Apartment parity: module structure, Ihara coefficients, and an exact thermal code

4472 — the protected 10-space is 1|8|1

The 10-dimensional nondegenerate apartment quotient carries faithful inner and outer projective symplectic actions of orders 25,920 and 51,840. Its invariant filtration has dimensions 1 < 9 < 10, hence composition dimensions 1|8|1. The middle 8-space is faithful and irreducible for PSp(4,3), with nonzero orbits of sizes 135 and 120.

4473 — exact four-spin Hamiltonian

Because ker(Hᵀ) is generated only by the global reversal vector, the exact ground energy is −1620 and there are exactly two ground states: all line signs + and all line signs −. A one-line flip has weight 162 and energy −1296. Random-line energy has exact mean 0, variance 1620, third moment −12960 and fourth moment 9891720.

A separately labelled 4,096-sample control gives corr(signed spectral radius, energy) ≈ −0.277526 and a Ramanujan fraction ≈ 0.26465. That row is empirical; the ground-state and moment statements are exact.

4474 — four-cycle semantics fixed

The old common-neighbour helper records every underlying simple cycle once from each diagonal. A repository advisory checker now flags unqualified uses of “1620 four-cycles” unless the intended induced/apartment meaning is explicit.

4475 — the apartment sum is an Ihara coefficient

For the signed 480-state non-backtracking matrix Bσ and S=Σℓσℓ:

The fixed 960 is 8 orientations times the 120 line-internal C₄s. Thus W₄ is exactly the signing-dependent part of the fourth logarithmic Artin–Ihara coefficient after the fixed geometric-line term.

4476 — apartment-code low weights

The code remains [1620,39]. The primal minimum distance is not proved: one line gives d≤162 and exhaustive coefficient supports through five line flips find nothing smaller. The dual beginning is exact:

The projective similitude action gives a certified automorphism subgroup of order at least 51,840.

4477 — a new route to the E8/2E8 carrier

The irreducible 8-core has a unique invariant plus-type quadratic refinement: 136 singular vectors including zero and 120 nonsingular vectors. Its two nonzero polar graphs are exactly

A nonsingular stabilizer has PSp suborbits 1,1,1,27,27,27,36; after adjoining the outer action they become 1,2,27,36,54. These reproduce the repository’s corrected code-embedding fingerprints rather than merely matching 135 and 120 numerically.

4478 — outside box: real four-qubit Pauli phase space

An explicit hyperbolic basis puts the same 8-dimensional quadratic space in canonical form

That is the standard real four-qubit Pauli quadratic: 135 nonidentity even-Y/symmetric classes and 120 odd-Y/skew classes. The witness verifies the isometry on all 256 vectors. This is a finite phase-space identification, not a claim that the full 1,620-coordinate code is a physical four-qubit device.

4479 — outside box: MacWilliams is a thermal transform

The apartment four-spin partition function is therefore exactly a specialization of the code weight enumerator, and MacWilliams duality gives a finite high/low-variable transform. No thermodynamic-limit phase transition is claimed.