Draft: Conditional κ = 7.10436534421701e-4: concave-descent frame retiming of 25 gates on #251's source527 banks

[rohanarun]

Conditional κ = 710436534421701/10^18 = 7.10436534421701e-4, +2.85 × 10⁻⁹ over #251 (7.10433687047883e-4). Stacked on #251's unmerged 37-entrance construction at 2b030c0.

Status: draft, not a leaderboard candidate. This is above #251, #252 and #253, but below the #254 lineage; #258 independently composes the same 25-gate retiming with #254, and #259 (518 entrances, 7.11029e-4) is now far ahead. This PR stays as the Python-verifier (eight-stage verify.py) form of the retiming on #251's word.

Full replay: PASS. verify.py --output /tmp/descent-source527-verification completed all eight mandatory stages with package integrity unchanged in 347 s: {"status": "PASS_IMMUTABLE_PARITY_REGAUGED_DESCENT_SOURCE527_FIVE_STAGE_BANKED_CONSTRUCTION", "manifest_sha256": "00119cd179dcc2c837c0c9bf53f32f5131a9c88e278f7aad41e84179be5e2a42", "files": 133, "kappa": "710436534421701/1000000000000000000", "fresh_stages": ["banks", "bit", "complex", "finite", "math", "primes", "raw", "scalar"], "inputs_unchanged": true}.

Change

A second exact frame retiming of #251's regauged source527 ADD word. Twenty-five internal forward_gate ADD frames (25 of 786,623 gates) are reassigned by local descent on the concave paid-rank objective Σ r·ln(120/r), using only frames already present on the operands' chains. The local paid histogram changes by {2: −2, 4: +24, 5: −48, 6: +25, 11: −1, 13: +2, 17: −1}: one local recursive call per helper invocation disappears at unchanged rank mass 433,953. The literal construction keeps 24 replicas, stock 521,425 and deficit 105,600 with 11,792,160 paid calls (was 11,792,280); the per-replica bank call count is 491,340 (was 491,345). Banks, charts, entrances, prime witnesses, the scalar word and every endpoint are unchanged.

Why this is the whole retiming budget: for the window (prev, cur, next) of a register the objective is concave in dim(cur), so each gate is optimal at either the minimal frame (join of the operands' previous frames and their source spans) or the maximal frame (meet of the next frames). On #251's word 99.7 % of gates have those two coincide with the current frame; the fully eager (greatest-fixpoint) assignment is worse (Σ r·ln(120/r) = 1.1817 × 10⁶ vs 1.1374 × 10⁶), and block moves (raising or lowering whole runs between ascents) are blocked by partner chains or source spans. The 25 gates are what remains.

Verification

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: nested chains, fixed source/dirty/target endpoints, nondegenerate endpoint bases, both reflected annihilator ledgers, unchanged COPY lifetimes, the byte-identical scalar/COPY projection, and the expected histogram delta and rank mass. In addition it recomputes the exact integer source span of every non-target operand from the actual scalar word and requires it inside the new frame (the inherited word satisfies this invariant for all 1.57 M non-target operand instances; targets live in cap coordinates and are excluded, as in the inherited word). raw_ledger.rebind_descent recounts the five-stage profile (493,656 calls, rank mass 2,830,840, deficit 4,400); bank_check and verify.py require the new counts and κ.

python3 -m pip install -r research/five-stage-source527-banks/requirements.txt
python3 -B research/five-stage-source527-banks/verify.py --output /tmp/descent-source527-verification

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

Before this PR was opened, the new transform stage passed inside the full verifier (PASS concave descent retiming 25 gates 95452 paid calls 885276 records) and the κ was reproduced with the package's own moment/outer engines (710436534421701/10^18); the complete eight-stage replay with the final package is in progress.

Scope and credits

Exactly #251'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.

eumemic (#251, #249, #244, #210): the source527 construction, entrance transport, common-frame retiming and verifier; hcg890 and Jacob Sussman (#234); Evan McKinney (#197); Chafik Boukhalfa (#200); Avi Eisenberg (#193); Dugongue (#216), sennemmi (#230) and the retained PR207/211/237 lineage. The concave-descent retiming and this package's rebinding were prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code