Conditional κ = 7.44728650221677e-4: second concave-descent retiming (89 gates, constructed frames) after #290's kernel entries on the gen5 word

[rohanarun]

Conditional κ = 744728650221677/10^18 = 7.44728650221677e-4, +1.42 × 10⁻⁸ (+0.0019 %) over #290 (7.44714499357628e-4), +0.082 % over #287, +0.262 % over #285's gen5 word. The bit side binds.

One self-contained package, research/five-stage-gen5-kernels-descent/ (#290's package with one added stage), with its own workflow. No existing file is changed.

Full replay: PASS on the committed package (77d92c5): all stages (virtual, raw, bit, kernel, scalar, primes, banks, complex, math, finite), package integrity unchanged (138 files, manifest 22495fab…), PASS_IMMUTABLE_GEN5_KERNEL_ENTRIES_FIVE_STAGE_BANKED_CONSTRUCTION, κ 744728650221677/1000000000000000000, ~5 minutes.

Change

A second concave-descent retiming after the kernel entries. #290's final word (gen5 → #287's 904-gate descent → 220 target-prefix squares → 450 collective response-kernel entries) is retimed once more by #287's descent_transform.py with its own frozen descent2-selection.json. The search is the one behind #287 with constructed candidate frames: for each internal ADD gate, the minimal join and the maximal meet of its operands' neighbouring chain frames are offered as candidates, not only frames the word already uses. It keeps 89 single-gate frame reassignments onto 32 new bases (explicit rational rows in the selection).

Local paid histogram delta {1: +41, 2: +154, 3: −82, 4: −48, 5: +24, 7: −48, 9: +24, 12: −24, 14: +24, 18: −41, 19: +41}: the local call count rises by 65 (five-stage calls 477,859 → 478,184; priced banked calls 475,180) at unchanged rank mass 2,746,720, but the concave weight Σ r·ln(120/r) that the moment certificate prices falls, so κ rises. Sources, targets, the 3,004 entrances, kernel entries, 545 charts, banks (literal stock 1,244,515), the scalar word (565,050 weighted ADDs) and every endpoint are unchanged.

one invocation, m = 120	#287	#290	this package
entrances	2,554	3,004	3,004
five-stage calls	473,599	477,859	478,184
literal stock, 60 replicas	1,247,070	1,244,515	1,244,515
κ	7.44120696397042e-4	7.44714499357628e-4	7.44728650221677e-4

Checks

descent_transform.py is unchanged from #287/#290. It rebuilds every MOVE from the retimed gate needs and independently checks the byte-identical scalar/COPY projection, nested chains, fixed source/dirty/target endpoints, unchanged COPY lifetimes, nondegenerate endpoint bases, both reflected annihilator ledgers and the exact integer source span of every non-target operand inside its new frame; the new bases are registered and prime_check evaluates their determinants with every other actual basis. raw_ledger.rebind_descent2 recounts the five-stage profile through #290's pins.pin; every count the stage changes is a recomputed pinned value (descent2_selected_gates, descent2_removed_calls, descent2_local_delta, descent2_five_stage_calls, priced_five_stage_calls, kappa, …), none dropped. Then every stage of #290's verifier runs fresh — virtual, raw, bit (descent, target, kernel, second descent), scalar, primes, banks, complex, math, finite — with its omitted-stage negative controls.

Verify

python3 -m pip install -r research/five-stage-gen5-kernels-descent/requirements.txt

python3 -B research/five-stage-gen5-kernels-descent/verify.py --output /tmp/gen5-kernels-descent-verification

Python 3.11+, SymPy 1.14.0, a new output directory outside the package; no network; about 5 minutes. Prints PASS_IMMUTABLE_GEN5_KERNEL_ENTRIES_FIVE_STAGE_BANKED_CONSTRUCTION with the pinned κ.

Scope and credits

Exactly #290's hypotheses and conditional interfaces (all-size compiler, weighted chart, restored rows, selectors, routing, prime supply, precision/recovery, complex symbolic correctness, analytic reduction). No new Lean build. #285, #287 and #290 are open; their stages are inherited unchanged and not re-proved.

chafreaky (#290, #282, #284): the collective response-kernel stage on gen5 and the pin plumbing this package extends; DreamingOfClouds (#285, #276): the gen5/gen4 bit words and their packages; eumemic (#268, #244, #234 lineage): the target-prefix mechanism, the common-frame retiming and the package machinery; Dugongue (#254/#259/#272/#283): response kernels and the tiling rule; hcg890 and Jacob Sussman (#234): the five-stage layout; Evan McKinney (#197): completed banks. Prepared by Rohan Arun with Anthropic Claude assistance; Apache-2.0; lineage in NOTICE.md.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @rohanarun. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Second concave-descent pass on collective gen5 kernels.

Review and limits: Source-bound complete Linux verifier and producer regeneration; preserves the paid complete profile at the revised head.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 5c296797e06101c74d15b0162dfe372c186ec330. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.