ℤ/m defines its own laws: for every modulus to 81 the units, the doubling orbit, the reflection and the Fibonacci period are computed from m alone and their laws hold
Theorem.ℤ/m defines its own laws: for every modulus to 81 the units, the doubling orbit, the reflection and the Fibonacci period are computed from m alone and their laws hold — ⟨2⟩ ⊆ (ℤ/m)ˣ ∧ |⟨2⟩| divides |(ℤ/m)ˣ| (Lagrange); d ↦ (m−d) mod m is an involution with pairs·2 + fixed = m; the Pisano period π(m) returns and is even for m ≥ 3.
Proof.ℤ/9 recomputes the sealed vortex sequence — its doubling orbit is the unit half of VORTEX_SEQUENCE, derived from the modulus alone. SEALED: the kernel decides the same five statements for the moduli to 47 (registry.lean, no axiom); the general laws behind them — Lagrange on (ℤ/m)ˣ and the parity of the Pisano period — are cited, not reproved.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/mountain/vortex/index.ts#theZModulesDefineTheirOwnLaws
1 · Classification
finite-complete — computed witness within a cited frame (the unbounded form leans on the cited literature)
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"ℤ/m defines its own laws: for every modulus to 81 the units, the doubling orbit, the reflection and the Fibonacci period are computed from m alone and their laws hold" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation theZModulesDefineTheirOwnLaws.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (theZModulesDefineTheirOwnLaws @ src/mountain/vortex) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes theZModulesDefineTheirOwnLaws (src/mountain/vortex/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.