Conditional κ = 7.44136146100942e-4: two concave-descent retimings (904 + 111 gates) and 220 target-prefix squares on #285's gen5 five-stage banks

[rohanarun]

Conditional κ = 372068073050471/(5·10^17) = 7.44136146100942e-4, +1.35 × 10⁻⁶ (+0.182 %) over #285 (7.42785663120535e-4) and above #289 (7.44127392470948e-4). Stacked on #285's branch at its head; three added stages after parity_transform.py: descent_transform.py (904 gates, frozen descent-selection.json), target_transform.py (eumemic's PR268 exact F₂ target-prefix compression, 220 squares, frozen target-selection.json), then a second run of descent_transform.py with its own frozen descent2-selection.json.

Full replay: PASS on the committed composed package (9f85de8): all nine stages (virtual, raw, bit, scalar, primes, banks, complex, math, finite), package integrity unchanged (115 files, manifest 0a31d05c…), PASS_IMMUTABLE_PARITY_FUSED_GEN5_FIVE_STAGE_BANKED_CONSTRUCTION, κ 372068073050471/500000000000000000, ~5 minutes.

Change

Stage 3 — second concave descent after the squares (111 gates, +87 calls, higher κ). On the word after the squares, the same concave-descent search with constructed candidate frames (the minimal join and the maximal meet of a gate's neighbouring operand frames, not only frames the word already uses) finds 111 further internal ADD reassignments onto 48 new bases. Local paid histogram delta {1: +63, 2: +198, 3: −126, 4: −48, 5: +24, 7: −48, 9: +24, 12: −24, 14: +24, 18: −63, 19: +63}: the local call count rises by 87 (five-stage calls 473,599 → 474,034) at unchanged rank mass 2,746,720, but the concave weight Σ r·ln(120/r) falls, which is what the moment certificate prices, so κ rises by 1.54 × 10⁻⁸ over the two-stage package (7.44120696397042e-4) and by 8.75 × 10⁻⁹ over #289. The new bases are explicit rational rows in descent2-selection.json; descent_transform.py registers them, checks nondegeneracy, both reflected annihilator ledgers and the exact integer source span of every non-target operand, and prime_check evaluates their determinants with every other actual basis (26,109 unique nondegenerate endpoint bases). raw_ledger.rebind_descent2 recounts the profile; bank_check (471,480 banked calls) and verify.py pin the new counts and κ.

Stage 2 — target-prefix compression (220 squares). After the retiming, every target reads rank-20 entrance helpers at a 20-dimensional frame; in each of 220 squares of four targets each helper is read by exactly two of the four, so the dependent target's prefix response is the F₂ sum of its three square-mates'. Its 1,100 frame-20 reads are omitted; it receives t −= p_j at the zero frame at the cut and t += p_j at the square's common nondegenerate 21-dimensional frame at the close (1,320 setup/restore additions): ZERO →(20)→(1)→ 21 becomes ZERO →(21)→ 21, delta {1: −221, 2: −8, 17: −1, 18: −7, 20: −212, 21: +220}. The transform is eumemic's target_transform.py from #268 unchanged apart from the stream count; it recomputes every literal dependency from the actual word, rebuilds every MOVE, checks nesting, endpoints, COPY lifetimes, nondegenerate bases and both all-column replays with an omitted-setup control, and rebinds the raw ledger. Downstream constants follow (575,052 weighted ADDs, coefficient counts, event hash; signed prefix bounds 46,393 / 2,282,970 unchanged); per-replica bank call count 471,045.

Stage 1 — retiming.
#285's gen5 pipeline (prepare → physical producer → parity fusion → lowering) carries no frame retiming. The concave-descent search behind #263/#279 finds 904 single-gate frame reassignments on gen5's parity-fused word — PR244's common-delivery-frame retiming for gen5's 880 unborrowed source-pair mixes (per pair the paid children of ranks 1, 1, 20, 22 become 21, 21, 2) plus 24 further internal gates. Local paid histogram delta {1: −1760, 2: +880, 6: +24, 7: −48, 8: +24, 20: −880, 21: +1760, 22: −880}; 880 local recursive calls per invocation disappear at unchanged rank mass 411,356. Five-stage profile: 474,744 calls (was 479,144), rank mass 2,746,720, deficit unchanged; per-replica bank call count 472,190 (was 476,590).

descent_transform.py consumes the frozen descent-selection.json (record bindings, old-basis hashes, explicit rational new bases) and, like PR244's retiming_transform.py, rebuilds every MOVE from the selected 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, the expected histogram delta and rank mass, and the exact integer source span of every non-target operand inside its new frame, recomputed from the actual word. raw_ledger.rebind_descent recounts the five-stage profile; bank_check and verify.py require the new counts and κ; prime_check retains the determinant obligation for producer frames the retiming removes (PR244's rule, as in #279).

python3 -m pip install -r research/five-stage-gen5-banks/requirements.txt
python3 -B research/five-stage-gen5-banks/verify.py --output /tmp/gen5-descent-verification

Python 3.11+, SymPy 1.14.0, a new output directory outside the package; no network. About 6 minutes.

Scope and credits

Exactly #285'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 or practical benchmark is claimed. The other levers the gen4 lineage accumulated (entrance regauge, kernel families, target-prefix compression, early restorations, fixed prime) are not yet applied to gen5 and compose on top of this.

DreamingOfClouds (#285, #276): the gen5/gen4 bit words and their five-stage completed-bank packages; eumemic (#244, #210, #234 lineage): the common-frame retiming this reproduces and the package machinery; hcg890 and Jacob Sussman (#234); Evan McKinney (#197); and the retained lineage. The descent search, the transform stage and this package's rebinding were prepared by Rohan Arun with Anthropic Claude assistance.

🤖 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: Two concave-descent passes and target-prefix compression on gen5.

Review and limits: Source-bound complete Linux verifier and producer regeneration. The later second pass is included in the frozen head, not confused with the earlier published checkpoint.

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: 9f85de891c5eec5de4e33a9185157c9e573bb315. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.