O33 — A Finite Spectral Architecture for Mass-Square Splitting and Mixing

Exact Doublets and Protected Sectors from Weil Admissibility

O33 is a mathematics-first paper in the Spectral Admissibility Programme. It studies the exact Fourier-mode filtration generated by the O12 three-character fingerprints and its operator-theoretic completion inside the finite Weil representation.

Main results

• The cumulative mode set is the explicit cyclic sumset K_n=c[-(n+1),n+1]+{0,+/-u}+{0,+/-v}.
• Boundary porosity is governed exactly by the parity of the cyclic spacings of the nine-point centre set.
• The complete stabiliser is expressed through the multiplicative symmetry group A_K; an orbit-divisibility sieve and uniform exact C_4 and C_6 families sharpen the generic value {+/-1}.
• An unconditional almost-periodicity lemma and a run-correspondence argument prove the rigidity theorem A_K = {+/-1} whenever every run and gap of the mode set is longer than the run count; completeness of the exceptional orders is reduced to the degenerate boundary regime.
• A third uniform family lives at the near-saturation end of that regime: a four-point complement {w, w+1, -w-1, -w} carries C_4 exactly when 2w^2+2w+1 = 0 (mod q).
• The degenerate regime hosts an infinite Gaussian-box C_4 family Q_r(i) generalising the depth-zero square. A fourth-moment theorem proves that every proper direct or wrapped Gaussian box has exact stabiliser C_4, while every free four-point complement orbit also has exact stabiliser C_4.
• An exhaustive audit through q=151 classifies all 75 exceptional sets into boxes, free four-point orbits, and Eisenstein C_6 sets, with no C_8 found at q=193,241,257. Completeness remains open, as does the uniform top-autocorrelation-level proof for the Eisenstein family.
• The porous depths of a uniform admissible block converge to an explicit limiting point process: count law (3/20,7/24,9/40,5/24,1/8), mean 28/15, piecewise affine intensity with breakpoints 1/10,1/8,1/6.
• Weyl quantisation realises the induced Mackey carrier internally in End(V_rho).
• Generic principal-series pairs carry canonical M_2(C) multiplicity algebras.
• Harper dynamics produces the universal finite split 1+/-q^{-1/2}.
• The eliminated residue produces a second axis exactly when a short boundary character sum over at most four hole pairs is non-zero: a Galois separation theorem removes the cyclotomic phase determinant, and the vanishing locus of the boundary sum is completely classified into antipodal-pair, cube-triple, and double-pair mechanisms.
• The odd Weyl-symbol sector is a protected doublet reservoir under the deposited parity-even operator algebra.

The physical discussion is deliberately calibrated. The paper derives finite two-level spectral and mass-square kinematics, not Dirac masses, Standard-Model generations, hypercharge, or a matter-sector Higgs mechanism.

Status

Published mathematics-first paper (v1.4) with the external-primary-reference audit complete. Since v1.1, the multiplicative stabiliser is proved to be {+/-1} outside a degenerate boundary regime; v1.3 closes the mixing criterion through the Galois separation theorem and the complete classification of the boundary-sum zeros. Version 1.4 adds the Gaussian-box family, refined degenerate mechanisms (withdrawing the earlier exhaustion reading and correcting the q=37 four-point adjacency sentence), and the exact asymptotic point process of porous depths. Branch 1.5 carries an unpublished draft adding unconditional exact stabiliser theorems for every proper Gaussian box and every free four-point complement orbit. The completeness of the three refined mechanisms and the Eisenstein top-level-set theorem remain open.

Reproducibility

scripts/porous_asymptotics.py machine-certifies the exact asymptotic law of porous depths: the 240-triangle rational polygon decomposition, the count law (3/20,7/24,9/40,5/24,1/8), the piecewise affine intensity, the total mass 28/15, and the conditional mean 491/10080. Standard library only; exact rational arithmetic; runs in under one second.

Compilation

bash compile.sh

Output: out/SpectralO33.pdf.