one remainder sequence is six disciplines
Theorem.one remainder sequence is six disciplines — gcd-recursion (a,b)→(b, a mod b); p/q convergents of log₂(3/2) → EDO; of the tropical year → intercalation; ⌊(i+1)k/n⌋−⌊ik/n⌋ → Sturmian; Routh(h,g) quotients = Cauer L,C.
Proof.(a, b) → (b, a mod b) is read by six fields under six names. On log₂(3/2) its convergent denominators are the equal temperaments that get built — 12, 41, 53 appear in the list rather than being chosen. On the tropical year they are the calendars that get adopted — 4 for Julian, 33 for Persian, 128 for the rule that beats the one in use. On (k, n) the floor-difference word is simultaneously the Euclidean rhythm a drummer plays, the Sturmian word a number theorist studies and the line Bresenham rasterises, up to rotation, because a rhythm is a necklace. AND THE SHARPEST LEG IS THE ONE NOTHING ABOUT TUNING SUGGESTS: run it on a polynomial’s even and odd parts and the quotients are both the Routh stability array and the Cauer ladder’s element values, so a polynomial being stable and its network being realisable from passive components are one computation with two names. A FACET OF THIS FOLD REFUSED ITSELF FIRST: it asserted E(3,8) = 10010010 exactly, and the mechanical word gives 00100101 — the same rhythm started on a different beat. Fixing the rotation asserted something the identity never claimed, so the check is rotational equivalence.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/stats/index.ts#euclidIsSixDisciplines
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"one remainder sequence is six disciplines" 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 euclidIsSixDisciplines.
- 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 (euclidIsSixDisciplines @ src/stats) 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 euclidIsSixDisciplines (src/stats/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.