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Summary

This adds research/descending-invariant-allp/. Together with the reduction lemma of my #4540, it answers open question 2 of Dugongue's #4534. For every controller size p, every keep mask, with or without thick-stage thinning, #4440's complex controller word has strictly descending stage words. Any scalar commutative factorization L W_ab R = diag is therefore also valid with noncommuting D_s (commuting only with Q). #4534 proved this only for the fixed p16 word.

Claim type: a written proof with a mechanized check. No new κ.

Proof idea

Every prepended stage letter exceeds every letter already on its target lines. Locals run in stage order 0…2p. The initial idle for pair j+1 lands on untouched lines. The middle idle {2j+2} lands on letters ≤ 2j+1. The terminal idles {2j+3…2p} land on pair j (letters ≤ 2j+2) before the final bridges. ADD and SWAP only merge or move words, and thinning or keep masks only delete letters.

Evidence

• Mechanized check. The constructor's actual event stream, with its polynomial arithmetic stubbed out, is tracked with per-line maximum letters for every p ≤ 60 × five keep masks × {full, thinned}: 590 cases. A rejection control is included.
• Recorded sweep. 440 cases up to p = 200.
• Direct check. Noncommuting controller identity for all p ≤ 14 (91 cases) via a support-injectivity reduction (answers #4534 Q2 empirically; no new κ) #4540 already verified the full free identity directly for p ≤ 14.

python3.12 -B scripts/research_package.py --package research/descending-invariant-allp --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/descending-invariant-allp
  PASS_REGISTERED_RESEARCH_PACKAGE research/descending-invariant-allp outcome=completed   (~1 min, stdlib only)

All constructor modules the verifier loads are pinned as inputs. Head: b63f675a2419b4560846884b3f92b08c90d0d817. Base: canonical main f626db5c.

Credit: Cinna Davis (#4440) and Dugongue (#4534). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code

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Update (head dd3a84d): #4440's BIT controller (bit.physical_controller over F2[(Z/2)^{2p+2}]) runs the identical schedule. The verifier now checks the same descending invariant on its event stream: 590 more cases, p <= 60, five keep masks, with and without thinning. All pass. Since no letter repeats within a word, the group-ring relation d^2 = 1 never acts on the support, so the corollary holds on the BIT side as well.