Paper 9: Logic Waves and Half-Wavelength Censorship — The Odd-Harmonic Ladder, Amplitude-Free Coherence Conditions, Kinematic Stability of Composites, and the Existence Ceiling for Single Entities
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).
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).
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.
Observational note (thought-experiment style; not a proof or claim paper). A sequel to the preceding note, which defined a single particle-like state as a phase-bearing local mode on the subjective geometry S^4(R_U) with spread W_a = 2 R_a and deferred the multi-particle case. Here we treat the minimal case: two particle-like states with the same spread phase, on a closed one-dimensional relational setting (period-1 phase line, a single great circle of S^4). Representing each particle as a rectangular standing wave (odd harmonics only), the n-th harmonic amplitude of the superposition is A_n = (2/n) cos(n pi f), where f is the relative position in fundamental wavelengths, and the total squared norm (energy-like quantity) is the triangle wave E(f) = (pi^2/4)(1-2f) for 0 <= f <= 1/2 (the autocorrelation of a square wave). Within the continuous f, discrete feature points appear: fusion (f=0), orthogonality / independent coexistence (f=1/4), and complete cancellation (f=1/2). Depending on whether the superposition is symmetrized or antisymmetrized, complete overlap (f=0) is allowed (boson-like) or makes the field vanish, i.e. exclusive (fermion-like). For unequal frequencies, interference arises only from shared odd harmonics (the irreducible ratio r/s with both r,s odd, i.e. matching 2-adic valuation); without sharing, interference vanishes and placement is free. All results are presented as observations of structural correspondence (with standing-wave quantization and the exchange symmetry of quantum statistics); no derivation of quantum statistics or the spin-statistics theorem is claimed. Notably the existence of the three feature points is universal to any half-wave-antisymmetric (odd-harmonic) profile, while only the linear (triangle-wave) envelope is specific to the square-wave (1/n) choice. This record contains Japanese and English versions (Markdown, LaTeX, PDF) and one figure.
A numerical experiment that simulates and observes the de Broglie relation lambda=h/p inside a single-source double-slit classical wave model. An electron is modeled as a spatially localized wave built from an equal-amplitude sum of odd harmonics (N>1). The central result is NOT the value of h (which is a near-tautological self-consistency, since lambda0=h/p is the input and the resonance lattice is dense ~1e-19 m): it is (i) that a highly localized odd-harmonic classical wave can survive two-slit interference as a sharp single peak only under an inter-harmonic alignment condition, and (ii) that the observed central fringe spacing DeltaX = lambda' D/W is set by the fundamental wavelength and is independent of the harmonic count N (N controls only the localization sharpness ~1/(N+1)). Scope is stated explicitly: this is neither a first-principles derivation of p*lambda=h nor a derivation of a Delta_x*Delta_p ~ h uncertainty relation. Each electron is given independent Monte Carlo draws of source position (cos^2, +/-lambda0/2) and initial wavelength (cos^2, +/-1%), with interference, alignment and exclusion recomputed exactly per trial. Bilingual (Japanese + English) drafts with source programs.
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.
Paper 16 of the series Dual Geometry of Wavelength Space and Frequency Space. The system supplies path amplitudes with no free parameters; what is not fixed is the aggregation rule that bundles them into probabilities — what standard theory installs as the Born rule, by axiom. This paper does not select the rule; it gives an exact reduction of the selection problem. The historical proximity 0.0007 between the derived and counting measures is shown to be an artifact of two-outcome compression (a one-line lemma; the divergence jumps to 0.24 at s=11). Derived measures do not classicalize to counting measures (exhaustive at s=9/11/13), yet the two share their support exactly. Interference granularity is not reducible to convention: the mirror parent has W=0 exactly under channel-consistent aggregation while decaying perfectly normally at configuration granularity (W'=768=W'(A)); at m=4 the unit of cancellation refines to single channels; the occurrence condition is reduced to the four-value lock — algebra that is the object of proof. Conditioning requires the branch-channel stratum of the readout hierarchy (pairs indistinguishable by intensity fringes have different P(X)). Event structure is exactly one bit (one-shot vs sequential), completing a 2x2 table. Main theorem: the inventory-defined candidates organize without duplication into the product granularity x event bit x conditioning, each axis independent — with a decision table of experiments computable inside the model (two executed before publication). Under the reduction, the Born rule has the status of an effective law to be merged with in a suitable limit (convergence / suppression below statistical censorship ~1/sqrt(N) / prediction). Reviewed in two rounds under a two-party independent verification protocol. Japanese and English full texts. No physical identification is made.
A two-mode quantum harmonic oscillator is reconstructed from the axiom that a single distinction is the primitive degree of freedom. The construction forces an internal reference witness, Euclidean geometry and a positive irrational remainder δ=2−1. The binary count remains I=1, while its complete account retains the identifying relations and succession that the count alone omits. Symmetry conservation and recoverable closure establish persistence, with a next lawful step always available. Circular closure supplies the exact area increment g=(π−2)/4. Its accumulation forces escapement: an additional complete unit is admitted with its witness and origin retained. Applying the primitive construction to this unit supplies the joint Archimedes trammel and its three-dimensional family. The completion event fixes the relative phase and recoverable shadow amplitudes. Subsequent comparisons complete the cell account, whose positive irrational remainder Ω=1−π/(32) permits further admission. Representing complete accumulated area as squared ladder weight uniquely fixes wn2=(n+1)g and the dimensionless action coefficient ℏ∗=g. Source and child quadratures satisfy the canonical relations on a common invariant domain. At escapement, the completed-unit contribution restores the commutator term missing from the source remainder. The Hamiltonian Htick=(πℏ∗/2)(nα+nβ+Id) has a complete discrete spectrum; ni counts oscillator occupations. Its unitary evolution reproduces the trammel's quarter-turn, and an explicit Hermite transform supplies the equivalent Schrödinger representation. The geometric birth comparison becomes an entangled source–child state. Its complete amplitudes and retained context recover the producing account. Each finite generated retained history admits a faithful quantum memory representation satisfying structural record coherence with sequential weak value one.