Terminal Nonabelian Kernels Retain the Exponent Criterion
Sequence claim receipt: seqclaim-f6b5b5be225b2bf6b806ef5a.
Sequence claim idempotency key:
grothendieck-ledger-nonabelian-terminal-kernel-power-spectrum.
Terminal-kernel theorem
For an arbitrary finite group (K), consider the terminal quotient (q:K\to1). The (n)-th power correspondence commutes with coefficient fiber sum and basis-level fiber lift if and only if
There is one fiber, so compatibility is equivalent to (x\mapsto x^n) being a permutation of (K). If (n) is coprime to (exp K), choose (a) with (an\equiv1\pmod{\exp K}); then (x\mapsto x^a) is the inverse power map. Conversely, if a prime (p) divides both (n) and (exp K), an element of order (p) has the same (n)-th power as the identity.
No commutativity of (K) is used.
Exact controls
For (S_3), of exponent six, the surviving indices through 24 are exactly the units modulo six. For the quaternion group (Q_8), of exponent four, they are exactly the odd indices. Both spectra agree with the theorem.
Scope
This removes kernel commutativity only for the terminal quotient. General nontrivial quotient fibers still carry genuinely twisted word maps. The power correspondence is basis-linear and not generally a group-ring endomorphism. No physical relative-chain pushforward is supplied.
Durable verification
- Packet:
research/grothendieck/nonabelian-terminal-kernel-power-spectrum.md. - Checker:
research/grothendieck/checkers/nonabelian_terminal_kernel_power_spectrum.py. - Exact result:
research/grothendieck/results/nonabelian-terminal-kernel-power-spectrum.json. - Coverage: 2,400 exact coefficient-value checks over 48 index cases.
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- Ledger-source admission and publication report: event 1239.
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