Fourth paper of the Dimensional Generation Structure series. Under a fixed antisymmetric generator, each rotation plane of the N-body complete pairwise relational-wave system is an invariant subspace carrying a single-frequency rotation, kinematically isomorphic to the two-body AB system. Five theorems are established: (1) constant-angle, constant-amplitude plane rotation with angle 2 arctan(gamma sigma_j), orientation fixed by the generator (K p = sigma q, residual gauge SO(2)); (2) under the in-plane SO(2) gauge the absolute phase is unreadable while signed progression, frequency, and amplitude are readable - Axiom 5 reproduced at the level of derived quantities; (3) exact energy decomposition H = sum H_j + H_ker with individual conservation; (4) the static kernel admits no progression-based readouts, while the projector, kernel squared quantity, and state projection are covariant (dynamical realization of Consequence 16.3); (5) the squared closure also decomposes, sum Q_j + Q_ker = R^2, with each term individually conserved - a double conservation register per plane, yielding a no-go: under a fixed generator neither closure nor energy-like quantities move between planes, so spontaneous state splitting requires dynamics beyond the fixed generator. The zero-order harmonic second-order structure Delta^2 c = -4 sin^2(theta/2) c holds on every plane at machine precision; the exponent -2 in the continuous-spectrum re-expression is a definitional consequence, explicitly distinguished from integer-harmonic closure. The plane counterpart of the AB same-time positional readout D_AB and the construction of the Axiom-17 observed acceleration on adopted planes are stated as connection problems. Experiments F-I (N = 3-9, 32 trials x 720 steps, fixed generator, no sequential renormalization) verify all claims, including frequency nondegeneracy as an independent general-position assumption. Japanese and English texts, TeX/PDF, four figures, experiment code, and reproduction data are included.
Fourth paper of the Dimensional Generation Structure series. Under a fixed antisymmetric generator, each rotation plane of the N-body complete pairwise relational-wave system is an invariant subspace carrying a single-frequency rotation, kinematically isomorphic to the two-body AB system. Five theorems are established: (1) constant-angle, constant-amplitude plane rotation with angle 2 arctan(gamma sigma_j), orientation fixed by the generator (K p = sigma q, residual gauge SO(2)); (2) under the in-plane SO(2) gauge the absolute phase is unreadable while signed progression, frequency, and amplitude are readable - Axiom 5 reproduced at the level of derived quantities; (3) exact energy decomposition H = sum H_j + H_ker with individual conservation; (4) the static kernel admits no progression-based readouts, while the projector, kernel squared quantity, and state projection are covariant (dynamical realization of Consequence 16.3); (5) the squared closure also decomposes, sum Q_j + Q_ker = R^2, with each term individually conserved - a double conservation register per plane, yielding a no-go: under a fixed generator neither closure nor energy-like quantities move between planes, so spontaneous state splitting requires dynamics beyond the fixed generator. The zero-order harmonic second-order structure Delta^2 c = -4 sin^2(theta/2) c holds on every plane at machine precision; the exponent -2 in the continuous-spectrum re-expression is a definitional consequence, explicitly distinguished from integer-harmonic closure. The plane counterpart of the AB same-time positional readout D_AB and the construction of the Axiom-17 observed acceleration on adopted planes are stated as connection problems. Experiments F-I (N = 3-9, 32 trials x 720 steps, fixed generator, no sequential renormalization) verify all claims, including frequency nondegeneracy as an independent general-position assumption. Japanese and English texts, TeX/PDF, four figures, experiment code, and reproduction data are included.
For a two-wave relational wave system, we design a single inelastic interaction operation that takes only the waveforms as input and contains no discriminating branches (no IF statements). Within the assumed class (two-wave, pointwise, cubic, common-phase invariant, no self-scattering, strength reusing the existing readout R = sin^2(theta)), conservation of the zero-square-sum closure (axiom) uniquely fixes the relative coefficient. The uniquified vertex rewrites exactly as an exchange flow driven solely by the imaginary part of the relative phase, is a pointwise two-channel rotation, and is exactly solvable in closed form (closure and power are conserved identically pointwise; measured drift 4e-14 over 3000 collisions). Measured consequences: (1) a parity theorem (generation is necessarily induced; a Z2 class conserved by the exponent arithmetic), (2) autocatalytic ignition rate = C f^2 with C = 11.45 constant to 0.4% over four decades of seed fraction, (3) no runaway: oscillatory relaxation to a statistical equilibrium f* = 0.4690, close to the mask phase-space fraction 0.494, (4) although no antiparticle degrees of freedom are input, products appear only in sum-rule partner bins (485 unpredicted bins at machine zero, ratio 1e-27), as correlated pairs settling the winding-number ledger (coherence 0.83), rising simultaneously from exact zero. This paper does not claim the interaction is the one realized in nature; assumptions, conditional derivations, and measurements are strictly separated, and the value of the assumption is measured by consistency and productivity. Japanese and English full texts included.
The preceding paper demonstrated shape-invariant translation of matter packets, but supplied the position axes (a 3D register lattice) and the clock (harmonic-closure dispersion) externally. This paper re-derives both from the observables of the three-direction metastable condensate itself, and re-derives the readouts of mass-like and charge-like quantities in the same framework, stating every readout method in the text. Measured results: (1) the substance of the condensate is not three equal axes but 2+1 - one complex eigenpair (a single rotation plane) plus a dependent rotation axis (agreement with the cross product 0.987 to 0.9965, improving with N); (2) the spatial x, y axes are derived as quadratures of the plane coordinate demodulated by the proper clock - orthogonality is a construction of the readout, not a property of the substance; (3) proper time tau is the tick of a single collective clock (omega about pi/72 per step, rigid rotation of all occupied planes), enforced collectively by entrainment: clockless coherent perturbations are synchronized to the condensate clock and brought to rest; displacements of the rest component persist undamped, and motion becomes the privilege of content carrying its own clock - mass; (4) mass reads as the non-coherence det Gamma = T^2 - X^2 - Y^2 - Z^2 >= 0 of a readout pair (mathematics known from the coherence-matrix theory of polarization; our contribution is the physical identification and in-model measurement: light-like eigenmode 1.2e-7 vs condensate 3.1e-3); (5) charge separates via transmission, relative phase, and winding number. Distributions are measured: sea modes concentrate at maximal non-coherence (the sea is heavy); charge distributes over multiple quantized values and does not stabilize at plus-minus one - recorded as a known open problem; the symmetric sea is gray (particle-antiparticle balanced) while charged seeds grow particle-like content; matter-packet motion is visualized in the 4-dimensional xyz+tau readout as uniform translation (all 10 points deviation 0.000). Organizing everything on the closure cone x^2+y^2+z^2 = t^2+R^2+Q^2 = R-prime^2: the full content of observation is the three spatial axes + proper time tau + mass m + charge q, and the uniquely unreadable quantity is the coordinate time t - it appears only as the arithmetic residue or as a wide-area selection over many local clocks. Consequently, the distinction general relativity postulates - proper time is physical, coordinate time is convention - is derived from the readout structure. Dimensional sharpness is finite (ambiguity about 1/M); at N=4 the third dimension does not crystallize - the few-body (baryon) domain is cut out as a separate subject. Japanese and English full texts included.
A finitist, relational axiomatization of physics: quantum theory and a relativity sector derived from four axioms about group-valued relations and closure. Contributions to the foundations of quantum mechanics (Born rule derivation, quantum reconstruction, local tomography, complex field selection), the emergence of time and causal structure in discrete networks, discrete Noether theorems, lattice gauge structure, and the Schrödinger equation derived by two independent routes. Every claim is graded in a Claim Ledger; every numerical claim is machine-verified with archived scripts; negative results are reported at full strength. What this paper does The universe is taken to be constituted by relations rather than objects. Stable relations form closures (self-consistent cycles) whose holonomies carry the physical content. Four axioms — primacy of relations (group-valued, edge-borne), closure stability, chirality from order, and observation as bidirectional group exchange — yield, at explicitly graded and conditional levels: Quantum sector Exchange-completeness theorems for finite label groups: abelian at all cycle ranks; non-abelian on the single cycle (Wilson readouts plus class-sum convolutions generate the full matrix algebra); the multi-cycle non-abelian case on the operational-equivalence quotient, with the obstruction completely characterized via class-preserving outer automorphisms — and the quotient's necessity now stratified by the source of word blindness (see New in Version 24.0). The Born rule, proved for finite abelian networks from a strictly weaker relational readout premise (statistics of realizable effects depend only on relational content); non-contextuality has this single source; additivity is basis-relative and declared as such. The Schrödinger equation by two routes of unequal premise grade, maintained side by side: a spectral route — the action operator is a weighted Cayley-graph Laplacian in the Pontryagin dual basis, the derived Gibbs state's unique canonical (modular) flow yields the Schrödinger equation with Hamiltonian equal to the action operator, no continuum limit, zero conjectures consumed, the single non-theorem step one flagged identification; and a Madelung route — physical and mechanism-rich (probability density derived from constraint gaps), valid exactly on the domain of the pseudo-continuity conjecture, with its full premise bill stated (continuity equation, Wallstrom circulation condition, m, V). A dedicated comparison subsection gives the line-by-line premise accounting and the machine-measured agreement region. Complex field selection: ℂ selected over ℝ and ℍ within the GPT comparison class in the covered sectors, with a second independent route from chirality (Frobenius–Schur dichotomy, quaternionic double exclusion); "uniquely selected" expressly not claimed. Positivity as pairing: all five in-text positive-semidefiniteness theorems traced to one mechanism (M = B†B with native factor); the Kraus form and complete positivity derived rather than assumed. The Noether sector on Noether's two theorems exactly: Noether I recovered on the holonomy action (compact Lie realization); Noether II identified as present (Bianchi as horn-filling, Schläfli as its Regge form); selection rules recovered for finite groups; the GNS bridge proved non-vacuously on the full exchange algebra; the sole potential extension — conservation as the kernel of exchange — isolated as a registered conjecture, consumed nowhere. A local–global representation theorem: six generative quantities (closure invariant, flux quantization, Weyl phase, non-abelian Stokes, orientation visibility, terminal normalization) are evaluations of cohomology classes against closed chains — one syntax, six exact machine probes. The gatekeeper no-go theorems registered and discharged: Kochen–Specker (the framework posits no global valuation — outcomes are resolutions of constraint gaps, and basis-relative additivity blocks state-independent variants), Pusey–Barrett–Rudolph (the statistical envelope is over resolutions yet to happen, not values already held; machine witness: identical configuration ensembles with opposite relative phase have opposite statistics), Wallstrom (registered as an explicit input, structurally discharged in the finite-abelian sector), Hegerfeldt (the Lieb–Robinson exponential tail identified as the only claimable front form for the continuum limit), Frauchiger–Renner (cross-agent certainty transfer presupposes kernel-free record transport, exactly what exchange-incompleteness excludes), and no-cloning (inherited in the covered sectors); Conway–Kochen and Colbeck–Renner noted as concordant. Relativity sector Time from finite capacity plus genuine randomness: the staging theorem (Θ(d) dependency depth, unbounded chains via the Moore bound; the quantitative failure law conditionalized on measure-level independence with exact minimal witnesses); the temporal quantum defined combinatorially, no Hamiltonian input. The speed of light grounded in an exact discrete dependency cone (combinatorial, no quantum input), the Lieb–Robinson bound consumed at cross-check grade only; a boost no-go theorem reducing Lorentz invariance to named statistical conditions, with the Collins fine-tuning objection recorded as a gate. The trace-descent theorem group: the growth-history measure forced onto the Mazurkiewicz trace quotient; chain reflection positivity at every depth with completely explicit spectrum. An equation sector of nine verification families (Schrödinger, Dirac, Maxwell/U(1), canonical commutator, Yang–Mills SU(2), Regge calculus, Klein–Gordon, Noether loop, Cartan curvature), where every machine-precision claim is the shadow of a proved backbone lemma — including the factor ½ of H = ½L + V, identified as the Wilson Fourier coefficient fixed by the action theorem. Four experimentally testable predictions, including a discriminating flat-versus-sigmoid visibility protocol. The Standard Model gauge group, ħ, and G remain open, with attack routes registered. How to read Three entry paths are marked in the document. Path A (derivational core): axioms → group structure → Noether sector → least action → Born rule → positivity. Path B (physical mechanisms): constraint propagation → time as determination → wave function as gap envelope → fields. Path C (scope and limits): the Claim Ledger, then Discussion and Open Problems — for the reader who wants to know what is claimed and what is not before reading anything else. The Claim Ledger is the single authoritative register of claim status; wherever body text sounds stronger, the Ledger governs. New in Version 24.0 (Version 23.0 is the immediately preceding public release; this version supersedes the internal V12–V15 and V17–V19 series, whose numbers are not used externally.) Version 24.0 resolves the structure of the multi-cycle non-abelian obstruction — the sector's longest-standing open front — following an external research exchange with three rounds of independent re-verification (fourteen core figures, three commutant instruments implemented independently, exact integer arithmetic on the obstruction side) and two rounds of counter-review that corrected one quantifier on each side: A dichotomy of the multi-cycle obstruction (new remark). Word blindness on multi-cycle networks has two identified sources, jointly covering every tested case, with opposite behaviour as cycle rank grows. Type I (global; graded): a class-preserving outer automorphism yields a commutant element by an argument free of cycle-rank reference; its non-triviality holds at theorem grade for b₁ ≥ d(G) via generating tuples, is monotone in b₁, and the threshold is measured at b_σ = 2 on the tested Wall group (at b₁ = 1 the permutation is the identity — no universal claim is made, and the existence of groups with threshold above 2 is registered open). Type II (local; measured): Out_c trivial but fusion fails on a proper subgroup, the blind pairs sitting in non-generating sectors — the same mechanism that defeats Wilson generation; this blindness heals: on every verified group (A₄, SL(2,3), A₄×ℤ₂) the full commutant collapses to trivial at b₁ = 3 by exact computation on two independently implemented instruments, although word readouts remain blind — the shift primitives separate, above a threshold, and the natural explanation (tuples becoming generating) is refuted by the persistence of blind blocks. Three boundaries at full strength: no exhaustiveness proof (a third source is not excluded); no general healing theorem (the verified groups all heal at 3, cause unidentified); the verified Type II sample is narrow. The quotient proposition restated in stratified form. The operational-equivalence quotient's necessity is now part of the proposition's own statement, in three strata: necessary where Out_c ≠ 1 at every b₁ ≥ b_σ(G), with b_σ ≤ d(G) at theorem grade; one-dimensional at b₁ = 2 on the fusion-type groups (verified cases); trivial at b₁ ≥ 3 on every verified fusion-type group and at every tested b₁ for word-separation-complete groups (a fourteen-group scan of orders ≤ 31 found no third source). Pre-registered validation preceded the restatement: a stratification-exhaustiveness scan (pass), a new-witness search (no new witness in range, downgrade clause applied as pre-registered), and a zero-conflict audit of the three strata against all archived data. The residual open problem is now precisely located. What remains of multi-cycle completeness: the rare Out_c ≠ 1 groups (54 in the exhaustive range through order 96) above their measured thresholds, the low-b₁ fusion stratum, a general healing theorem, and whether a threshold above 2 occurs. The multiplicity-freeness conjecture's mechanism sentence is corrected with the two regimes stated separately (at b₁ = 2 the shifts do not separate — the blind-pair transposition commutes with every primitive, integer-exact — and reduce the commutant to
This Theorem extracts the phase-bearing 2-plane from the T23–T25 ambient structure and identifies the scalar phase defect Δφ(τ) with the rotational coefficient of the induced collapsed operator on that plane. Two supporting lemmas are established. Lemma 26.A performs the branch-axis decomposition: the T22 triplet T₃ splits canonically into the branch-selected zero-weight axis span{u₃} and the transverse phase plane P⊥ = span{u₂, u₄}. Lemma 26.B establishes the readout kernel: every Hermitian operator on P⊥ decomposes uniquely in the real basis {I, σ_z, σ_x, R}, and the U(1)-holonomy readout ρ_phase annihilates I, σ_z, and σ_x, retaining only the coefficient μ of the rotation generator R. The main theorem establishes that the collapsed branch object C_A induces an operator Q⊥ on P⊥, the holonomy readout factors through P⊥, and (after fixing the T15 normalization convention) Δφ(τ) = μ exactly. The coefficients α, β, γ drop out of the readout regardless of their values. Status: Parts (1)–(3) solid by direct structural argument. Part (4) inherits the conditionality of T20; the identification of Δφ(τ) as the scalar shadow of the collapsed kernel is a load-bearing assumption carried forward from T20. Dependencies: T15, T18, T19, T20, T23, T24, T25.
What quantum mechanics calls "observation" has lumped together three interactions of different character: one that switches the discrete state of the partner, one that leaves the partner's state unswitched and shifts only the internal phases, and one that severs the mutual phase relation. This paper places the substance of observation in the second, the readout of the interference of a phase difference, and adopts only three premises. (P1) The distance between two parties is set by the ratio of a shared viewing angle to the partner's size; only ratios are readable and the absolute scale is not. (P2) An interaction is a single relation in which emission and absorption hold simultaneously; the speed of light appears on the readout side as a conversion factor between local clocks. (P3) The phase of a relation carries a harmonic order n that depends on the kind of wave. From these, the fact that a probability is the product of an amplitude and its conjugate (the form of the Born rule), the two-party correlation E = ∓cos nθ, the upper bound 2√2 of the CHSH quantity (the Tsirelson bound), and no-signaling follow in a few lines. The normalization that turns this into a probability is the division that discards the size ratio, i.e. the choice of the η = 0 section. As a consequence, the bound is 2√2 independently of the kind of wave, and the detector angle giving the maximal violation shrinks in proportion to 1/n (45° for spin-½, 22.5° for photons, 11.25° for gravitational waves). For a localized relation that is a sum of harmonics the correlation is the autocorrelation of the pulse shape; the bound holds for any weights, equality is attained only for a single harmonic, and a rectangular pulse reduces to a local hidden-variable model with S ≤ 2. Japanese original (authoritative) and English translation, with TeX sources, PDFs, five figures (Japanese and English labels) and the figure-generating program.
On a homogeneous space M = Sp(2n,R)/U(n) carrying a compact group action, a kinematic no-go establishes that no quantity derivable from the manifold, the metric, the base point and the group can label the members of a certain triple. This paper measures in the layer that premise does not reach: the layer in which a clock and a system are separated and one is conditioned on the other. The central result is structural — the triple and the clock are one consequence, not two. A single input, an irreducible of the compact group occurring with multiplicity m, forces at once that the sector is Sym2(Cm), that the commutant is u(m), and that the available clock is the Cartan of su(m) inside it, of dimension m−1. At m = 2: three lines, a four-dimensional commutant, a torus of rank two, a clock unique up to scale, weights {+1, 0,−1}. At m = 1: one line and no clock at all, measured in the same block, as the arena’s own falsifier. The clock is not assumed periodic; its compactness and minimal period are measured, and its overlap function, its three orthogonal readings and its sensitivity to the clock-state phase freedom follow in closed form from covariant-POVM formulae. With a distinguished tangent direction admitted, one exact measurement lifts the degeneracy: the projection of the adjoint square, restricted to the sector, is exactly diagonalwith integer equispaced spectrum. Two limitation theorems then show that admitting the direction buys classes but not one invariant direction, and that no spectral invariant can supply the one remaining bit. A family index is therefore a reading: the value a line takes on a clock derived from the same multiplicity that produced the line, with a residual bit gauge-fixed relative to an orientation the construction does not supply.