Blueprint: the Schur package, the dimension bound, the open sector

The second part of the axiom audit. It pins the symmetric-group input the forward direction rests on, the auxiliary ⌊2eR⌋ dimension bound, the total bound k + 2ℓ ≤ R, and the open-sector Proposition 3. Read Blueprint.lean first: it audits the forward proof itself.

The Schur package

Jacobi–Trudi characters: the Frobenius identity, orthonormality, the branching containment, the block faithfulness and the square growth bound — the fields of SchurPackage, and the package.

The forward theorem on Deligne alone

The trace zeta function

The zeta function of a Frobenius tower is the Newton generating series of its super power sums, and rational when the characters are hook-confined.

Corollary A.2 with the sharp threshold

The appendix's own statement: a real dimension bound A, every side s > 2e√A, and degrees at most s − 1.

Hook-confined sequences

Lemma A.9 for arbitrary hook dimensions, with numerator degree at most b and denominator degree at most a.

The separate-sector dimension bound

The ⌊2eR⌋ bound: a square diagram past it is dead, its idempotent acts as zero, and the surviving sector is bounded.

The total dimension bound

Native blocks bound constituent dimensions; simultaneous word orbits bound the commutant by a polynomial. The transported colour action then forces k + 2 * ℓ ≤ R by comparison of exponential bases.

The open sector: Proposition 3

Repair connectivity of a pairing fibre, the canonical frame and its re-canonicalization, the per-move ledgers and the paired step — together, the signed value depends on the boundary pairing and nothing else. Independence across pairings is false.

The appendix, for an object

Corollary A.2 as the appendix states it: for an arbitrary object of a rigid symmetric ℂ-linear category whose tensor powers have exponentially bounded endomorphism dimensions, the trace zeta function of every endomorphism is rational of the stated degree.

The appendix's theorems, by type

The axiom audits above fix what these rest on; the pins below fix what they say — the hypothesis on the object, the threshold, and the degree bound.