This packet contains exact finite-dimensional algebra and finite permutation computations. It does not promote a continuum, particle-physics, or hardware claim.
Let (\mathcal H_{\mathbb Z}) be the primitive integral form of the literal 26-dimensional (W(E_6)/S_5) Hecke algebra. Primitive idempotents are lifted inside (\mathcal H_{\mathbb F_p}), and the corner dimensions directly reproduce [ C_p=D_p^{\mathsf T}D_p. ] Here the displayed (D_p) are dimension-compatible Cartan factorizations. The checked-in verifier does not reconstruct the claimed ordinary-to-modular trace congruences or prove uniqueness of those factorizations, so neither claim is promoted in this packet.
The Cartan matrices are [ C_2=\begin{pmatrix}1&0\0&22\end{pmatrix}, ] [ C_3=\begin{pmatrix} 5&1&3&0\ 1&3&2&0\ 3&2&5&0\ 0&0&0&1 \end{pmatrix}, ] and [ C_5=I_6\oplus \begin{pmatrix} 2&1&1\ 1&1&0\ 1&0&2 \end{pmatrix}. ] The projective indecomposable dimensions are respectively [ (2,22),\qquad (9,6,10,1),\qquad (3,2,1,1,1,1,4,2,3). ] Thus the modular block algebra dimensions are ((4,22)) at (p=2), ((25,1)) at (p=3), and ((9,4,1,1,1,1,9)) at (p=5).
The frozen Pass-1341 artifact records the central-character-projector provenance [ N_{20}=20\sum_{g\in W(E_6)}\chi_{20}(g)\rho(g) ] with reported rank 20 and identity [ N_{20}^2=51840N_{20}. ] A deterministic pivot basis produces exact rational matrices (C,D\in\mathrm{GL}_{20}(\mathbb Q)) for the Atlas standard generators with [ C^2=D^9=(CD)^{10}=I. ] Their class-trace vector is [ (20,4,4,2,5,-1,0,0,0,-2,1,1,1,-1,0,10,2,2,2,1,1,-1,0,0,-1), ] which is the frozen degree-20 row. The matrices are committed in machine-readable form under SHA-256 [ \texttt{8d0c52cf1f962471be1ab6dc4d98af5bc397fe003cbf9660a819ac0572689deb}. ] The representation is faithful because the character takes its full degree only on the identity class. The local GAP/AtlasRep run matches the unique CTblLib row and independently affords faithful images of order (51840). This is a character-level comparison, not a basis equality. The checked-in packet does not independently rebuild the rational matrices from the literal 480-edge carrier, so that origin remains provenance rather than a promoted derivation.
Literal dihedral cycle enumeration through length six gives a smallest nontrivial cycle orbit: the length-four orbit represented by [ (0,1,2,3), ] of size 120 and stabilizer order 432. Length-seven and length-eight representatives retain the exact Pass-1332 stabilizers 2 and 1, but those two lengths are not exhaustively enumerated.
The separate GAP witness closes the global gap without enumerating those large cycle sets. It classifies ordered simple paths under the full (W(E_6)) action: there are (19) five-vertex path orbits with pointwise stabilizer at most (6), and (133) six-vertex path orbits with pointwise stabilizer at most (3). The stabilizer of a cyclic order maps into (D_{2n}), with kernel fixing every cycle vertex. Thus a length-five cycle has stabilizer at most (10\cdot6=60), and a cycle of length (6\le n\le40) has stabilizer at most (2n\cdot3\le240). Their orbit sizes are therefore strictly larger than (120). Together with the literal length-three and length-four census, this proves that the length-four orbit is globally minimal over all simple cycles.
Every directed-edge cycle operator transports to the three species-20 copies as (C\otimes I_3). Therefore a cycle does not itself choose a multiplicity coordinate; one must also choose a primitive idempotent in the internal (M_3) block. Such idempotents form an (S_3)-orbit of size three. Hence the smallest combined selector orbit under (W(E_6)\times S_3) is [ 3\cdot120=\boxed{360}, ] with stabilizer order [ 432\cdot2=\boxed{864}. ] Any additional cycle constraints only shrink the stabilizer, so this is the global simple-cycle-plus-copy minimum. It is still a gauge choice, not a canonical internal selector.
Complete orthogonal primitive idempotent systems lift by Newton–Hensel to precision (p^6) for all three bad primes: [ 2^6=64,\qquad3^6=729,\qquad5^6=15625. ] No primitive-idempotent lifting obstruction is observed through this certified precision. This finite computation does not by itself execute an infinite (p)-adic tower.
The Smith saturation and Loewy radical filtrations are distinct. Their cumulative dimensions are cross-checked directly against the owning Pass-1326 Smith certificate and Pass-1330 Loewy certificate: [ \begin{array}{c|l|l} p&\text{Smith cumulative ranks}&\dim(\mathcal H/J^k)\\hline 2&(5,12,17,20,22,25,25,25,26)&(5,9,13,19,24,26)\ 3&(13,21,23,26)&(4,10,16,22,26)\ 5&(24,26)&(20,24,26). \end{array} ] At (p=5), the Smith ranks agree with the shifted stages (\mathcal H/J^2) and (\mathcal H/J^3), but the raw filtrations are not indexwise identical.
The theorem insert is checked into both w33_paper.tex and
photonic_holonet.tex. The idempotent integrator places it before the
bibliography when one exists, otherwise before \end{document}.
Local gates:
- four isolated exact component generators: PASS;
- merged deterministic certificate: PASS;
- focused tests: PASS;
- source integration and idempotency checks: PASS.
Full historical-paper builds and the independent GAP/AtlasRep comparison remain explicit GitHub Actions surfaces until their runs are observed.