Conditional κ = 7.49982567267028e-4: 1,734 kernel entries (restart-packed), second descent, 440 restorations and 8 sinks on #296's gen5b word

[rohanarun]

Conditional κ = 187495641816757/(25·10^16) = 7.49982567267028e-4, +0.438 % over #296's gen5b word (7.46709720962478e-4), +0.011 % (+8.3 × 10⁻⁸) over #299 (7.49899215319498e-4), +0.599 % over #295, +0.789 % over #287. The bit side binds below #233's complex supplier.

One self-contained package, research/five-stage-gen5b-full/, with its own workflow. No existing file is changed.

Full replay: PASS on the committed package (aea3dc9): all twelve stages (virtual, raw, bit with descent and targets, kernel, second descent, restore, sink, scalar, primes, banks, complex, math, finite), package integrity unchanged (149 files, manifest 7f5feb21…), PASS_IMMUTABLE_GEN5_KERNEL_RESTORE_SINK_FIVE_STAGE_BANKED_CONSTRUCTION, κ 187495641816757/250000000000000000, ~7 minutes. (The first version of this PR, fd51617, was the plain greedy packing with 7 sinks at 7.49917931696438e-4, also a full PASS.)

#299 (chafreaky, 7.49899215319498e-4) composes the same three stages on gen5b and was posted while this package was being built; the two re-derivations agree on every census (1,720 entries of rank 2,192, 440 restorations, 7 sinks, 910 charts, literal stock 1,229,750) and on the pivot-bank workaround. This PR adds the second constructed-frame descent (+1.87 × 10⁻⁸).

What it is

Every bit-side stage of the gen5 lineage composed on chafreaky's gen5b word (#296), each re-derived on gen5b with its own discovery script and frozen selection, in #295's pipeline (parity → descent → target → kernel → second descent → restore → sink → lowering, scalar, primes, banks, complex, math, finite):

stage (credited)	on gen5 (#287/#290/#291/#295)	on gen5b (this package)
concave-descent retiming (#287; #296's re-derivation)	904 gates	904 gates (#296's selection)
target-prefix squares (eumemic #268; #287; #296's re-derivation)	220	220 (#296's selection)
collective response-kernel entries (chafreaky #290; Dugongue #254/#283)	450 entries, rank 1,022	1,734 entries (920 twin pairs, 431 triples, 383 quadruples), entrance rank 2,196 — the greedy packing improved by 250 randomized-order restarts (φ −3,799 → −3,860)
second concave descent with constructed frames (#291)	89 gates	87 gates (32 new bases)
early restoration of cleanup helpers (#280; #295)	440 helpers	440 helpers
terminal sinks (#283; #295)	8	8 (R 15,434 → 15,426)

Kernel entries on gen5b. #295's discovery (discovery/coll/load.py → families.py → pack.py → build_selection.py) on the gen5b word after the descent and squares: 13,200 plain dirty helpers, 1,632 triangles and 10,041 quadruples at class level, 6,863 families with a common cut and a nondegenerate common entrance; the greedy packing under the deficit-fixed φ ledger keeps 1,720 (gen5b's alternative local design makes far more helper responses coincide than gen5's 450); 250 randomized-order restarts of the same greedy with the same exact shared-donor marginals (discovery/coll/pack_restarts.py, discovery/pack_restarts.md) keep 1,734 at φ −3,860. The φ ledger reproduces every κ exactly; the second descent, restorations and sinks add the rest.

Bank tiling. With 1,574 residual-23 pivots, #283's (r⁴, 4³⁰⁻ʳ) pivot banks would need ~188k rank-4 blocks against the ~104k the width-4 helpers supply, so bank_template.build_patterns gives each pivot residual r the banks (r⁴, 24ᵃ, 4ᵇ) with 120 − 4r = 24a + 4b (a = ⌊(120−4r)/24⌋), taking plain 24-blocks from the (24⁵) banks. Every bank still partitions its 120 coordinates; all 4,627,800 role/replica/stage assignments are enumerated, the per-pattern endpoint controls run as before (807,280 banks; 910 exact charts incl. 440 (σ, E) charts; selector calls 905,935,487,400 < 2⁴⁰).

Everything else is #295's and #296's code unchanged: #296's word, generator and word constants (R = 15,434; 18,954 formal columns; 753,472 parity ADDs; entrances 20 × 2,182, 21 × 34, 18 × 16, 17 × 2); #295's transforms, endpoint-aware completions P_σ + I − P_E, pins plumbing and #233 complex stage; #291's descent_transform.py run a second time with descent2-selection.json and raw_ledger.rebind_descent2. Every changed count is a recomputed pinned value in expected/kernel-pins.json; none dropped.

one invocation, m = 120	#296 (gen5b)	this package
helper registers R	15,434	15,426
independent entrances	2,234	3,968
literal stock, 60 replicas	1,236,750	1,229,680
priced calls / rank mass	—	482,890 / 2,691,520
κ	7.46709720962478e-4	7.49982567267028e-4

Verify

python3 -m pip install -r research/five-stage-gen5b-full/requirements.txt

python3 -B research/five-stage-gen5b-full/verify.py --output /tmp/gen5b-full-verification

Python 3.11+, SymPy 1.14.0, a new output directory outside the package; no network; about 7 minutes. Prints PASS_IMMUTABLE_GEN5_KERNEL_RESTORE_SINK_FIVE_STAGE_BANKED_CONSTRUCTION with the pinned κ. gen5bit/producer/regenerate.py --work /tmp/gen5b-regeneration rebuilds #296's pinned bit files byte for byte.

Scope and credits

Exactly #295's/#296'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, #290, #291, #295 and #296 are open; their word, generator and stages are inherited unchanged and not re-proved.

chafreaky (#296: the gen5b word and its re-pinned package; #295: the restoration/sink stages, the pins plumbing and the #233 complex stage; #290/#282/#284: the collective kernel stage and discovery); DreamingOfClouds (#285, #276): the generator and gen4/gen5 word design; utcorvusvolat-dotcom (#280): early restorations; Dugongue (#283, #254/#259/#272): terminal sinks, response kernels, endpoint-aware banking and the pivot tiling rule; eumemic (#268, #244, #234 lineage): the target-prefix mechanism, the common-frame retiming and the package machinery; hcg890 and Jacob Sussman (#234): the five-stage layout; Evan McKinney (#197): completed banks; #233/#256 and jacobalansussman/wht-power-saving-lean#2: the complex supplier. This package contributes the composition, the three re-derived selections, the second descent on gen5b and the generalized pivot 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: Composition of the gen5 mechanisms on the gen5b word.

Review and limits: Source-bound full Linux twelve-stage pipeline and producer regeneration. Each selection is re-derived on this word; contributor mechanisms are credited cumulatively.

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: 56eb64125a76b3427986f4a3fcfc4b6f4cf55591. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.