research: exclude 15-call five-b UT4 splits across dirty workspaces

[steph4n-gh]

Summary

Resolves the explicit 15-call five-$b$ weighted-tariff handoff formulated by Gareth Davidson (@bitplane) in PR #2314 (research/ut4-mixed-weighted-boundary) and PR #2343 (research/register-pairing-compression, Discussion #2019 comment) for $\operatorname{diag}([z], [uz], I_{\text{rest}})$ in $\mathbb{F}_2[\mathrm{UT}_4(\mathbb{F}_2)]$ across arbitrary finite restored dirty workspaces ($r \ge 2$):

1. Nilpotent Same-Label Dirty Compression for (6,5,4) and (4,5,6) to Two Roles (Lemma 1): Because $n_b = 5 \implies e_b^x = e_b^y = 0$ and $n_c = 4 \implies e_c^x = e_c^y = 0$ (or $n_a = 4 \implies e_a^x = e_a^y = 0$), only one outer generator label can have nonzero dirty row or column components. Any dirty interior contraction $v_{i_m}(u_{i_{m+1}}) \ne 0$ in a product subsequence therefore occurs between two adjacent calls of the same outer label, which vanishes identically in $R = \mathbb{F}_2[\mathrm{UT}_4(\mathbb{F}_2)]$ by $([a]+1)^2 = ([c]+1)^2 = 0$. Active-corner projection preserves the $2\times 2$ active block over $R$, and the predecessor exclusions at length $\le 14$ (PR Research: prove the mixed UT4 z/uz endpoint needs fourteen calls #2175, PR Prove a sharp weighted region for the mixed UT4 target #2314) prevent any projected coupling from vanishing.
2. Three-Role Active-Plus-Dirty Quotient Decomposition for (5,5,5) (Lemma 2): All common-orientation excess profiles compress to two roles. For the remaining two opposite-orientation three-role modes (Case I: $(e_a^x, e_a^y, e_c^x, e_c^y) = (1, 0, 0, 1)$, Case II: $(0, 1, 1, 0)$), in $M_3(Q)$:
   • the 14 basis monomials without adjacent ${A, C}$ decouple completely from the dirty role on $W_{HH}$, restricting the active projections to the exact 5,664 compatible (5,5,5) profile pairs;
   • the off-diagonal blocks $W_{KH} = 0$ and $W_{HK} = 0$ receive zero dirty interior contractions (by $ACA = CAC = 0$ and $\deg \ge 4$ vanishing in $Q$), yielding exact homogeneous linear equations on the dirty prefix profiles;
   • $W_{HH}$ on AC, ACB, BAC receives only the rank-one cross-role contraction.
3. Complete 5-Gap Rank-Sum Profile Characterization and Exact Censuses (Lemma 3):
   • A 5-gap prefix profile $P \in M_2(\mathbb{F}2)^5$ with gap increments $S_g = P{g+1} \oplus P_g$ is realizable by $n \in {4, 5, 6}$ nonzero rank-one couplings iff $\sum_{g=0}^5 \operatorname{rank}(S_g) \le n$ (9,721, 47,521, and 174,670 profiles; 55,566, 907,200, and 15,471,918 realizations).
   • Two-role censuses: Reproduces PR Research: prove the mixed UT4 z/uz endpoint needs fourteen calls #2175 (4,5,4) (5,536 words) and PR Prove a sharp weighted region for the mixed UT4 target #2314 (5,5,4) / (4,5,5) (211,968 / 209,920 words) byte-for-byte, and exhausts (6,5,4) (3,376 profile pairs, 6,049,824 words), (5,5,5) (5,664 profile pairs, 8,655,168 words), and (4,5,6) (3,280 profile pairs, 5,874,848 words) — 20,579,840 new 15-call words, 0 survivors.
   • Three-role (5,5,5) censuses: Across all 5,664 active profile pairs, both Case I and Case II leave 17,864 dirty profile pairs passing $W_{KH} = W_{HK} = 0$ and 42,447,616 expanded 3-role words per mode (84,895,232 total), 0 survivors.
4. Main Theorem & Weighted Price Cone:
   • Every 15-call realizing word has $n_b \ge 6$, and every realizing word with $n_b = 5$ requires $n_a + n_c \ge 11$ ($L \ge 16$) on any finite dirty workspace $r \ge 2$.
   • For all $(\alpha, \beta, \gamma) \in \mathbb{R}_{\ge 0}^3$, every realizing word costs at least $\min(4\alpha + 6\beta + 4\gamma,; 4\alpha + 5\beta + 4\gamma + 3\min(\alpha, \gamma))$.
   • The direct 14-call (4,6,4) word is globally optimal on the expanded cone $\beta \le 3\min(\alpha, \gamma)$; in particular, at $(\alpha, \beta, \gamma) = (1, 3, 1)$, cost $25$ is impossible and $26$ is the exact minimum.

Scope & Attribution

• Scope: Formal $\mathbb{F}_2[\mathrm{UT}_4(\mathbb{F}_2)]$ register-model theorem and computer-assisted necessary-quotient exclusion. Does not claim a native bit supplier, physical conversion, or $\kappa$ improvement.
• Task / Claim: Drawn via scripts/choose_research_task.py (seed = 09aa8c1ab0ddd939f9b2d42697e50a37, task_id = bit-new-register-model, priority_revision = 5); tracked in Discussion #2019 comment 18961496.
• Credit: Full predecessor attribution to Gareth Davidson (@bitplane, PR Research: sharp mixed UT4 root tariffs with arbitrary dirty workspace #2027, PR Prove a sharp weighted region for the mixed UT4 target #2314, PR Research: body-dependent BIT stock cutoff and fixed-point counterexample #2343), Nick Petalas (@petalas, PR Research: prove the mixed UT4 z/uz endpoint needs fourteen calls #2175), ToikY (PR Research: UT4 central-target tariff bound with arbitrary dirty workspace #1662, PR Research: UT4 two-target unit count is independent of dirty workspace #1680), CinnaDavis / 1337isnot1337 (PR Scoped negative theorem: three coordinate swaps need nine calls with arbitrary dirty workspace #1541), and Aaron Wang (@navinor, PR Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690).

Verification

python3.12 -I -B scripts/research_package.py --package research/ut4-fifteen-call-five-b
python3 -B -m unittest discover -s tests -p 'test_research_*.py'

Both verify.py (PASS_UT4_FIFTEEN_CALL_FIVE_B, receipt 9b7ad621553bb1517c52128298e5ea66f3841f51cebed67dc9839d0b6ffb65bf) and controls.py (PASS_UT4_FIFTEEN_CALL_CONTROLS) complete in ~8 seconds total.