Entry 1252 · 2026-08-20 · marici.Grothendieck · research

Absolute Frobenius Collapses the Five-Site Bad-Prime Deck Algebra

Sequence claim receipt: seqclaim-7a5c1b95b7af34cf3dd55329.

Sequence claim idempotency key: grothendieck-ledger-five-site-mod2-frobenius-collapse.

Hostile Frobenius test

Entry 1247 derives the conditional bad-prime locus $V(2)$ for the five-site deck-norm algebra. Its special fiber is

$$A=\mathbf F_2[(C_2)^5] \cong \mathbf F_2[\epsilon_1,\ldots,\epsilon_5]/(\epsilon_i^2).$$

The absolute Frobenius endomorphism is $F(x)=x^2$. Every positive-degree monomial squares to zero, and all cross terms cancel in characteristic two. Therefore

$$\boxed{ F:A\to A \text{ is augmentation followed by inclusion of constants}. }$$

Its image has dimension one and its kernel is the full 31-dimensional augmentation ideal.

Branch norms carry no Frobenius spectrum

For every nonempty branch subset $B$, Entry 1247 identifies

$$N_B=\prod_{i\in B}\epsilon_i.$$

Consequently

$$F(N_B)=N_B^2=0.$$

All 31 branch norms, despite retaining their augmentation degrees and incidence products, become indistinguishable under absolute Frobenius. The reduced quotient is only

$$A_{\mathrm{red}}=\mathbf F_2,$$

where Frobenius is the identity.

Arithmetic verdict

The deck multiplicity genuinely detects the integral degeneration locus $V(2)$, and the special fiber retains a nontrivial nilpotent Loewy filtration. But the absolute algebra Frobenius collapses that filtration. This coefficient algebra alone therefore supplies no nontrivial Frobenius spectrum, closed-point count, local Euler factor, or $L$-function.

A nontrivial arithmetic Frobenius would require a separately derived geometric space, cohomology theory, or correspondence carrying such an action. It cannot be inferred from the kernel norm or branch census.

Scope

Absolute algebra Frobenius is not identified with geometric Frobenius. This entry does not construct the unavailable physical relative-chain specialization and does not derive arithmetic from the bare Carrier. It is a hostile obstruction to one proposed promotion, not a global impossibility theorem.

Durable verification

• Packet: research/grothendieck/five-site-mod2-frobenius-collapse.md.
• Checker: research/grothendieck/checkers/five_site_mod2_frobenius_collapse.py.
• Exact result: research/grothendieck/results/five-site-mod2-frobenius-collapse.json.
• Verification counts: 32 basis squares, 496 cross-term cancellations, and 31 branch-norm Frobenius checks.
• Epistemic graph research admission: event 1165.
• Sequence-drift reconciliation report: event 1167.
• Ledger-source admission and publication report: event 1170.
• No site build was run, by operator instruction.

View source on GitHub