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

We give an exact arithmetic and representation-theoretic description of a finite Weil filtration generated by three-character Heisenberg fingerprints. For every odd prime $q$ and non-zero total character $c=c_1+c_2+c_3$, the cumulative Fourier mode set at depth $n$ is the cyclic sumset \[ K_n=c[-(n+1),n+1]+\{0,\pm(c_2+c_3)\}+\{0,\pm c_3\}. \] This closed form turns the filtration into a union of at most nine cyclic intervals. It yields an exact spacing criterion for boundary porosity and reduces the full Weil stabiliser to the multiplicative symmetry group of $K_n$. Weyl quantisation then identifies the induced Mackey carrier internally as a direct summand of ${\mathrm{End}}({\mathbb{C}}^q)$. Its generic principal-series constituents occur in canonical multiplicity-two pairs with commutant $M_2({\mathbb{C}})$. The finite Harper operator selects one axis in every such doublet and produces the universal normalised split $1\pm q^{-1/2}$. The eliminated-sector residue supplies a second axis precisely when a short boundary character sum over at most four hole pairs is non-zero: a Galois separation theorem eliminates the cyclotomic phase determinant for every generic character, 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 itself a pure doublet reservoir for $q\equiv1\pmod4$, but it is exactly protected by the parity-even algebra generated by the deposited projectors and Harper dynamics. An orbit-divisibility sieve constrains every multiplicative stabiliser, and an unconditional almost-periodicity lemma, a run-correspondence argument, and a Diophantine descent prove that the stabiliser is $\{\pm1\}$ whenever every run and gap of the mode set is longer than the run count. Completeness of the exceptional orders is thereby reduced to a degenerate boundary regime, where a fourth-moment argument proves that every proper Gaussian box has exact stabiliser $C_4$, and a free four-point complement orbit has the same exact stabiliser. An exhaustive audit through $q=151$ finds $75$ exceptional sets, all of order four or six and all within three refined mechanisms, with no order eight at $q=193$, $241$, $257$. Finally, the porous depths of a uniform block converge to an explicit point process, with count law $(3/20,7/24,9/40,5/24,1/8)$, mean $28/15$, and piecewise affine intensity. Interpretive status. The construction supplies finite two-level spectral splitting, mixing frames, and mass-square kinematics at an operator stratum. It does not derive Dirac masses, an absolute mass scale, Standard-Model generations, hypercharge, or a matter-sector Higgs mechanism.