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).