Generation of Fermionic Structure Is Induced, Autocatalytic, and Pair-Correlated: Assumptions and Consequences of a Universal Inelastic Map Acting on Waveforms Alone
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.
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.
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.
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 study asks which structures can be derived from a single finite relational system to which no prior physical meaning, coordinates, or time are assigned. A selector-free relational lift and its minimal self-sustaining orbit generate an intrinsic separator order and grading, providing a directed order axis without an external clock. Boolean-diamond incidence defines transverse locality, and five-K4 incidence produces an exact S3 minimal persistent complex whose canonical continuation yields a four-dimensional associated realization. An explicit finite witness gives a nonzero commutator between longitudinal incidence and transverse locality, ruling out a simple product of fixed space with independent time. A complementary stable-quotient readout contains a refinement branch that closes as a one-dimensional metric-graph continuum. Combined with the S2 fiber of the standard S4 representation, this gives a regular spatial sector locally of the form (0,1) x S2(R). The S2 Laplace spectrum provides a branch-independent nonzero rest gap for every l >= 1; under the stated internal second-order wave representation this gap is read as a rest-mass parameter. Gravity is represented by an exact finite source-to-field map using the Moore-Penrose Green operator, and the regular three-dimensional branch yields the local 1/r^2 acceleration exponent in the weak-static regime. The same relational development also supplies three distinct internal Hilbert representations. Conditional on the full orthogonal-projector event structure and standard noncontextual probability consistency, Gleason's theorem fixes the Born form in the spatial representation. The incidence representation is noncommutative and yields the Robertson uncertainty relation, while procyclic reaction gives exact discrete unitary dynamics by Haar-measure invariance. No canonical identification of these Hilbert spaces, unique continuous Hamiltonian, measurement-update rule, spin-1/2 structure, or Bose/Fermi statistics is claimed. Complete derivations and branch audits are provided in a Supplementary Formula Atlas.
As the final paper of the series, this paper constructs the minimal system in which two composites coexist — the twin minimal space model — and examines how much of the behavior that looks quantum mechanical is derived as a reading of hierarchy interfaces in a discrete relational model that assumes neither amplitudes nor continuous time. External face of the stable species: The handles (hairs) by which a composite is observed from outside divide into three ledgers: the face fundamental wave ν0 = 1/(2√s) (the dominant line of the self-container spectrum), the parent-record line νobs = |k|, and the energy-like norm R' = √s. The exact identity ν0R' = 1/2 (the zero point) links the first two, and R' never appears as a line — it is readable from outside as the spacing 1/R' of the odd comb (the particle-level realization of the record theorem). The external face is a minimal-uncertainty quasi-rectangular wave packet that carries a built-in odd comb of spacing 1/R' and is localized within an extent 2R' (half-width × fundamental wave = 1/2). Construction of the twin system: The initial value Ω0² = 2ν1² = 18 is an even label and, by the classification theorem (Paper 5), cannot exist as a single state — being born as a pair is forced by parity (the first event requires no dynamics). The allowed lattice separations of the twins (each s = 9) are quantized to |Δ|² ∈ {2, …, 20}, and |Δ|² ≡ 1 (mod 4) is entirely forbidden. The relational classes (21 under B4 × swap) are completely separated once the branch (cos/sin) channels are read in addition to the fringe intensity data (intensity-only fringe data degenerates to 18 kinds as ordered data, and to 11 kinds under the operationally available unordered reading) — the scale marks of space are born as chord sets of equal-frequency shells. The exact form of measurement: Inspecting the two conditions for measurement (affecting the system; detecting the change), the detection of change requires knowledge of the prior state and regresses infinitely; in this model, however, the record of the prior state is absent by censorship, and the λ ledger is append-only (the monotonic increase theorem, Paper 6), so the regress is structurally cut. Measurement = an append to the record (absence → presence); "collapse" is the appearance of a record, and its irreversibility reduces to the same single bit as the arrow of time. Computing the first decay of the twins explicitly, the appearance of the reference wave (s = 1) promotes the information class of the record from interferometric to holographic — the measuring apparatus is generated by the event itself. We also show that the final-state statistics depend on the choice of measure (one-shot equal weight over configurations 60/61 versus sequential conditional counting 396/403) — measure bifurcation, with an exact difference of +0.00098. Layer resolution of entanglement: The self-spectrum of a single fragment lies on all-components-even vectors and is therefore identically invisible in the logical-wave (odd) sector — only counts and relations can be read (the relational-only readout theorem). The decay of one twin instantaneously rewrites the branching statistics of the other (total variation 0.774), but this shift does not depend on the twin separation at all, and the influence does not propagate through the space of the child layer — instantaneity is not a speed but a mismatch of levels. Since order and attribution do not exist in the final record, operational no-signalling follows. An entangled pair, being an even label, cannot become a single particle at the layer above (parity protection). The same map applies to two-path interference of a single wave packet. Vertex deficits and the two hidden axes: An exhaustive inspection of the record continuity rule across an event (that the parent's identity line persists in the cross spectrum of the daughter system) derives, as the persistence condition, a conservation law of position vectors v5 ± v3 = ±v9 (a momentum analogue), which holds for parents in the (2,1,0,0) orbit but fails for parents in the (1,1,1,1) orbit by exactly one unit (one axis / one quantum). Furthermore, in the bookkeeping of the excitation degree ε = (−1)Σ|k| there exists a second one-bit deficit independent of this one (specific orbits at s = 7, 9). The visible four-axis ledgers demand two invisible directions — R on the curvature side and Q on the bare-parity side — and intermediate displacements along the invisible directions, being unrecorded, are permitted (the licensing principle for virtual displacements). This licence virtually unlocks two-body splittings that are forbidden in the visible ledgers, and decomposes every decay into sequences of cubic vertices (multiple paths to the same final state). Finally, we organize the correspondence between the obtained structure and standard theory as a transport principle (a theorem is transported whenever the counterparts of its premises are verified), and state explicitly the candidate answers this model gives to questions that remain postulates on the standard-theory side (the measurement problem, the location of the cut, the problem of time, dimensionality, the measure), together with falsifiability (gravitationally mediated entanglement experiments, etc.). Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus four figures (PNG). v0.3 (2026-06-12): Corrects Sec. 3.2 and the abstract: the twin pairs decompose into 21 relational classes under B4 × swap (not 18); intensity-only fringe data degenerates to 18 kinds (ordered) / 11 kinds (unordered, operational), and complete separation is recovered by the branch-channel readout. All other results are unchanged.
We develop the quantum layer of the finite relational algebraic substrate: observation as the interaction of a Subject and an Object, phase-coherent cyclic subsystems of a common finite Carrier shell. Four identifications carry the construction, each a derivation rather than an interpretation. The wave function is phase: pure states are windings of the Object's cycle, selected by stationarity under the drive, with E=hf an identity. The probability amplitude is Object-to-Subject projection onto a forced orthogonal basis, so that completeness, the selection rules, and the Lüders update are finite character theory rather than postulates. Probability is a coincidence count in the Subject's ledger, the cyclotomic ring Z[i] of the shared quarter-turn core. This derives the Born rule within the coherence horizon, with the quarter-turn core supplying the canonical complex sector of the amplitudes. Composition is conservation: the single drive conserves the relative offset of a pair, synchronised orbits are the Bell states, and the framework saturates the Tsirelson bound exactly, S=2(ζ8+ζ8−1)=22, while reproducing the network behaviour of the real-versus-complex discrimination. The same conservation makes gravity an entangling channel. Its distinctive parameter-free signature is equal force but unequal entanglement: a source gravitates by its full mass yet entangles only through its coherent fraction, and entanglement forms at the Newtonian rate under the complementarity V2+C2=1. Every claim is https://github.com/gamayos/frc-numerics/tree/main/22-quantum, verified by exact finite arithmetic in three worked configurations, and the experimental confrontation closes in a single falsifiability ledger on one committed carrier scale.
In this paper, we verify the existence conditions and stability of composite states (clusters consisting of many cells) in the reciprocal dual model by explicit construction of interference patterns. First, consolidation of the logic wave hypothesis: the per-axis value of the counting condition, |ki| + 1/2, is an odd-harmonic sequence with the zero point 1/2 as its base, and a spectrum containing only odd harmonics is the fingerprint of a binary symmetric signal (square wave, logic wave). That the phase wave carries no amplitude is not an approximation but a coherence condition demanded both by the theorem layer (the integer partition structure) and by curvature consistency. Information resides in edge positions, i.e., in the relative phases of the harmonic ladder, and a parallel translation of a square pulse is exactly equivalent to a linear phase multiplication on the coefficients (position = phase). Second, kinematic stabilization by half-wavelength censorship: the maximum of the curvature-induced anharmonic shift of the ladder is exactly 1/2 (the lowest mode, verified in rational arithmetic), and the censorship condition "shift ≤ half the resolution" is satisfied exactly at the limit. Without censorship, if the shift is taken as real dynamics, a composite dephases and collapses within 3–6 fundamental periods; under censorship, the continuous decay channel is kinematically closed, and the only remaining decays are the discrete channels (Paper 7). The stability of composite particles is not a dynamical accident but a quantization of the deformation space. As a by-product, we observe that the maximum dimension in which this protection holds is d = 4 (the fixing of normalization is deferred to Paper 11). Third, interference closure tests and the existence ceiling: a single cell can be constructed exactly from odd harmonics alone (width/period = 1/2 is the unique filling fraction at which the even harmonics vanish identically). The condition for odd-harmonic ladders to nest is that the scale ratio be odd (the odd nesting theorem — a wave-level rederivation of the odd-k rule of the genealogy). Computing, by exact Fourier analysis over the entire odd-s ladder, whether an occupancy structure can be certified by its own waves alone (the eye opening in the fundamental-wave band), we have established that the stable species s = 1, 3, 5 certify themselves with strong margins (≥ +0.22), the decay threshold s = 7 cannot be certified, s ≥ 9 is critical, and for s ≥ 25 certification at the composite scale disappears, while for s ≥ 49 certification fails in every band of the odd sector (verified down to band depth 31; the margin deteriorates monotonically with s). A control experiment shows that allowing even harmonics restores certification, so the closure is intrinsic to the logic wave (odd, binary) structure. As a consequence, a large content S cannot exist as a flat single entity, but only as a nesting of certifiable small entities — hierarchization is not a choice but a compulsion. Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus three figures (PNG).