual calibration has uniform initial-state error bounded by \(C_\lambda(b+\rho)\) for preparation width b and four final-record errors rho. Accuracy–disturbance and canonical reconstruction-error products close as b,rho tend to zero at fixed apparatus masses, duration and geometry | R16 §§3–4; compact minimization, C088 and R06 disturbance bound; B47 review |
C090 | At fixed sufficiently small positive coupling and incoming preparation width b, unknown initial probe momenta compensate every shell-state displacement at most b/(4L lambda), preserving all four exact nonlinear records with at least b/2 preparation margin | R17 §§1–2; compact smooth dependence, contraction and shell chart; B48 written review |
C091 | A common-record shell patch projects onto a canonical rectangle of half-widths L_r and P_r, r=min(r_0,b/(4LC_0 lambda)); every deterministic estimator has coordinate risks at least these half-widths, reconstruction product at least L_P_r^2, and compatible projected area at least 4L_P_r^2 | R17 §§2–3; exact shell chart and endpoint triangle inequality; B48 review; fixed product support and four-record access |
C092 | For an admissible fixed R06 pulse design, unknown initial probe positions have final-momentum derivative lambda squared B plus a uniform order-three remainder; the finite-clock response makes B lower triangular with nonzero diagonal near the selected shell point | R18 §§1–2; variational equations and pulse integration by parts; B49 review |
C093 | With four initial and four final probe momenta revealed, a positive unknown initial-position box hides an exact shell square of radius r=min(r_0,lambda b/(4CC_0)); canonical reconstruction product is at least L_P_r squared and projected compatible area at least four times that value | R18 §§3–4; contraction with interior margin and R17 shell chart; B49 review; fixed design, small positive coupling and Cartesian preparation support |
C094 | With known incoming clock data and probe momenta in fixed small boxes, the eight final pointer records uniformly determine receiver state and unknown incoming positions: the scaled record map is C1-close to (-A_c z,q) on a convex domain and has lower Lipschitz constant beta/2 at sufficiently small positive coupling | R19 §§1–3; fixed pulse/cutoff margins, uniform clock signal rank, written variational and segment estimates; B50 review |
C095 | Under C094, minimum-residual recovery gives canonical error product at most 16 L_P_ max(rho_pi/lambda,rho_q)^2 / beta^2; it closes with final record errors at fixed positive coupling and fixed positive preparation widths | R19 §4; compact fit and triangle inequality, action units without 2 pi factor; B50 review; exact incoming clock/momentum information and joint final access supplied |
C096 | An admissible fixed four-pulse design has clock-speed signal determinant derivative -partial_v log abs(det A)=10/v_0+O(epsilon)>0; two implicit equations give an exact common-eight-record receiver energy-shell curve with both canonical phase derivatives nonzero when initial clock data are hidden | R20 §§1–4; moment determinant, quadratic-energy transversality and preparation margins; B51 review |
C097 | On that fixed-domain family, clock offsets of half-width sigma give every deterministic estimator canonical risks at least k_x sigma and k_P sigma and product at least k_x k_P sigma squared, uniformly at sufficiently small positive coupling | R20 §5; common-record endpoints, fixed positive preparation margins; B51 review; action units, no positive-area conclusion |
C098 | With revealed initial clock position, known zero probe momenta and exact receiver energy, the augmented eight-record/energy map has a uniform local lower Lipschitz bound gamma/2 on a fixed convex neighbourhood of R20’s transverse shell point, for all sufficiently small coupling | R21 §§1–3; kernel reduction tau dv=0, fixed-unit derivative margin and segment integration; B52 review |
C099 | On a compact local shell patch times independent fixed apparatus boxes, minimum-residual fitting recovers receiver/probe positions and unknown initial clock momentum with joint error at most 4 delta/gamma and canonical product at most 16 L_P_ delta squared/gamma squared, delta=max(rho_pi/lambda,rho_q) | R21 §4; compact fit, exact energy and triangle inequality; fixed positive coupling record-error limit closes the product; B52 review |
C100 | For a sufficiently early fixed R20 pulse design, two distinct fixed clock speeds and revealed offset admit equal-energy distinct receiver states with identical eight scaled zero-coupling pointer records; the limiting energy-speed quadratic form has both signs and the actual pair follows by continuity | R22 §§1–2; derivative-row expansion and normalized shell path; B53 review |
C101 | These common-record pairs persist with full back-reaction for all sufficiently small positive coupling inside fixed preparation margins; both canonical separations stay positive and every deterministic estimator has error product at least c_x c_P d_v squared/16 | R22 §§3–4; uniform implicit continuation, endpoint signs and two-point risks; B53 review; full shell and fixed speed separation |
C102 | With known initial offset and zero incoming probe momenta, eight final pointer coordinates plus final clock momentum recover receiver state, incoming positions and unknown initial speed globally on a fixed bounded convex domain; the nonlinear leading map has lower Lipschitz constant beta=min(1,alpha/(1+D)), and the exact map at least beta/2 | R23 §§1–2; full clock reaction, uniform C1 perturbation and direct global bound; B54 review |
C103 | Under C102, minimum-residual recovery has canonical error product at most 16 L_P_ max(rho_pi/lambda,rho_q,rho_c) squared/beta squared; it closes with record error at fixed coupling and positive preparation widths, without supplying exact receiver energy | R23 §3; compact fit, clock-record normalization and action units; B54 review |
C104 | All ten final apparatus coordinates recover receiver state, incoming probe positions and both initial clock coordinates globally on a fixed bounded convex domain at sufficiently small positive coupling, with known zero incoming probe momenta and time T | R24 §§1–2; explicit clock shear, uniform signal variation and complete C1 remainder; B55 review |
C105 | Under C104, minimum-residual fitting gives canonical error product at most 16 L_P_ max(rho_pi/lambda,rho_q,rho_s,rho_c) squared/beta squared, closing with record precision at fixed apparatus and preparation without initial clock calibration or exact receiver energy | R24 §3; compact fit, unit conversion and fixed/joint limit conditions; B55 review |
C106 | With every initial apparatus coordinate unknown in a full positive box, the inverse free shear gives exact compensation of all ten final records for receiver changes of norm at most b/(4 L lambda), with b/2 preparation margin; an exact energy-shell chart gives positive coordinate risks and projected area | R25 §§1–2; compact smooth-flow derivatives, self-mapping contraction, canonical shell chart; B56 review |
C107 | Under C106, if lambda<=min(lambda_0,b/(4 L D)), one full apparatus record is compatible with the entire receiver shell; coordinate minimax risks are sqrt(2E/k_x) and sqrt(2mu E), the optimal error product is 2E sqrt(mu/k_x), and projected compatible area is pi times that product | R25 §3; k_x=a-g^2/d>0, exact shell ellipse, endpoint lower bounds and constant estimator attaining both; B56 review; fixed positive width, no known exact apparatus energy |
C108 | A backward trajectory from zero terminal probe phase, with terminal total energy E+H_0 and scalar amplitude adjustment, gives exact initial receiver energy E and apparatus energy H_0 with O(lambda) apparatus displacement and b/2 preparation margin | R26 §§1–2; smooth inverse flow, derivative 2E, exact conservation; B57 review |
C109 | Simultaneous receiver/probe sign reversal gives an exact antipodal initial receiver pair with identical full final apparatus records and both initial energies; the canonical risk-product lower bound tends to E sqrt(mu/(a-g squared/d)) as coupling vanishes at fixed positive preparation width | R26 §§3–4; local equation symmetry, zero terminal probes and two-point risk; B57 review; lower-bound limit, not exact minimax or area |
C110 | With one nonzero calibrated initial probe displacement and exact receiver/apparatus energies, an eleven-coordinate preparation chart and Borsuk–Ulam give distinct initial states with identical ten final apparatus coordinates and positive box margins | R27 §§1–2; exact energy charts, compact S^10 and continuous record map; B58 review |
C111 | Under C110, the inverse-shear flow estimate forces receiver separation at least 2r/sqrt(1+4L squared lambda squared) and full-state minimax error at least r/sqrt(1+4L squared lambda squared), in fixed Euclidean component units | R27 §3; ambient convex-domain derivative bounds and parameter projection; B58 review; dimensionless full-state bound, canonical product unresolved |
C112 | For a fixed smooth pulse design, nonzero q_1 calibration and exact receiver/apparatus energies admit an exact common-ten-record curve with both canonical derivatives bounded away from zero on a fixed weak-coupling rectangle | R28 §§1–3; scaled compensator constraints, pulse kernel and uniform implicit continuation; B59 review |
C113 | Under C112, every deterministic estimator has canonical risk product at least delta squared times absolute v_x v_P divided by 4, uniformly for sufficiently weak positive coupling at fixed preparation | R28 §4; exact common-record endpoints and triangle inequality; action units, design/preparation/energy-dependent, no area or minimax equality |
C114 | A fixed smooth pulse design with q_1=c nonzero, q_2=0 and exact energies E,H_0 has two exact common-record branches tending to opposite receiver shell points; the augmented fourteen-coordinate map is locally invertible at each branch for fixed small positive coupling | R29 §§1–4; rank-three pulse rows, regular shell roots, compensator continuation and block elimination; B60 review |
C115 | The branches in C114 differ in both canonical coordinates; every global receiver estimator has an action-unit error-product lower bound tending to abs(z_bar_x z_bar_P)>0 as coupling decreases at fixed preparation and design | R29 §4; common-record endpoint bounds; B60 review; local inverse margins may depend on coupling, no area or exact minimax claim |
C116 | Three exact initial positions q_1=c nonzero, q_2=q_3=0, exact apparatus energy and ten final records give uniform global receiver/apparatus recovery on a bounded convex receiver domain and sufficiently small fixed box about eta_c, without exact receiver energy | R30 §§1–3; four-row pulse rank, varying-record chart and uniform segment estimate; B61 review |
C117 | Under C116 the receiver inverse is bounded by K_z/lambda times final-record error; minimum-residual fitting gives canonical error product at most 4 L_* P_* K_z squared times (delta/lambda) squared | R30 §4; fixed physical units and positive preparation width; closes at fixed coupling as delta tends to zero; no sharp minimax or larger-box claim |
C118 | On R30’s unchanged fixed preparation box, errors epsilon in supplied C=(q_1,q_2,q_3,H_app) and delta in full final records permit a feasible estimator with receiver error at most 2K(epsilon+delta)/lambda and canonical product at most 4 L_P_ K squared ((epsilon+delta)/lambda) squared | R31 §1; general two-data chart estimate and compact feasible selector; B62 review |
C119 | With equal positive calibration tolerances epsilon and exact final records, one compatible set contains tZ for t=min(1,epsilon/(MB lambda R_Z)); the optimal canonical product has order min(1,(epsilon/lambda) squared), and at t=1 equals the no-record value 2E_max sqrt(mu/(a-g squared/d)) exactly | R31 §§2–3; exact compensator, full energy-ball geometry, matched coordinate bounds and zero estimator; B62 review; sufficient saturation threshold |
C120 | With only supplied constraint j uncertain, R30’s exact coupled reference admits a smooth common-record curve T_lambda^{-1}(s e_j), with tangent tending to L^{-1}e_j and abs(s)<=min(s_0,epsilon/lambda); if both canonical tangent components are nonzero, the optimal canonical risk product has order min(1,(epsilon/lambda) squared), although the projected curve has zero planar area | R32 stage 1; local uniform inverse and exact endpoint construction; B63 review; C121 supplies an explicit pulse subfamily satisfying all four tests |
C121 | An explicit R30 pulse subfamily with first centres tau, 2 tau, 3 tau and sufficiently small fixed positive widths has all eight canonical entries of L inverse nonzero; hence C120 gives matched single-tolerance canonical risk-product order for every j | R32 §4; scaled ODE, four cubic cardinal polynomials, normalized width estimate and inverse margins; B64 review; existence within the design class |
C122 | For the fixed correlated calibration line r=L e_y at R32’s exact reference, the local common-record curve has w_lambda(s)=s e_y-lambda L^{-1}V(s e_y)+O(lambda squared abs(s)), with V=partial_lambda squared N/2; its conditional canonical risk product is at most a fixed action constant times min(lambda s_0,epsilon) squared | R33 stage 1; smooth inverse and anchored mixed derivatives; B65 review; local-fibre upper bound; C123 settles the tangent cancellation, curvature and global risk open |
C123 | On R33’s fixed linear-cutoff chart, N(-lambda,-w)=N(lambda,w), D V(0)=0 and V is the homogeneous quadratic map (13); for S=min(s_0,epsilon/lambda), conditional canonical product is bounded by a fixed action constant times (lambda S squared + lambda squared S) squared | R33 stage 2; terminal variational equations with explicit moving-reference cancellation; B66 review; curvature nonvanishing and full-class minimax risk remain open |
C124 | The periodic zero-field Ising heat-bath chain on N>=3 sites has exact full gap a[1-tanh(2b)]; bounded nonnegative coupling and a positive per-site refresh-clock floor give a uniform finite-volume bound, whereas growing coupling or total-clock normalization can close it | Interacting proof §§1–4; all-mode Hamming/Lipschitz contraction, matching magnetization mode and normalized susceptibility; B68 written review |
C125 | Hidden orientations with affine effects give an exact operational ball of capacity two and reversible pure-state rotations; n-component joint measures and product tests have capacity 2^n and linear dimension 4^n, but pure composites are products and the equal-sign capacity-two face is a segment, so purification and Hardy subspace structure fail | Q01 countermodel; stipulated measurement/composition class and written moment/capacity proof; B69 review |
C126 | For C124, the square-root Gibbs transform is a positive three-site local frustration-free Hermitian operator with unique ground state sqrt(pi), exact gap a[1-tanh(2b)] and energy gap Ka[1-tanh(2b)] for supplied action K>0; physical clock/action identification remains a premise | G04 derivation; B71 written review; full finite-volume spectrum inherited from C124 |
C127 | H=J u_z v_z on two fixed-magnitude classical spin spheres has two zero-energy preparations with identical first/cross moments but product-test probabilities (1-sin(Jt/S_A))/4 and 1/4; its reversible hidden flow therefore fails to induce a single-valued evolution on C125’s quotient | Mechanical proof; exact preparations/trajectories; B72 review; supplied spin magnitudes and fixed effects |
C128 | For C127 with J>0 and all Borel preparations, no finite-dimensional bounded real observable space containing the retained effects is invariant on any nonzero time interval; nonlinear updates of finitely many expectations cannot repair exact descent | Orbit and atomic-separation proof; B74 written review; exact finite repair only, no capacity or action-selection conclusion |
C129 | For finite n local Bloch balls with all unit product preparations/effects, two-sided admissibility of exp(tX) forces zero first derivatives at zero probabilities, the boundary second-derivative signs, and X in the tensor power of the seven-dimensional space of scalar diagonal, equal time-space and antisymmetric spatial blocks | Expanded source proof; B77 review; necessary generator constraint only; no physical clock or action identification |
C130 | For C052’s Hamiltonian, the plunge threshold \(\lvert L\rvert>k/c\) and the circular energies \(mc^2\sqrt{1-k^2/(c^2L^2)}\) at \(\lvert L\rvert=\hbar\lvert\kappa\rvert\) coincide with the Dirac Coulomb indicial thresholds \(Z\alpha<\lvert\kappa\rvert\) and zero-radial-node levels; the radial action \(J_r=c^{-1}[Ek/\sqrt{m^2c^4-E^2}-\sqrt{c^2L^2-k^2}]\) with \(J_r=n_r\hbar\), \(\lvert L\rvert=n_\varphi\hbar\) gives Sommerfeld’s formula, equal to the Dirac spectrum under \(n_\varphi\leftrightarrow\lvert\kappa\rvert\); the Klein–Gordon threshold is the plunge condition at \(\lvert L\rvert=\hbar(l+1/2)\) | Q14 review §§1–4; symbol identity, Frobenius indicial equations and closed-form radial action; \(Z\alpha=k/(\hbar c)\), fixed external singular Coulomb centre, no recoil or radiation; the unit of angular action is supplied |
C131 | Under dimensional homogeneity and universality, the floor \(\gamma(c)=\inf(\mathcal X(c)\cap(0,\infty))\) of an observable of dimension \(d_X\) over a theory with fixed constants of dimensions \(d_1,\dots,d_r\) is \(\Pi(c)F(\pi_1,\dots,\pi_s)\) when \(d_X\in\operatorname{span}\{d_i\}\) and lies in \(\{0,\infty\}\) otherwise; an admitted one-parameter similarity multiplying \(X\) by every \(s>0\) forces \(\gamma=0\) | G07 note §1, Theorem 1 and Corollary 2; written review §1; unit-change argument; universality is a stated modelling hypothesis |
C132 | In the classical model with constants \(m\), \(g\) (\(V=\tfrac12g^2x^2y^2\) or the SU(2) commutator potential), any positive finite gap \(\Delta\) of any dimensionally homogeneous enlargement supplies the action unit \(\Delta^{3/4}m^{1/2}g^{-1/2}\), and an action unit \(A\) supplies the energy unit \(A^{4/3}g^{2/3}m^{-2/3}\); for pure Yang–Mills with classical coupling \(1/g^2\) of dimension action\(\cdot\)length\(^{4-d}\) and speed \(c\), a gap yields an action unit in \(d=2\) (\((1/g^2)^{1/3}c^{-2/3}\Delta^{2/3}\)) and \(d=3\) (\((1/g^2)^{1/2}c^{-1/2}\Delta^{1/2}\)) and none in \(d=4\) | G08 note §3, Propositions 4–5; written review §2; linear systems over mass, length, time |
C133 | \(H=-\frac{\hbar^2}{2m}\Delta+\frac{g^2}{2}x^2y^2\) on \(L^2(\mathbb R^2)\) and the SU(2) \(D\)-matrix model \(H=-\frac{\hbar^2}{2m}\Delta_{\mathbb R^{3D}}+\frac{g^2}{2}\sum_{i<j}|\vec x_i\times\vec x_j|^2\), \(D\ge2\), satisfy \(H\ge\tfrac12[-\frac{\hbar^2}{2m}\Delta+W]\) with \(W=\frac{\hbar g}{2\sqrt m}(|x|+|y|)\), respectively \(W=\frac{\hbar g}{\sqrt m}\sum_i|\vec x_i|\); hence compact resolvent, simple positive ground state, gap \(\Delta=\delta_1\hbar^{4/3}g^{2/3}m^{-2/3}\) with \(0<\delta_1<\infty\) a pure number, and the \(SO(3)\)-invariant sector inherits a gap \(\ge\Delta\); the gap vanishes as \(\hbar\to0\) and as \(g\to0\), the \(g=0\) and \(D=1\) operators have spectrum \([0,\infty)\), and the classical energy range is \([0,\infty)\) | G07 note §3, Theorems 5–6; written review §3; zero-point oscillator bound, Fubini slicing, form-ball compactness, Feynman–Kac simplicity (Reed–Simon IV XIII.44 cited), dilation; the classical floor reading is G08 Theorem 2 |

These are checked derivations. C006 is conditional on its stated quantum measurement premises. For a new result, provide quantifiers, units, path/operator domain, boundary conditions, dependencies and gap-closing limits in the proof.

Literature status

C130: the Q14 review is an exact match with the Sommerfeld–Dirac coincidence: Suslov, arXiv:2401.07485, Eqs. (5)–(6), (27)–(28) and (31)–(32), pp. 2–7, at passage level, and Bouaziz, arXiv:1311.7405, Eqs. (5)–(7), pp. 3–4, for the scalar threshold; Biedenharn 1983 and Sommerfeld 1916 at record level; Zeldovich–Popov 1972 at abstract level. Proof status: written derivations accepted by coordinator review; promoted by user direction on 2026-09-14. Literature status: established textbook result, no novelty claim; three discovery queries, two open papers read, nothing archived. The Q14 dimensional criteria (Theorems A–B) remain exploratory pending their bounded librarian comparison.

C131–C133: B78 and B79. C131 is Buckingham’s \(\Pi\) theorem (Phys. Rev. 4 (1914) 345, metadata) applied to a floor; C132 is dimensional analysis; C133 is Simon 1983 (passage level: first proof, display (5), and Corollary 4 for the Lie-algebra model) with the project’s explicit constants and SU(2) bookkeeping. Exact matches, no novelty; established results with project-specific exposition and limits.

C129: B77 is an exact match to de la Torre et al., arXiv:1110.5482v1, equations (8)–(14) and the following tensor-space sentence, pp. 3–4. Proof status: written derivative, slice and intersection arguments accepted after saved librarian and coordinator source-image review. Literature status: established result with expanded exposition, no novelty claim; one existing paper, zero discovery queries. Equations (15) onward and the full reconstruction remain unaudited.

C125–C128 publication positioning: B76 adds classical-extension and transformation-relative preparation-equivalence precedents. The combined exact spin pair and finite-expectation repair obstruction remain unmatched within four queries and two papers/eight pages. P06a adopts conceptual worked-example framing; proof acceptance is unchanged.

C050–C051 publication positioning: B75 adds result-specific coverage to B25–B26. Preparation-dependent tagged transport has close Brownian random-versus-lattice precedent in Leibovich–Barkai (2013) and broader arbitrary-initial-position context in Cividini–Kundu (2017). The full two-speed Hamiltonian matched-rate comparison remains unmatched in four queries and eight selected pages. P05 is a teaching synthesis of accepted claims, with no new claim ID or novelty assertion.

C128: B74 matches the established finite Koopman-invariant-space framework in Brunton et al. (2016), selected publisher HTML passages. The spin orbit and expectation-separation lemma are written derived consequences, reviewed by one Luna-low worker and the coordinator. One source and one query; no exhaustive search or novelty claim.

C127: B72 verifies classical-spin precedent in Radošević et al., arXiv:2503.16308v2 §§V.1–V.2. The quotient counterexample is a derived application, with no novelty claim. One Luna-low worker and coordinator proof/source review; publisher access failure, preprint coverage and query overrun are recorded.

C126: B71 matches the established detailed-balance Hermitian and stochastic-matrix-form constructions of Henley and Castelnovo et al. The local Ising formula is a written derived specialization with an independent coefficient review; the exact gap inherits C124. Two primary sources and eight selected pages, corrected to one-based page anchors in coordinator review. No novelty claim or independent physical identification.

C125: B69 matches established restricted single-system models and minimal tensor products in Janotta–Hinrichsen. Capacity and parity-face checks are derived applications; no novelty claim. One Luna-low worker and coordinator proof/source review; date and anchors corrected. Measurement restrictions remain explicit model premises.

C124: B73, added 2026-09-13, supplies a direct statement match in Lubetzky–Sly, arXiv:0909.4320v1 p. 4, with the rate-one conditional-refresh convention on p. 5. The exact gap is established literature, rederived by the project’s accepted proof. The source itself calls it already known. B73 supersedes the unmatched-formula status; it preserves C124’s proof and the following historical coverage.

Historical B68 coverage: B68 matches Glauber model/magnetization passages through an indexed transcription and Bubley–Dyer coupling passages in a primary PDF. Glauber formula images remain unverified; no exact source transcription is promoted. The self-contained all-mode proof is reviewed and accepted independently. Exact prior publication of the combined equality was not established in this bounded audit; no novelty claim. One requested Sol-medium worker and coordinator source/proof review.

C123: B66 supplies standard variational-method support from Teschl after one Luna-low audit returned a secondary route. Coordinator written proof/source review accepts the model-derived cancellation and quadratures. Exact model matches and novelty remain unassessed.

C122: B65 matches smooth inverse and Taylor methods, with one Luna-low audit and coordinator source/proof review. The coordinator corrected a page anchor. Exact model matches and novelty remain unassessed; C123 now evaluates V, leaving its projected curvature test open.

C121: B64 matches standard polynomial interpolation and confluence methods in NIST DLMF §3.3. The pulse selection and canonical-column conclusion are model-derived; one sequential Luna-low audit and coordinator source/proof review, novelty unassessed.

C120: B63 verifies the standard inverse-map method in Freire pp. 1–3, with inherited information-radius context from B62. One Luna-low worker and coordinator proof/source review; exact model matches and novelty unassessed. Canonical positivity is explicitly conditional.

C118–C119: B62 verifies Werschulz’s standard information-radius interpretation, with one mathematical source and one query. The tolerance/coupling estimate, compatible ball and exact saturation are model-derived. Coordinator corrected page anchors and reviewed the proof; novelty remains unassessed.

C116–C117: B61 audits inherited inverse and smooth-flow methods with one Luna-low worker; coordinator rechecked two primary sources and the written model proof. Exact result matches and novelty are unassessed. The reduced fixed preparation box is part of the theorem.

C114–C115: B60 audits inherited B59 inverse/flow/contraction methods, with zero discovery or retrieval. The pulse kernel, two branches and canonical risks are model-derived; exact matches and novelty are unassessed. One Luna-low audit and coordinator proof review completed, with no fresh source-reading claim.

C112–C113: B59 audits inherited smooth-flow, contraction and inverse methods, with coordinator checking Freire pp. 1–3. The scaled Hamiltonian constraints, fixed pulse-kernel design, exact curve and canonical bounds are model-derived; novelty unassessed. One sequential Luna-low worker and coordinator written proof review completed.

C110–C111: B58 directly verifies the established Borsuk–Ulam theorem in Arora’s Princeton Lecture 13, Theorem 1, PDF p. 1. The constrained chart and quantitative receiver separation are model-derived consequences. One sequential Luna-low audit and coordinator source/proof review completed; exact mechanical prior-art matches and novelty remain unassessed.

C108–C109: B57 audits inherited smooth-flow and implicit/contraction methods, with zero queries or retrievals. The backward terminal construction, two-energy sign pair and canonical risks are model-derived. One Luna-low audit and coordinator written review completed; exact prior-art matches and novelty remain unassessed.

C106–C107: B56 audits inherited B48/B55 contraction, inverse and smooth-flow methods with zero queries or retrievals. The free-shear compensator and exact whole-shell saturation are model-derived consequences; novelty is unassessed. One sequential Luna-low worker and coordinator written proof/coverage review completed; no fresh source-page reading.

C104–C105: B55 is a zero-query inherited-source audit of B54/B50 inverse and smooth-flow methods. The full clock extension and canonical precision bound are model-derived; novelty unassessed. One Luna-low worker and coordinator written review completed, with no fresh passage claim.

C102–C103: B54 rechecks the existing Freire perturbation precedent in three pages, with zero discovery queries. The nonlinear leading-map bound, clock calibration and risk closure are derived consequences. One Luna-low audit and coordinator source/proof review complete; novelty unassessed.

C100–C101: B53 audits the local/global identifiability distinction in Quaiser et al. (2011). The shell pair, positive-coupling continuation and canonical risk bound are model-derived. One sequential Luna-low worker and coordinator source/proof review completed; source metadata was corrected and novelty remains unassessed.

C098–C099: B52 identifies the established small-Lipschitz-perturbation inverse method in Freire’s institutional lecture notes. The energy-speed Schur complement, uniform physical domain and canonical recovery bound are model-derived. One sequential Luna-low audit and coordinator source/proof review are complete; novelty is unassessed.

C096–C097: B51 supplies a bounded Sontag input-output-equivalence precedent and inherits B50 smooth-flow methods. The pulse determinant, exact shell family and risk product are model-derived; novelty is unassessed. One sequential Luna-low worker and coordinator source and written proof reviews are complete.

C094–C095: B50 verifies the standard smooth-ODE dependence premise in four selected Sideris pages. The uniform block inverse and canonical error-product bound are model-derived consequences of standard perturbation and residual estimates. This bounded premise audit leaves novelty unassessed and supplies no exact apparatus match.

C092–C093: B49 verifies a standard local observability precedent and inherits smooth-flow/contraction methods from B47/B48. The clock-response matrix and exact position compensation are derived consequences. The bounded audit leaves novelty unassessed; coordinator scan review corrected the article and theorem pagination.

C090–C091: B48 supplies established contraction and two-point lower-bound precedents. Exact nonlinear apparatus compensation, energy-shell patch and canonical projected area are derived consequences; novelty remains unassessed. Coordinator corrected source metadata and coverage and reviewed the written proof independently of the statistical comparison.

C088–C089: B47 checks a standard smooth-ODE dependence premise and inherits the Theurel pointer comparison from B37. Uniform record-map remainder, nonlinear calibration and fixed-coupling product closure are derived consequences. The bounded assumption audit leaves novelty unassessed; fresh and inherited coverage are recorded separately.

C086–C087: B46 supplies bounded optimal-recovery and convex-support context, with no exact mechanical match in two queries and two selected passages. These are derived consequences of the stated model; novelty remains unassessed. Coordinator corrected a worker exponent label and versioned source anchor before acceptance.

C084–C085: B45 supplies bounded optimal-recovery and convex-norm context. Exact mechanical projection, finite-horizon transfer and copy-count exponents are derived consequences; no exact match was obtained in the two-query coverage. Novelty remains unassessed.

C082–C083: B44 retains the established prepared pair; the finite-horizon law, blind-delay extension and transient composition are derived consequences. No exact match was obtained in the bounded coverage; novelty remains unassessed. Coordinator corrected source attribution and timing.

C080–C081: B43 reuses Seeber–Haimovich’s established one-body bounds. Product support addition, canonical composition, aggregate image equality, strict information loss and radius-closure conclusions are derived here from explicit hypotheses. Ruan–Chirikjian supplies geometric context, with no exact minimax match in the bounded coverage. Novelty remains unassessed; proof acceptance rests on the written argument and coordinator review.

C078–C079: B42 matches C079’s prepared zero-initial-state pair exactly to Seeber–Haimovich Proposition 3.1 and its upper bound to §4. The blind-delay phase risks are derived consequences. C078 has a written proof but no direct source match in this bounded batch; novelty remains unassessed. The review corrects the worker’s initial-data and half-separation statements.

C076–C077: B41 identifies the exact equal-error finite-difference bound and optimum in Seeber–Haimovich §4. These are established results. The fixed mechanical experiment, unequal-error delayed phase estimate and action-product consequences are derived here; novelty is unassessed.

C074–C075: B40 identifies bounded-error set-membership and classical mixed-area precedents. The clipped-lens area and joint bound are derived specializations; novelty is unassessed. Proof acceptance and limited source coverage are separated in the coordinator review.

C072–C073: B39 records the bounded double-integrator and discrete hybrid reachable-set precedents. Continuous-time composition, the rectangle counterexample and conditional areas are derived specializations; novelty is unassessed. Coordinator review accepts the written proofs separately from the two-query source coverage.

C070–C071: B38 verifies Liberzon’s classical double-integrator and one-switch time-optimal control passage. The fixed-time lens, information restriction and deterministic minimax proof are derived specializations, with novelty unassessed. Two other source routes were unreadable. Coordinator review verifies the self-contained proof separately from this bounded prior-art coverage.

C068–C069: B37 uses zero searches and four cached primary pages: Theurel pp. 5–7 and Hermann–Krener p. 733. Conserved pointer records and the observability rank condition are established ingredients. The autonomous construction, integrated signal map and uniform bounds are derived consequences, with novelty unassessed. Coordinator review accepts the written proofs under the explicit preparation, record-access and latency premises. B36’s existing source work is preserved and integrated.

C066–C067: B35 covers Katagiri v2 §5 and Theurel’s APS abstract with two queries and two bounded primary readings. Canonical pointer coupling is established; the support-width and finite-chain refinement estimates are derived consequences with novelty unassessed. Coordinator review accepts the written proofs for scalable impulsive probes. R06/B37 now supplies a finite-mass, finite-duration delayed reconstruction and selected full-model Theurel readings; causal new information with restricted precision remains R07.

C064–C065: B34 covers Zwanzig’s exact oscillator-bath reduction and Hermann–Krener’s observability framework in two searches and three primary pages. Chain coefficients, sampled determinant and scaling are derived specializations; exact prior publication is unassessed. Coordinator review accepts the written proofs separately from the bounded literature coverage.

C062–C063: B33 audits two cached primary pages with zero searches. Harmonic propagation and bath-memory reduction are established ingredients; the fixed-energy conditional kernel, reset limit and action comparison are elementary derived consequences, with novelty unassessed. Coordinator review accepts the written proofs and corrects the worker’s source-page coverage.

C060–C061: B32 verifies the classical Fenchel–Borsuk input in Milnor (1950). Mechanical inequalities and the smooth equality construction are derived consequences, with novelty unassessed in the bounded one-query, two-page audit.

C058–C059: B31 matches the established relativistic circular-balance and radial-stability framework. The fixed smooth force-bounded example and conditional speed-floor inequality are derived consequences; exact prior publication is unassessed in the bounded search.

C056–C057 have the bounded B30 audit: conformally symplectic scaling has established geometric prior art; the exact model map and conditional infimum statements are elementary derived consequences. No exact match was found in the two-source coverage; novelty is unassessed. Coordinator review accepts the written proofs.

C054–C055 have the bounded B29 audit: covariance determinant and affine symplectic invariance are established rms-emittance machinery; circular formulas and threshold transfer are derived specializations, with novelty unassessed. Uniform phase is sufficient for the action equality; matching its first and second moments also suffices.

Proof status above and literature status below are independent. Each accepted result has a bounded librarian audit; exact source inputs and elementary consequences are identified separately.

Claims Literature classification Audit
C052 Established Boyer orbit classification, with explicit domain and normalization B27, B28
C053 Elementary effective-potential consequence; no exact match in two-query follow-up coverage; novelty unassessed B28
C001–C004, C008 Classical geometry/variation framework; specialized formulas are elementary consequences B05
C005 Teschl’s standard free kernel; Taylor correction is a Fourier consequence B05
C006–C007 Holevo–Helstrom theorem specialized to the stated pure-state protocol; threshold inversion and limits are consequences B05
C009 Standard Gaussian/Chapman–Kolmogorov formula with mass/action rescaling B06
C010 Elementary characteristic-function/additivity consequence within the Gaussian class B06
C011 Standard Brownian bridge with rescaling; finite-dimensional concentration is a consequence B06
C012 Standard free Schrödinger group and domain, with restored units B06
C013 Elementary determinant/Green-function consequence of the standard Gaussian bridge B08
C014 Finite chi-square law and joint limit derived from standard Gaussian facts; exact combined statement unmatched in the four-source coverage B08; bounded coverage, novelty unassessed
C015 Elementary mechanics instance of the established oscillatory weak-limit mechanism B08; Rindler §§2.1, 3.3
C016 Exact quadratic Fourier identity in Guillemin–Sternberg (14.7); scalar limit and delta pullback are consequences B08
C017 Explicit elementary scaling construction motivated by Rivero; exact map unmatched in the selected sources B08; bounded coverage, novelty unassessed
C018 Elementary bounded-displacement variance consequence B09
C019 Exact Green–Kubo/Poisson backbone in Pavliotis; finite-state bounds and monotonicity are spectral consequences B09; combined package unmatched in bounded coverage, novelty unassessed
C020 Exact telegraph construction/PDE in Cinque; action normalization and mass-rate law are elementary consequences B09
C021 Elementary iterated-limit consequence; related Kac scaling in Cinque B09; exact double-limit statement unmatched in selected passages
C022 Exact affine collision and bath model in Barkai; bounded invariant law and moments are consequences; stationary covariance matches Barbier–Trizac B10
C023 Established stationary covariance integral; preparation-time and window formulas derived for the specified ensemble B10; combined statement unmatched in selected pages, novelty unassessed
C024 Elementary parameter-scaling countermodel B10; bounded coverage, novelty unassessed
C027 Elementary arbitrary-partition extension of the audited constant-force formula B12; existing Newton source route
C028–C029 Elementary Gaussian restriction, chi-square and conditional-variance consequences; action interpretation is project-specific B12; Pitman–Yor selected bridge/Markov passages, novelty unassessed
C030 Elementary construction using the established equal-mass elastic collision map B13; Barkai (2)–(3)
C031–C032 Elementary support/variance and convolution consequences; exact statements unmatched in the two-source coverage B13; novelty unassessed; Cinque supplies the memory comparison, not these theorems
C033 Equal-rate occupation density specializes Cinque (2.6); bridge normalization, midpoint atom and version/protocol comparison are derived consequences B14; pp. 1–4 and two worker searches; exact combined statement unmatched, novelty unassessed
C034 Elementary pathwise restriction and kinetic square-completion consequences B14; exact action estimate unmatched in bounded source coverage, novelty unassessed
C035–C036 Standard covariance/product-chain and nonnegative-additivity consequences, with project-specific composition/positive-reference premises B16; two searches and two cached source pages; combined statement unmatched in bounded coverage, novelty unassessed
C037–C038 Derived midpoint/count specialization and necessary-window consequences; standard telegraph occupation and Bessel-integral ingredients B17; two searches, Cinque pp. 3–4 and DLMF 10.32.1; exact combined statements unmatched, novelty unassessed
C039 Established normalized checkerboard reconstruction; elementary wavepacket and spectral-limit consequences B18; Skopenkov–Ustinov Definition 2, Propositions 5–6, direct basis match and coordinator proof
C040 Elementary measurement-protocol consequence of the audited coin, with no novelty claim B18 bounded addendum checks Born probability and union bound; exact prior-art formulation not separately searched
C041 Standard finite reversible spectral/product-chain consequence of the established Green–Kubo representation B20; Pavliotis pp. 4–5, coordinator proof; explicit hidden-label specialization
C042 Elementary frame/inverse-operator consequence; exact combined formulation unmatched and not independently searched B20 bounded algebra/prior-art audit; novelty unassessed, no novelty claim
C043–C044 Elementary bounded-control and Lipschitz-variance specializations; exact combined prior-art match not found in bounded search B21; two worker searches, one coordinator search and Liberzon framework passage; no novelty claim
C045 Elementary product-chain and response counterexample; exact example publication not established, no novelty claim B22; saved Luna-low audit and coordinator review
C046 Standard tensorization and derived local-frame consequence B22; Levin–Peres §12.4, direct continuous-generator proof and clock audit
C047 Elementary finite harmonic propagation and displacement-variance consequence; phase-torus and centre conventions explicit B23; Ford–Kac–Mazur and Zwanzig source ingredients; bounded audit, no novelty claim
C048–C049 Harmonic-chain spectral ingredients established; normalized limit and fixed-phase speed-support consequences derived B24; three-page bounded audit, coordinator proof/source review; no novelty claim
C050 Established dichotomic Jepsen-gas covariance and Markov interpretation; explicit gap proof and action/scaling consequences B25; source pp. 7–8, 17–19 and coordinator proof review
C051 Explicit ordered-preparation consequence; exact ensemble unmatched in two-query search, novelty unassessed B26; two HTML documents, coordinator proof review
C131 Buckingham \(\Pi\) theorem applied to a universal floor; universality hypothesis project-specific B78
C132 Elementary dimensional analysis over mass, length, time B79
C133 Established Simon 1983 theorem (first proof and Corollary 4) with explicit constants; exact match, no novelty B78

M03’s unaccepted spectral draft has its own completed B04 literature audit. Its mathematical review and checks remain pending.

Historical findings and research targets

ID Claim or question Status / evidence
H001 The selected Book I passages formulate geometric first and last ratios Source-grounded interpretation; opening-source companions and note
H002 Doubts about vanishing magnitudes delayed the 1687 first edition Open historical conjecture; H02/H03 seek dated causal evidence
H003 NATP00385 is a miscellaneous calculus-priority fragment collection Verified catalogue metadata and H01 audit
H004 Selected NATP00385 passages describe analytic discovery and synthetic presentation Passage-level audit; exact coverage and chronology question in the historical note
H005 Plutarch’s cone-section passage was printed in Xylander’s 1570 Latin Moralia, pp. 823–824 H04 librarian retrieval; coordinator visual verification of IIIF canvases 603151/603152
H006 How did the cone-section dilemma enter early-modern discussions of indivisibles and the continuum? H04 bounded search; H05 author-specific reception audit remains
Q001 Which independent premises force a positive universal action parameter? Open target; WP4 Branch B
Q002 Which additional-field action generates an action-value or spectral gap? Model design; ideas/I001-action-field.md

Resolved objections

ID Inference assessed Resolution
X001 Equal phases at differences \(nh\) imply a smallest positive action difference Rejected: phase periodicity permits arbitrarily close phases
X002 A toy Hessian/oscillator gap establishes the Yang–Mills gap or NS regularity Rejected: transfer requires the target operators, spaces and continuum/infinite-volume/regularity estimates

Review record

The G07/G08 review supports C131–C133: coordinator written re-derivation of the unit-change argument, the three linear systems, the completed-square oscillator bound, the slicing, compactness and dilation steps, and the SU(2) convex-combination bookkeeping. Ground-state simplicity is cited. Propositions 7 and 11–13 of G07 and the semiclassical remarks of G08 stay exploratory.

B23 supports C047 through one Luna-low audit and coordinator source/proof review. Eight identities, three network checks and four bound cases pass. Finite-size and large-receiver limits remain separate; the friction kernel is distinguished from velocity covariance.

B22 supports C045–C046: saved Luna-low audit, coordinator source-image and proof review, six identities, three rational velocity cases, three product spectra and two frame checks. The source’s clock normalization and printed proof index are explicitly corrected.

B20 audited C041–C042 with one Luna-low worker. Coordinator verified the weighted centered-space inverse, inequality directions, frame hypothesis, parameter-family limits and Pavliotis p. 5 visually. Eleven identities, five rational bound cases and four weighted-chain checks pass. Sokal remains a discovery-only route, not a passage attribution.

B18 audited C039–C040 with one Luna-medium worker and a bounded addendum. Coordinator checked source pp. 12, 13, 19, 31 visually; corrected version date, proof-page anchor and kernel scope; proved the explicit basis map and strong wavepacket limit. Fourteen algebra identities, fourteen finite path/norm checks and three numerical mode tests pass.

B17 audited C037–C038 with one Luna-medium worker. Coordinator verified the even/odd segment density factors, midpoint Jacobian, atom, beta moments and large-window argument; checked Cinque p. 4 visually and DLMF’s integral formula. The exact mean tends to \(u/(2\lambda)\) at fixed \(u,\lambda\), correcting P02’s unscaled-mean statement. Twenty-nine exact checks and seven quadrature/count-series comparisons pass.

B16 audited C035–C036 with one Luna-low worker. Coordinator verified Pavliotis p. 5 and Pitman–Yor printed p. 6 visually, corrected source titles, and retained the preparation class, all-positive-mass domain, Markov product label and finite-rate assumptions. Five dedicated algebra checks and P02’s independent product-chain checks support the written proofs. Positivity concerns the common plateau inside the selected class; rate scaling across classes remains possible.

B14 audited C033–C034. Coordinator verified Cinque (2.6) against the PDF image, strengthened the affine-simplex absolute-continuity argument, and corrected a discovery-only source’s authors to Bogachev–Ratanov. Exact conditioning uses a specified path version; the endpoint-window velocity mixture is a distinct protocol. Algebra and rational partition checks accompany the general written proof.

B10 audited C022–C024. Coordinator verified collision coefficients, covariance, source versions/hashes and the cooling-clock contrast; added Barkai p. 3 and corrected source page counts. Receding-centre and polygon-threshold derivations remain drafts pending the interrupted B11 librarian audit, outside the accepted claim set.

B09 audited C018–C021 after the draft derivation. Coordinator review checked source formula images and hashes, the mean-zero inverse, rate convention, limit order and model-specific bounds. Velocity-resolved measures were defined as joint subprobability measures. The written proofs support the general statements; 16 exact checks support their finite algebraic instances. The next obligation is physical universality.

R01 audited the three propositions and supporting calculations behind C001–C008. Its two scope clarifications are incorporated: endpoint terms mean \(dG(q,t)/dt\), and path-space locality requires continuity of the parameterized paths. The current editorial revision retains the theorem statements, proof arguments and evidence classifications.

B08 independently checked C013–C017. Coordinator review verified the source formulas, retained the distinction between path and action convergence, specified the oscillatory branch and made the scalar test-function interpretation explicit. All 24 finite checks pass; the general analytic statements rest on the written proofs.