[petalas]

Result and claim type

Answers the first open question of unpaidmoon's #6642: which shortest paid words realize the plane braid endpoint [w], w=sts=tst=3214, in the alphabet D_r(N)=I+([r]-I)N, r ∈ {s,t,u}, N rank-one on K^2.

• Scoped proof, every field. Sixteen explicit families of six-factor words equal [w]: four with three free projectors (e.g. s(P) s(P') t(X) t(X') s(Y) s(Y'), s(P) t(X) t(X') s(Y) s(Y') t(P')) and twelve with two (e.g. s(P) t(X) s(X) t(X') s(X') t(P')), X'=I-X. The proof uses only a line lemma for factors sharing one splitting and conjugation by a pure permutation; verify.py replays the rewrite on the symbolic words. Over F_q the families contain 4n^3+12n^2 distinct words, n=q(q+1).
• Corollary, every field. Every tight two-factor suffix allowed by the Record tight mixed peels for the braid endpoint 3214 #6642 peel table extends to a shortest word. The concern in Record tight mixed peels for the braid endpoint 3214 #6642 that a free second peel may fail to extend does not arise.
• Finite certificate, F2 and F3. These families are every six-factor word equal to [w]: 1296 over F2, 8640 over F3. Two engines agree: a tight-peel search and an unpruned meet-in-the-middle join, both testing exact operator equality on A⊗U.

Not claimed: completeness over other fields, any caller invoice, a κ change. Selected value 781451/250000000 unchanged; all five framework assumptions remain.

Evidence and reproduction

CPython 3.12.x, standard library only, from the repository root:

~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/braid-3214-shortest-words
~/git/integer-mult-bounds/.venv/bin/python3.12 -B research/braid-3214-shortest-words/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/braid-3214-shortest-words --check-source-history

Local results: PASS_BRAID_3214_SHORTEST_WORDS (about 13 s), PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, PASS_RESEARCH_COMMIT_BYTES for c3398c5.

Limitations and unrun obligations

• Completeness is proved only over F2 and F3; larger primes are an opt-in verifier argument, not run to completion and not claimed.
• Inherited and not replayed: spectator elimination (JoyboyBrian Prove spectator elimination and the all-workspace S4 cost-21 bound #4765), the six-factor bound and peel table (Record tight mixed peels for the braid endpoint 3214 #6642), the ordered-pair defect (Prove all-field rigidity of shortest permutation circuits (no κ) #6449), the st classification (Classify all shortest words for the rank-two st endpoint #6550).
• Scalar conjugation is algebraic, not a physical basis change. No Lean, Docker or selected replay was run.

Open next questions

• Are the sixteen families complete over every field? First step: derive the tight third-peel table symbolically after each Record tight mixed peels for the braid endpoint 3214 #6642 second-peel branch.
• Do other length-three rank-two endpoints share the same building blocks (complement pairs, single-splitting blocks, conjugation)? First step: change the target constant and rerun over F2 and F3.
• What does a two-splitting realization cost physically against a three-splitting one? Blocked on a caller tariff.

Related open work

• Record tight mixed peels for the braid endpoint 3214 #6642 (unpaidmoon): source of the question and the peel table; this package extends it to whole words and agrees with its length 6 and its exclusion of u-factors.
• Classify all shortest words for the rank-two st endpoint #6550 (ToikY): shortest words for the different endpoint st; method analogue, no shared number.
• Prove all-field rigidity of shortest permutation circuits (no κ) #6449 (Nick Petalas): ordered-pair defect used by the search; not re-checked.
• No other open PR by this author created since the claim studies this endpoint or its word count.

Task

Card formal-interface-lemma, scope changed as in #6642 (recorded in TASK.json). Claim: #1166 (comment)

CI notes

At head c3398c5, both hosted checks passed in https://github.com/CrocSwap/integer-mult-bounds/actions/runs/41661364802: plan and replay (research/braid-3214-shortest-words).

Credit

Produced by Nick Petalas. Question and peel table: unpaidmoon #6642. Predecessors: ToikY #6550, #6449, JoyboyBrian #4765, Ross Richardson #1989, Adam Gall's cut lineage.