Scoped obstruction: commuting reorders plus free frames cannot lift the selected k8 ceiling by 5% (no new κ)

[thakrarsagar]

Scoped obstruction with an exact certificate. No new κ, and the selected record is unchanged.

Reordering commuting gates within segments, combined with any legal frame assignment, cannot lift the selected k8 word's packing ceiling by 5%, or even by 0.7%.

• Track: complex supplier, seed 26.
• Card: complex-rank-shape-decoder, with the scope refined.
• Base: 0fdd7d82.
• Package: research/k8-commutation-ceiling-seed26/.

Why this is needed

Two merged scoped results bound the selected gauge44/k8 complex word (c2b29430):

• Research: exact complete frame ceiling for the selected k8 word #1671 (bitplane) frees every frame but keeps the chronology fixed.
• seed27 (woklerx) frees commuting-ADD order but keeps frame slots fixed.

Neither covers both at once, and #1671 lists that combination as open question 1. The rank-shape card asks for a decoder whose ceiling clears 5%, and this joint class is the natural place a cheap gain could hide.

Result

At a* = (21/20)·a_hi, the exact lower bound on every member's bill is 1 + 2.146289027469049 × 10⁻⁴ > 1. A second certificate gives 1 + 4.86 × 10⁻⁶ > 1 at 1.007·a_hi.

Bound at a*	Value	Verdict
The selected word itself	1.000245413040	(member)
Stage 1, chronological intervals	1.000227868002	rejects (#1671's case)
Stage 1, commutation intervals (decoupled)	0.999971231189	does not reject
Stage 2, Lagrangian coupling	1 + 2.146 × 10⁻⁴	class excluded at 5%
Stage 2 at 1.007·a_hi	1 + 4.86 × 10⁻⁶	excluded at 0.7%

The decoupled per-register bound leaves room once the chronology is relaxed. Coupling the two operands of each ADD gate is what closes it.

How

1. Commutation.
   • The rule: two ADDs t←t+cs and t′←t′+c′s′ commute when t ≠ s′ and t′ ≠ s; two SCALEs commute; any other pair commutes when it shares no register.
   • It is checked against exact rational composition, in both orders, on all 109,388 same-register pairs.
   • Of the 9,092 adjacent within-segment pairs, 4,519 commute.
2. Intervals. A dependency DAG gives every gate an interval [lo2, hi2] that contains its frame in every member under every legal assignment.
   • Two implementations agree: propagation, and a closed form over ancestor and descendant closures.
   • In chronological mode both reduce to Research: exact complete frame ceiling for the selected k8 word #1671's intervals exactly.
   • Forced gates drop from 3,672 to 2,979.
3. Stage 1. For each register, the cheapest chain of subspaces from its start to its end that passes through every item's interval, ignoring the order of items within the register. Each register's paid climbs in any member are at least this.
4. Stage 2.
   • Coupling: each free ADD gate's two copies get dimension-linear multipliers, +λ on the target copy (charged at entry) and −λ on the source copy (charged at exit).
   • Why it stays a lower bound: operand agreement cancels the multipliers in any real member, so the sum of the per-register optima remains a valid lower bound for every λ.
   • Exact arithmetic: costs use ceiling.child_cost lower ends on the 2⁻²⁵⁶ grid, and λ = k/2⁴⁰. The bound is therefore computed in exact integers.
   • The certificates store the integers k: 1,008 in LAMBDA.json and 962 in LAMBDA-1007.json.
   • discover.py regenerates such certificates; the verifier never imports it.

PROOF.md gives the validity argument.

Controls

• Commutation rule: exact on every pair, and a wrong rule (one that ignores t = s′) is caught.
• Intervals: the chronological intervals equal Research: exact complete frame ceiling for the selected k8 word #1671's; the two trace implementations agree; the word's frames lie inside.
• Subspace enumerators: two enumerators agree on every distinct interval, with counts 1, 2, 5, 16, 67, 374 and 2,825 for F₂^0 … F₂^6.
• The given word: it reproduces Research: exact complete frame ceiling for the selected k8 word #1671's profile and ceiling bracket, and the Stage-2 bound stays below its own bill, as it must.
• Toy exhaustive check (toy.py): 150 random small words, 93 with several orders, checked over every linear extension and every frame assignment.
  • The intervals contain every legal assignment.
  • The Lagrangian bound never exceeds the exact minimum.
  • Charging negative multipliers at the entry point (an invalid rule) is caught.
• Certificate files: a multiplier on a forced gate is refused, and so is a certificate for a different word.

Checks run

• verify.py: PASS_K8_COMMUTATION_CEILING, in about 30 s, with RESULT.json reproduced byte for byte.
• 13 tests pass.
• research_package.py --check-only passes.
• The registered python3.12 replay: PASS_REGISTERED_RESEARCH_PACKAGE.
• Run on Python 3.12.15 and 3.14.8 (macOS 26.5, arm64).

Scope

Covered.

• Every within-segment linear extension of the A, B and tail dependency orders, with coefficients unchanged.
• Every frame assignment under Research: exact complete frame ceiling for the selected k8 word #1671's contract.
• Every exponent at or above the certified one.

Not covered.

• Cross-segment moves. 217 of the 485 adjacent pairs that straddle a segment boundary commute.
• Coefficient-rescaling rewrites.
• Changes to the cut, compensation, pins or endpoints.
• New child interfaces: fanout (open Research: nonconsecutive commuting C3 fanout schedule #1865) and C3 gathers (Discussion 2032).
• Open Tooling: exact frame solver and delivery-rewrite search for gauge44/k8 local words (no new κ) #1186's different cost model.
• Other words.
• A lift below 0.7%, which is neither shown nor excluded.

No κ.

Reproduce

python3 -I -B research/k8-commutation-ceiling-seed26/verify.py        # about 30 s; PASS_K8_COMMUTATION_CEILING
python3 -I -B -m unittest discover -s research/k8-commutation-ceiling-seed26 -p test_package.py
python3.12 scripts/research_package.py --package research/k8-commutation-ceiling-seed26

Credit

• Research: exact complete frame ceiling for the selected k8 word #1671 (bitplane, Gareth Davidson): its model, code and word are imported with digests pinned.
• seed27 (woklerx).
• Research: diagonal rank-16 batches under register commutation, 134 not 8 (audit of #1779; no κ claim) #1817 (JosieElliston).
• navinor (Aaron Wang): the frame method and frame_model.py.
• complex-local-word-ceiling: ceiling.py.
• Related open work: Tooling: exact frame solver and delivery-rewrite search for gauge44/k8 local words (no new κ) #1186 (mdumitrean), Research: nonconsecutive commuting C3 fanout schedule #1865 (1337isnot1337) and Discussion 2032.

Disclosure

Prepared by Sagar Thakrar (thakrarsagar) with substantial AI assistance from Claude (Anthropic) via Claude Code.

• Parallel read-only scout and judge subagents selected the question; both judges ranked it first.
• Reader subagents and a synthesizer specified the model and found that operand coupling decides it.
• The package code, certificates and controls were then written and checked.

No claim was posted to Discussions. New material is Apache-2.0.