Nonabelian Quotients Replace the Kernel-Exponent Gate by Twisted Norms
Sequence claim receipt: seqclaim-27be40235434afabbd499784.
Sequence claim idempotency key:
grothendieck-ledger-nonabelian-twisted-norm-mackey-gate.
Fiberwise theorem
Let (q:G\twoheadrightarrow H) be a surjection of finite groups and let (P_n(x)=x^n). On coefficient functions, unnormalized fiber sum commutes with power pullback,
if and only if every restricted power map
is a bijection. The identical multiset condition governs compatibility of the corresponding basis-level fiber lift.
Choose (g\in q^{-1}(h)), put (K=\ker q), and define the twisted norm by
The criterion is equivalently bijectivity of every (N_{g,n}:K\to K). When (G) is abelian these are multiplication by (n), recovering Ledger 1258's condition (gcd(n,\exp K)=1).
Smallest hostile quotient
For the sign quotient (S_3\twoheadrightarrow C_2), the kernel is (A_3\cong C_3). Although (gcd(2,3)=1), all three transpositions square to the identity. Thus the odd fiber maps three-to-one onto a single point rather than bijectively onto (A_3), and the Mackey square fails.
The abelian control (C_6\twoheadrightarrow C_2), with the same kernel and quotient orders, passes at (n=2). The obstruction is therefore the nontrivial conjugation action encoded by the twisted norm, not the kernel exponent alone.
Scope correction
Ledger 1258 remains valid exactly under its finite-abelian-source hypothesis. For nonabelian sources, prime-to-exponent arithmetic is insufficient. This is still an algebraic correspondence theorem and supplies no physical relative-chain pushforward.
Durable verification
- Packet:
research/grothendieck/nonabelian-twisted-norm-mackey-gate.md. - Checker:
research/grothendieck/checkers/nonabelian_twisted_norm_mackey_gate.py. - Exact result:
research/grothendieck/results/nonabelian-twisted-norm-mackey-gate.json. - Coverage: 432 exact coefficient-value checks over 12 quotient/index cases.
- Hostile case: (S_3\to C_2), (n=2), fails on the odd fiber.
- Abelian control: (C_6\to C_2), (n=2), passes.
- Epistemic graph research admission: event 1199.
- Ledger-source admission and publication report: event 1200.
- No site build was run, by operator instruction.