Scoped S4 result: four roles do not compress noncommuting 3-cycles below four calls

research/s4-noncommuting-threecycle-r4-seeda27b/CLAIM.md

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+# Cycle 4 research claim
+
+- Seed: `a27baaa734817e395db88b840bf4a089`
+- Track and task: `bit-mechanisms` / `bit-new-register-model`
+- Priority revision: 3
+- Exact base: `0b1cccf86e2f5132ed286a26ef0a681a495740ea`
+- Selected-record SHA-256: `10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`
+- Scope status: refined to a four-register nonabelian address model.
+
+## Question
+
+For the ordered noncommuting pairs `([a],[b])` drawn from
+`a,b in {s1*s2, s2*s1, s2*s3, s3*s2}`, can one added restored dirty
+scalar role let the complete paid alphabet realize
+`diag([a], [b], [e], [e])` with at most three calls? Here
+`s1=(12)`, `s2=(23)`, and `s3=(34)`. Each paid call is
+`I+([s]+[e])N`, for `s` one of the three adjacent transpositions and any
+rank-one idempotent `N` in `M4(F2)`; free scalar `GL4(F2)` mixing is included
+through this full idempotent alphabet. The finite family has eight ordered
+noncommuting pairs.
+
+## Changed assumption, method, and bounded domain
+
+The three-register abelian result excludes extra roles and arbitrary free address maps. This scope adds one restored role and uses the explicit nonabelian address group `S4`. Enumerate the exact 360 factors (3 paid labels times 120 rank-one idempotents), all 129,600 ordered two-factor products, and all 2,880 target-side products for the eight targets, giving a complete meet that excludes words of length at most three. Verify the explicit four-call diagonal construction for every target. Stop after this exact radius-three meet.
+
+## Cost screen
+
+Optimistically price every paid transposition call at one and scalar mixing, register hosting, and restoration interface overhead at zero. The direct identity costs four; a three-call identity would save one call (25%) in this abstract model. No selected supplier, bit primitive, recursive moment, or `kappa` change is inferred without a physical realization.
+
+## Nearest work and overlap
+
+- PR #1720, head `edb4a289a5f32c8209140144e384dcd3fa8cbfd1`: two identical 4-cycle targets at three roles.
+- PR #1734, head `48855b620b23af3a7fe87d53bde05cd2211342c8`: two identical double-transposition targets at three roles.
+- PR #1705, head `ad6b7d641b3e8576a95add7981fa75b6a8278fbc`: four coordinate-SWAP targets over a different `F2[GL2(F2)]` alphabet at four registers.
+- PR #1742, head `679f3624a0a41ed14942825dd7f413132412d1e3`: three-role `UT3(F2)` D8 central-commutator pair with labels `{x,y}` and a seven-call lower-bound search; the group, alphabet, target, role count, and call floor differ from this four-role S4 family.
+
+The pre-submit REST refresh at 08:37 UTC covered 424 open PRs. The complete Discussions overlap snapshot covered 383 records at 08:33:44 UTC; this claim and its two update comments are the only exact S4 four-role matches. Discussion #1738 / PR #1742 is a structurally related D8 third-role result with a distinct group, target, alphabet, and search depth. The S4 predecessor PRs above are related but use different targets and only three roles. The initial focused target was `diag([s1*s2],[s2*s3],[e],[e])`; the exact radius-three search was then extended over the full eight-pair family without changing the alphabet or method.
+
+## Expected handoff
+
+A small registered package with exact group-algebra arithmetic, state counts and hash, the direct word, omission/target mutation controls, source pins, cost accounting, and an explicit scoped conclusion. Claim expiry: `2026-10-14T10:23:00Z`.

research/s4-noncommuting-threecycle-r4-seeda27b/README.md

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+# Four roles do not share two noncommuting S4 three-cycles in three calls
+
+## Result
+
+For every ordered noncommuting pair \((a,b)\) from
+\(\{s_1s_2,s_2s_1,s_2s_3,s_3s_2\}\), the target
+\(T_{a,b}=\operatorname{diag}([a],[b],[e],[e])\) has minimum paid length
+**four**. There are eight such ordered pairs. Each has the direct four-call
+construction below. The exact radius-three search excludes every word of
+length at most three for all eight targets, even with the additional dirty
+scalar role and free scalar \(\mathrm{GL}_4(\mathbb F_2)\) mixing included
+in the alphabet. This is a scoped negative result: that extra role does not
+produce the proposed one-call sharing for these targets and this operation
+alphabet.
+
+With right-to-left permutation composition, \(s_1=(12)\), \(s_2=(23)\),
+and \(s_3=(34)\), so \(s_1s_2=(123)\) and \(s_2s_3=(234)\); the set also
+contains their inverses.
+
+## Exact model
+
+Work over the noncommutative group algebra \(\mathbb F_2[S_4]\). Its 24
+basis elements are the permutations of four points, and matrix entries are
+24-bit coefficient masks. A paid call is
+
+\[
+F_{s,N}=I+([s]+[e])N,
+\]
+
+where the paid address label \(s\) is one of the adjacent transpositions
+\((12),(23),(34)\), and \(N=xy^T\in M_4(\mathbb F_2)\) ranges over all
+rank-one idempotents, with \(x,y\ne0\) and \(y^Tx=1\). There are 120 such
+idempotents and hence 360 paid factors. Each factor is an involution. The
+complete idempotent inventory includes the normalized free scalar
+\(\mathrm{GL}_4(\mathbb F_2)\) mixing allowed by this formal model.
+
+The equality to \(T\) restores roles three and four exactly and applies the
+two stated transformations to arbitrary values in roles one and two. All
+four roles may start dirty. Each adjacent-transposition call costs one; scalar
+mixing, hosting the fourth role, and interface transitions are optimistically
+priced at zero.
+
+## Exact lower bound and upper bound
+
+The verifier enumerates every one of the \(360^2=129{,}600\) ordered products
+\(F_iF_j\), including repeated factors. This gives 87,016 distinct pair
+product matrices, including the identity. For each of the eight ordered
+noncommuting target pairs, it checks all 360 target-side meets, for 2,880
+complete misses. The canonical row-major encoding uses a three-byte
+big-endian coefficient mask for each of the 16 entries; the sorted
+complete-state stream has SHA-256
+
+`cc18ab4f9e58731f43f42ed0fd791531696836385f9435c6328cbbe36651f2f7`.
+
+No target-side meet hits this set. For each target \(T_{a,b}\) and each of
+the 360 possible final factors \(F_k\), the verifier checks
+\(T_{a,b}F_k\notin\{F_iF_j\}\). Since every factor is an involution, any
+three-call identity \(T_{a,b}=F_iF_jF_k\) would imply
+\(T_{a,b}F_k=F_iF_j\). This complete meet excludes length three; direct
+membership checks also exclude lengths zero, one, and two for all targets.
+
+Every target pair has a two-letter factorization for each cycle. If
+\(a=u_1u_2\) and \(b=v_1v_2\), its four-call word uses the diagonal
+idempotents \(E_{00}\) and \(E_{11}\):
+
+\[
+F_{u_1,E_{00}}F_{u_2,E_{00}}F_{v_1,E_{11}}F_{v_2,E_{11}}=T_{a,b}.
+\]
+
+The verifier also checks a one-call positive control, all 32 one-call
+deletions across the direct words, a changed-target control, all 360 factor
+involutions, and noncommutativity of every target pair. Arithmetic is exact
+in \(\mathbb F_2[S_4]\); no sampled columns or floating-point checks are used.
+
+## Cost and limits
+
+The favorable comparison is four paid calls versus a sought three-call word,
+so the proposed saving would have been one call (25%) before any other charge.
+This experiment found no such call reduction for any of the eight declared
+targets. The fourth role and free scalar mixing were not charged, which makes
+the failed three-call search a useful obstruction inside this abstract model.
+
+The result does not cover other address groups, other paid labels, more than
+four roles, different register dimensions, or richer call types. It does not
+map these formal calls to the selected bit supplier, prove a native circuit
+cost, change the selected recursive moment, or improve \(\kappa\).
+
+## Provenance, overlap, and reproduction
+
+The cycle used seed `a27baaa734817e395db88b840bf4a089`, track
+`bit-mechanisms`, task `bit-new-register-model`, priority revision 3, base
+`0b1cccf86e2f5132ed286a26ef0a681a495740ea`, and selected-record SHA-256
+`10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+`SOURCES.json` pins the task catalogue, selected record, selected review,
+source manifest, assembly certificate, and the abelian three-register
+predecessor. Current related work at the final overlap refresh included PR
+#1720 (two identical 4-cycle targets at three roles) and PR #1734 (two
+identical double-transposition targets at three roles); this package changes
+both the target pair and register count. PR #1705 uses a different
+`F2[GL2(F2)]` address alphabet. The live pre-submit refresh also found
+[PR #1742](https://github.com/CrocSwap/integer-mult-bounds/pull/1742), which
+settles a three-role \(UT_3(\mathbb F_2)\) D8 central-commutator target with
+labels \(x,y\) and a seven-call lower-bound search. That result has a
+different address group, alphabet, target pair, register count, and baseline;
+this package searches four roles over \(S_4\), with three adjacent
+transpositions and distinct noncommuting 3-cycle targets. The scoped
+coordination claim is
+[Discussion #1740](https://github.com/CrocSwap/integer-mult-bounds/discussions/1740).
+
+Run the exact finite search directly:
+
+```sh
+python3.12 -I -B research/s4-noncommuting-threecycle-r4-seeda27b/verify.py
+```
+
+Expected marker:
+
+```text
+PASS_S4_NONCOMMUTING_R4 rank_one_idempotents=120 factors=360 ordered_pairs=129600 pair_states=87016 target_pairs=8 radius3_misses=2880 pair_sha256=cc18ab4f9e58731f43f42ed0fd791531696836385f9435c6328cbbe36651f2f7 direct_calls=4
+```
+
+The registered repository package gate is:
+
+```sh
+python3.12 scripts/research_package.py --package research/s4-noncommuting-threecycle-r4-seeda27b
+```
+
+Prepared with substantial OpenAI Codex assistance. No subagents or
+independent human review are claimed.
+
+## Open next questions
+
+- **Question:** Does a fifth restored dirty role or another explicitly priced
+  paid-label set permit a sub-four-call identity for this family of
+  noncommuting targets?
+  **Why it matters:** That would separate a four-role obstruction from a
+  limitation caused by the three adjacent-transposition labels, and could
+  expose an actual sharing mechanism in a larger address model.
+  **First step:** Fix five roles and the same target
+  `diag([s1*s2],[s2*s3],[e],[e],[e])`, enumerate its complete rank-one
+  idempotent alphabet through radius three for the eight target pairs, and
+  test the exact four-call baseline.
+  **Blocked by:** A new bounded scope and source/base refresh; this package
+  deliberately stops at four roles. Any abstract saving would still need a
+  separate implementation and full register/interface cost.

research/s4-noncommuting-threecycle-r4-seeda27b/SOURCES.json

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