A Finite Spectral Architecture for Mass-Square Splitting and Mixing: Exact Doublets and Protected Sectors from Weil Admissibility

Main results

• Closed mode-set formula: the cumulative depth-$n$ mode set is the cyclic sumset $K_n=c[-(n+1),n+1]+\{0,\pm(c_2+c_3)\}+\{0,\pm c_3\}$; in particular $K_n=-K_n$.
• Spacing-parity criterion: before saturation, a flanked hole exists iff some cyclic spacing of the nine-point centre set equals $2n+4$; odd spacings never produce one.
• Complete Weil stabiliser: $\mathrm{Stab}(W_K)=\{\left(\begin{smallmatrix}a&0\\t&a^{-1}\end{smallmatrix}\right): t\in\mathbb{F}_q,\ a\in A_K\}$, of order $q|A_K|$. The generic value is $A_K=\{\pm1\}$; exact exceptional families give $C_4$ (from $a^2=-1$) and $C_6$ (from $a^2+a+1=0$), with enlargements to $C_8$ and $C_{18}$ excluded analytically.
• Additive rigidity (v1.1): every $a\in A_K$ satisfies $|K\,\Delta\,(K+a)|=2m$ (unconditional almost-periodicity); if every run and gap of $K$ is longer than the run count $m$, a run-correspondence permutation gives $ma\equiv t$, $0<|t|\le m$, and a Diophantine descent forces $A_K=\{\pm1\}$. Exceptional stabilisers are thereby confined to the degenerate boundary regime.
• Near-saturation $C_4$ family (v1.1): a four-point complement $K^c=\{w,w{+}1,-w{-}1,-w\}$ carries $A_K\cong C_4$ exactly when $2w^2+2w+1\equiv0\pmod q$ (solvable iff $q\equiv1\pmod4$), with generator $(w{+}1)/w$; realised by deposited blocks at $q=73,109,173$. For any $|K^c|=4$, $|A_K|$ divides four and the order-four case is a single free orbit.
• Gaussian-box mechanism (v1.4, exact stabiliser v1.5): for $i^2=-1$ and $q>8r^2$, the box $Q_r(i)=\{x+iy:|x|,|y|\le r\}$ is proper with $C_4\subseteq A_K$ and is realised by admissible blocks at depth $r-1$ for $r\le4$; near saturation, wrapped boxes with $q<8r^2$ also occur. A fourth-moment theorem (v1.5) proves $A_{Q_r(i)}=\langle i\rangle$ exactly for every proper box, direct or wrapped, so no case is left to individual certification. The earlier exhaustion reading of the family statements is withdrawn, and the false claim that all audited four-point complements are adjacent is corrected: at $q=37$ the orbit $\{3,18,19,34\}$ has one adjacent pair and two singleton gaps, and all three centrally symmetric gap profiles occur.
• Asymptotic porosity law (v1.4): the porous depths of a uniform admissible block converge to an explicit limiting point process; the number of porous depths equals $0,\ldots,4$ with probabilities $(3/20,7/24,9/40,5/24,1/8)$, its mean is $28/15$, the intensity is piecewise affine with breakpoints $1/10,1/8,1/6$, and the conditional mean normalised depth is $491/10080$; the exact rational-polyhedral computation is machine-certified by an archived script in the paper repository.
• Internal Mackey carrier: $\mathrm{End}(V_\rho)\cong\mathbf{1}\oplus\mathcal{D}_+\oplus\mathcal{D}_-$ as a conjugation $\mathrm{SL}(2,\mathbb{F}_q)$-module; the induced carrier is already part of the bosonic operator algebra, not an external addition.
• Hecke doublets: generic principal-series pairs $\{\theta,\theta^{-1}\}$ carry canonical multiplicity spaces $\mathbb{C}^2$ with full commutant $M_2(\mathbb{C})$; the even-sector commutant is $\mathbb{C}^4\oplus M_2(\mathbb{C})^{(q-5)/4}$ for $q\equiv1\pmod4$ and $\mathbb{C}^2\oplus M_2(\mathbb{C})^{(q-3)/4}$ for $q\equiv3\pmod4$.
• Universal Harper split: the finite Harper operator has non-scalar reduced density in every generic doublet, with normalised eigenvalues $1\pm q^{-1/2}$ set by a Gauss sum.
• Exact mixing criterion: the residue $E_K=-P_KT_c(1-P_K)T_cP_K$ supplies a second axis in the $\theta$ factor iff the cyclotomic phase determinant $\Delta_{\theta,c}(K)=C_\theta(K)\overline{\tau_{\theta,c}} -\overline{C_\theta(K)}\tau_{\theta,c}$ is non-zero; a canonical $\mathfrak{su}(2)$ frame then follows.
• Galois separation (v1.3): $\Delta_{\theta,c}(K)\ne0\iff C_\theta(K)\ne0$ for every character of order greater than two: the cyclotomic automorphisms fixing $\zeta_{q-1}$ and moving $\zeta_q$ make a non-trivial phase alignment impossible.
• Complete zero classification (v1.3): the hole pairs are geometrically bounded by four, and $C_\theta(K)=0$ occurs exactly through an antipodal pair $\theta(h_1/h_2)=-1$, a cube triple (ratios equal to the two primitive cube roots), or a double antipodal pair; each mechanism is an explicit character condition on rational functions of the block parameters.
• Parity protection: the unital star-algebra generated by the deposited projectors and Harper dynamics is parity-even and has exactly zero projection onto the odd sector $\mathcal{D}_-$, which is itself a pure doublet reservoir for $q\equiv1\pmod4$.

Arithmetic exceptions and computational certification

The exact formulas are guarded by three audit layers: direct BFS enumeration at $q=29$ and the eight deposited O-series cases; a seeded sweep of 1,170 blocks and 19,153 pre-saturation depth cases over 40 primes through 307 with no spacing failure; and an exhaustive normalised audit at $q\in\{13,17,29,31,37,41,43,53\}$.

The stabiliser audits find only the orders two, four, and six: exhaustively for all primes $13\le q\le73$, and in 12,234 divisor-sieve survivors of a seeded exact audit through $q=499$. The uniform $C_4$ and $C_6$ families are verified for every applicable prime through $q=997$.

The zero classification (v1.3) is verified by two independent exact computations on every instance: exhaustively over all normalised blocks and pre-saturation depths for $7\le q\le73$ and by a seeded sweep through $q=311$, totalling 31,166 porous cases and 1,069,192 exact zero tests with no mismatch. The zeros split into 3,620 antipodal-pair, 1,030 cube-triple, and six double-pair factors; porosity alone therefore does not imply mixing, but every failure mechanism is now classified.

The degenerate regime (v1.4) is audited exhaustively through $q=151$ with exceptional sets deduplicated as exact sets: $75$ sets, $62$ of order four and $13$ of order six, classified without remainder into $52$ Gaussian boxes, $10$ non-box four-point orbits, and $13$ Eisenstein $C_6$ sets; targeted order-eight searches at $q=193,241,257$ cover 4,540,161 pre-saturation cases and find none. The original pre-registered kill-switch fired on its mechanism branch (the Gaussian boxes); the residual, post-refinement kill-switch stayed silent. The asymptotic law is machine-certified by the archived exact script and matched by exact enumerations at $q=4001$ and $q=8009$.

Interpretation (calibrated)

The proved structure is a parity-graded operator architecture: an even sector carrying a universal finite split, arithmetic mixing frames indexed by the filtration, and two-level mass-square kinematics; and an odd sector of protected doublets. It is structurally tempting to compare the even sector with bosonic symmetry breaking and the protected sector with a missing chiral datum — that comparison is interpretive only.

The paper derives no Dirac masses, no absolute scale, no Standard-Model generations, no hypercharge, and no matter-sector Higgs mechanism. The residue $E_K$ remains of $-M^*M$ type, and the splitting scale $q^{-1/2}$ vanishes as $q\to\infty$ without a separate scaling prescription. A multiplicity doublet in an operator algebra is not by itself a weak-isospin doublet of fermionic matter.

Relation to the Cosmochrony programme

O33 is the arithmetic completion of the O12 filtration and uses the radial ordering perspective of projective temporal ordering only to state what the deposited operator class cannot do: the directed frontier does not orient generator pairs, so the odd Harper lift stays outside the deposited even algebra.

Downstream, the canonical $M_2(\mathbb{C})$ doublets, the Harper orientation, and the parity-protected odd reservoir are the exact objects on which the fermionic-matter doublet fronts are gated; transporting them to a matter carrier requires new structure (an ambient doubled representation and a chiral odd datum) that the paper explicitly leaves open.

Open questions

• Prove or refute that the three refined mechanisms (Gaussian boxes, free four-point orbits, Eisenstein $C_6$ sets) and the orders $C_4$ and $C_6$ exhaust the degenerate boundary regime for arbitrary odd primes. Outside that regime, $A_K=\{\pm1\}$ is proved (v1.1); through $q=151$ the audit finds nothing else (v1.4). The unit groups of the Gaussian and Eisenstein integers are an organising principle for this question, not a completeness theorem.
• Prove uniformly that the top autocorrelation level of the Eisenstein $C_6$ sets is $\{\pm1,\pm\omega,\pm\omega^2\}$; the stabiliser value itself is proved, and the level identification is audit-certified only.

References

Jérôme Beau. A Finite Spectral Architecture for Mass-Square Splitting and Mixing: Exact Doublets and Protected Sectors from Weil Admissibility. Preprint. doi:10.5281/zenodo.21360722