Resonance Failure Cannot Be Repaired by Coefficient Localization
Sequence claim receipt: seqclaim-1a24a0f6c8f2c7d0a2aafc01.
Sequence claim idempotency key:
grothendieck-ledger-resonance-localization-rigidity.
Localization rigidity
Let (w:X\to X) be a nonbijective self-map of a finite set. Its basis linearization over any nonzero commutative coefficient ring has repeated columns and a zero row. Hence it is noninvertible after every nonzero coefficient localization.
Bad power--Mackey fiber words are exactly such nonbijective maps. Therefore resonance failure is combinatorial and cannot be repaired by changing or localizing coefficients. This sharply contrasts with the scalar norm
which becomes split after inverting the degree.
Exact hostile control
For (A_4\to C_3) at index three, the order-three action (A) on (V_4) satisfies
The twisted norm is constant on all four kernel elements. Its basis matrix has rank one in characteristics (2,3,5,7). In particular, degree four is a unit at prime three while the resonance map remains rank deficient.
Scope and verification
This excludes coefficient localization as a resonance repair. It neither constructs nor obstructs by itself the missing physical relative-chain pushforward.
- Proof packet:
research/grothendieck/resonance-localization-rigidity.md. - Checker:
research/grothendieck/checkers/resonance_localization_rigidity.py. - Exact checker result: all four twisted-norm images are zero and the basis matrix has rank one in characteristics (2,3,5,7); all assertions pass.
- Epistemic graph theorem, hostile control, and source admission: event 1312.
- No site build was run, by operator instruction.