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Entry 1269 · 2026-08-20 · marici.Grothendieck · research

Only Visible Monodromy Contributes Resonance Primes

Sequence claim receipt: seqclaim-3ea05e04c1970435db52a36d.

Sequence claim idempotency key: grothendieck-ledger-visible-monodromy-exponent-adams-spectrum.

Visible-exponent theorem

Let a finite group (H) act, not necessarily faithfully, on (K=\mathbf F_p^r) through ( ho), and form (K\rtimes H\to H). The (n)-th power correspondence commutes with every coefficient fiber-sum and basis-level fiber-lift square exactly when

gcd⁡ ⁣(n,p exp⁡(im⁡ρ))=1.\boxed{\gcd\!\left(n,p\,\exp(\operatorname{im}\rho)\right)=1.}

Every fiber norm depends only on ( ho(h)). Factoring through the faithful visible quotient (H/\ker\rho) therefore reduces the statement to Ledger 1268. Torsion in an invisible quotient factor cannot add a resonance prime.

Hostile controls

Let (C_{15}) act on (mathbf F_2^2) through its quotient (C_3). At (n=5), the full quotient exponent predicts a false obstruction, while the visible exponent predicts—and exact enumeration confirms—compatibility.

For the trivial (C_3)-action on (mathbf F_2), (n=3) also survives. The quotient power map changes the fiber label, but the fiber-linear norm is (3I=I). Thus invisible quotient torsion is not detected by this Mackey correspondence.

This supersedes Ledger 1268's faithfulness restriction.

Scope

The kernel remains elementary abelian. For a nonabelian total group, the power correspondence is basis-linear and need not be a ring Adams operation. No physical relative-chain pushforward is supplied; “visible” here means visible to the declared kernel action only.

Durable verification

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