Conditional kappa 0.001587212454640692: trace refinement with a curvature-cone certificate

[Dugongue]

Result

Conditional kappa = 0.001587212454640692, exactly 396803113660173/250000000000000000: +0.143940% over #535 at its pinned head 0d343e2f813f962405481f83d8935299812a7a1c.

This combines a new bit word with #535's unchanged complex supplier. The bit supplier still binds:

• New bit saving: c = 0.001589735702947998836661.
• Cited Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535 complex saving: b = 0.001589991533606044174031.
• Public snapshot frozen through Conditional kappa 0.001530327580902932: kernel92 plus optimized delivery frames #536; this is a comparison to that snapshot, not a claim about later submissions.

New numerical mechanism: move the scalar event, then relabel its frame

On the R480 bit word from #503/#510, factor 60 two-source helper additions. Move one summand across its explicitly checked commuting prefix and execute it at the original source's rank-one frame. Leave the other summand at its old location, then regenerate literal inverse cleanup. The complete operator on arbitrary source, target and dirty-helper inputs stays unchanged.

The helper climb splits into two rank-one climbs, while a source's formerly separate rank-one and rank-two climbs coalesce into rank three. The exact paid local difference is

60 * (e1 - 2*e2 + e3).

The complete local profile, including all nine retained rank-eight copies, is

{1:1260, 2:1320, 3:60, 4:180, 6:120, 8:9}.

The 480 helper residuals keep their width-eight signature. They fit the unchanged literal whole-owner selection in nine stages and 32 replicas. The complete global profile, including every remainder, is

{1:362880, 2:395520, 3:17280, 4:67200,
 6:49920, 7:23040, 8:2592}.

Live stock 30,720, physical stock 30,729, rank deficit 9,984, total child count and primitive scalar-add count are unchanged. The gain comes from the rank distribution, not omitted work or added numeric bonuses. The complete two-supplier assembly is recomputed.

Four reusable sufficient arguments

1. Frame-sensitive trace refinement. Scalar factorizations and certified commuting exchanges preserve the full dirty operator; explicit predecessor/common/successor inclusions certify the new frame chains. The optimizer searches event order together with frame assignments, exposing changes unavailable to fixed-order frame optimization.
2. Complete-context resource substitution. Equal normalized endpoints and boundary/residual signatures allow replacement of any subset of complete supplier calls. Actual owner maps, pending residual operators and spectator restoration are retained. Exact profile differences compose through the controller; scalar-body restoration is not confused with a physical helper identity.
3. Curvature-cone calculus. With equal rank mass and child count, the finite certificate C_j = sum_{r<j}(j-r)*delta_r characterizes domination for every discrete concave cost by C_j >= 0. Its second difference reconstructs the exact profile delta. Here only C_2 = 17,280 is nonzero, proving strict moment improvement for every exponent 0 < a < 1, including the unchanged fallback envelope. The proof is independent of decimal tuning.
4. Filtered and versioned response compilation. Reusable-donor elimination requires reversed physical stage order and checked frame chains; compensation moved past a background prefix must become D A^-1. The package gives a concrete neutral 720-column transport example and a separate 40-to-44-pivot complex presentation checked on 728 columns in both orientations with 44 omission controls.

The fourth argument's examples are research extensions, not extra final exponent gains. The 44-pivot complex candidate is local-only and still needs global physical admission; the numerical result uses unchanged #535. These are explicit sufficient proofs for this application, not claims that transvections, concavity or conjugation are globally new mathematics. No new Lean certification is claimed.

Direct dependencies and inspiration

• Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510 (64edee18af02aab0c1696b677a0287dd0146381b): nine-stage R480 controller, exact whole-owner selection and bit invoice.
• Bit word R480: totals on dead registers, c = 0.001574753352953808650916 (+0.595% over #493) #503: R480 dead-total source word; Conditional κ = 0.001303530097714685: an E8 bit word with 489 helpers, sources as carriers (0.001325 with #484's draft complex supplier) #493: E8 skeleton, mutable-source/native legality, formal-column, prefix and corruption checkers. Their exact versions are vendored from Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510.
• Post-center rank-four kernels for conditional κ = 0.001584931111785787 #535 (0d343e2f813f962405481f83d8935299812a7a1c): unchanged complex supplier and assembly lineage. It is cited and repriced, not redundantly reconstructed.
• Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463: variable-stage global emission and invoice; Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442/Conditional κ = 0.001246073505487942: R740 replicated kernel with idle-data hosting #444: whole-residual hosting; Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386: nine-stage geometry. Existing source manifests retain their subordinate references.
• Direct delivery with 504 helpers for conditional κ = 0.001517760977820331 #531: the 504-helper complex foundation; Conditional κ = 0.001524207950793244: 92-pivot direct504 supplier #533/Conditional κ = 0.001526158885117467: frame-chain search on #531's 504-helper word #534/Conditional kappa 0.001530327580902932: kernel92 plus optimized delivery frames #536: response-presentation and frame-chain comparisons informing the local appendix. Their gains are not stacked into this bound.
• Source-read analogies: Mathlib transvections; OpenAI ordered products, machine substitution, elementary congruence clearing and Jensen-gap proofs. Exact paths, declarations and source hashes are documented. These informed the mathematics rather than serving as unproved imported algorithms.

Verification and reproduction

From research/trace-curvature-preload, with Python 3.11+, NumPy, SymPy, C++17 and genuine Boost headers:

python -B verify.py --work /absolute/new/output --cxx c++ --boost-include /usr/include

Offline; no search, optimizer, network, GPU or Lean is required. The replay checks:

• 720 dirty-input local columns and exact rational frame incidence;
• freshly compiled native 960-column five-stage and 1,440-column nine-stage checks;
• all 18 omitted-bridge, 6 structural and 3 mutable-source controls;
• all 672 frame Grams;
• 10,631,808 emitted global events and 138,240 whole residual occurrences, with 17 parent and 16 hosting mutations rejected;
• fresh finite invoice: prefix 193 bits charged 256, two-level coefficient 425 bits charged 512, row coefficient 18,355 below 20,161;
• exact curvature certificate, both independent rational moment engines, all 47 strict outer inequalities and four invalid-parameter controls;
• neutral version transport and the explicitly local-only static44 appendix.

Root manifest hashes normalize text line endings; the new bit-word and geometry gzip files fix level 9, mtime 0 and OS 255. Inherited generated templates are identified by uncompressed contents. The final upload is split into small browser-safe batches, and its reconstruction is compared against the checked source tree. User-identifying attribution is redacted only from inherited metadata, while PR numbers, source hashes, other authors and licenses are retained.

Scope

This is a conditional finite-witness improvement under the retained all-size compiler, common-label/mutable-basis, whole-call hosting, restored-work, precision/tape and analytic-transfer contracts. It is not an unconditional sub-n-log-n multiplication theorem. Proofs, bounded unsuccessful routes, constructor recipes, upstream lineage and exact evidence are included for future work.

[Mister-Meeseeks]

Thank you @Dugongue. This contribution at 8725c162e4ffdbada8847e235ccdbcaedef87af9 is incorporated in the reviewed family through #588, published in 9594b61.

Trace curvature and context substitution alter the local frame ledger; finite replay compares the reconstructed profile and exact cost against the stated curvature certificate. No global optimality is inferred.

Preserved implementation and research:

• research/trace-curvature-preload

The family review and individual disposition record the scope, evidence and attribution. Parallel and intermediate contributions remain credited even where a later composition has a stronger bound. The published multiplication bound remains conditional on the documented compiler, realization, hosting and analytic assumptions; these finite checks are not a full Lean multiplication proof.