[intqwq]

The joint helper/native-map construction, with a smaller finite compiler reserve, supports conditional κ = 15108879/5000000000 = 0.0030217758. This improves this PR's frozen 0.00302176 checkpoint by 0.0000000158, its original 0.00302152 checkpoint by 0.0000002558, and PR4535's 0.003020165 by 0.0000016108. The selected certificate remains unchanged.

The native construction changes arbitrary helper entrances, scalar contacts and native maps jointly. Its local paid profile is {1:2201,2:412,3:510,4:127,5:40,8:8}, replacing the original checkpoint's {1:2201,2:412,3:506,4:135,5:36,8:8}. The productive helper census stays {4:28,7:7,8:47,9:326}. Four pairs of rank-four paid jobs become rank-three/rank-five pairs; strict real concavity proves old(t)-new(t)=4(2·4^t−3^t−5^t)>0 for every 0<t<1.

The native witness changes 26 helper entrances, 316 scalar contact frames, 20 warp frame pairs and 20 native G maps relative to the original checkpoint. All 408 helper inputs remain arbitrary and independent. Public main endpoints, helper exits, scalar opcodes/coefficients/order, owned jobs and the 4,429-entry frame registry are retained. Every changed helper receives its actual whole relative operator and exact pre/post monomial normalizers. The full p16 controller has 64,404 paid calls and 10,928 productive fields, including all 18 full calls, 15 query shears, idles, diagonals, owned centres, routing and completion bills.

The latest increment keeps that native word, complete profile and raw crossing fixed. It reduces the finite contraction reserve from 1e-9 to 1e-11, retaining power1024, cutoff20, floor21, tail offset1/5, tail coefficient97/100, complement factor63/64 and logarithm power1000. The separately larger tested saving is 0.00302177581. Two exact interval implementations place the unchanged raw crossing in (0.00302177587077, 0.00302177587078). Complete recomputation gives outer gap 1/700000000000, j=55099, H=38569300000000000 and N=2^38569300000000000. Lean checks (H+1)^1000 < 2^55099; the expanded word is algorithmically defined and not materialized.

Validation:

• Guarded, solver-free regeneration reproduces canonical native payload SHA256 cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd.
• Both retained native interpreters pass all 632 arbitrary persistent input columns in both orientations, including owned restoration. Exact phase admission covers every changed native map on 512 columns, all 180 actual warp classes, 105 newly occurring paid pairs and all 26 changed helper interfaces, in both orientations.
• The frozen native/profile package completed 35 Lean modules and 45 exported axiom reports. Its independent AI audit includes five semantic corruption controls and eleven controls against the native Lean wrapper.
• The new tightening completed four fresh Lean modules and 31 exported axiom reports: actual real-power contraction and both charged tails; real compiler capacity; 54 strict plus four functional source outer predicates; finite integer compiler bounds; and an aggregate binding the actual native census/helper endpoints to the complete numerical profiles.
• The tightening explicitly reuses 31 native/concavity dependency modules. Its replay checks their runtime, source/log/object/dependency hashes, compile commands, axiom reports and all 49 external input pins. These are recorded separately from fresh compilations.
• A separately implemented AI audit reconstructs the complete invoice and checks independent exact intervals, tails, capacity, integer bounds and all 58 outer formulas. Three isolated corruptions against the actual new replay tool reject a wrong saving, wrong paid profile, and a modified numerical proof with its prepared hash also changed.
• All printed axiom reports use only propext, Classical.choice and Quot.sound. No sorry, custom axiom, native_decide or kernel bypass is used. Lean is pinned to 4.21.0 (6741444a63ee), mathlib to 308445d7985027f538e281e18df29ca16ede2ba3. The new larger integer proof uses a 256 MiB thread stack.

Packages and replay instructions:

• research/kappa-round3-warp-20261022/: frozen changed native witness, exact native/phase replay, complete original arithmetic and bounded discovery screens.
• research/kappa-round3-warp-lean-20261022/: frozen 35-module proof package and original genuine replay receipt.
• research/kappa-round3-audit-20261022/: independent AI native audit and adversarial wrapper controls.
• research/kappa-round6-compiler-tightening-20261022/: new exact arithmetic, readable source derivation, four fresh Lean modules/logs/receipt, independent AI arithmetic audit, corruption controls and a checked publication inventory. Its README includes clean-checkout replay commands with explicit native evidence reuse.

The original 0.00302152 and 0.00302176 packages remain frozen. Python validation uses the standard library; discovery solver status is not trusted for admission. No toolchain, compiled object or solver cache is published.

Built against canonical main f626db5c3a3d781d8f7695038776c621651f75e1; main was refreshed at fbc9ea0249746acca4f504d4fda2b84e0e36cf0d. Credit Cinna Davis's PR4470 supplier and cumulative lineage, Dugongue's transported frames, zxv's native/compiler work and Jacob Alan Sussman's mixed-target recursion. Pierre-Loic Doulcet's concurrent PR4626 at c665f908fc991aa558ea8a8cd33e9ad26f97a22d supplies a different public helper621 adjustment; it is credited. Concurrent PR4635 at 898b0c8e2ca23897c312230988515c2553154f30 uses this PR's paid profile for its conditional Word RAM assembly; this PR retains its existing compiler/model and does not adopt that assembly as a proof.

These are scoped Lean certificates for a conditional construction, not a complete Lean integer-multiplication theorem. JSON transcription, native amplitudes/absolute phases and free-group endpoint semantics remain separate Python/proof boundaries. Uniform physical execution, whole-job normalization, permitted owned erasure, tape, precision, stock, analytic decomposition, prime avoidance and exact recovery retain the predecessor hypotheses. No practical speedup, independent human review or hosted-CI proof replay is claimed. The latest increment is compiler tightening; it establishes neither a new native profile nor a larger raw crossing.

The PR4610 comment has been addressed with credited scope and acceptance notes. The new one-edge phase classification covers all 34,638 edges of its declared fixed graph through 2,008 exact full-system deletions. All 100 helper-changing cases, including their entire surviving form spaces, are single-helper boundary preprocessing. All 100 full candidate invoices lose. Two fresh analytic Lean modules expose seven standard-axiom reports, using independent negative-exponent and strict-concavity proofs; the finite graph classification remains Python. The independent audit also preserves the overlapping-window route rejection control.

Concurrent-work credit: Cinna Davis's PR4696, at def8293, already gives a broader written rank-debt argument against paid symmetric-phase changes on the retained graph. Our narrower finite classification does not replace or claim priority over it. PR4690, at ddb7ead, supplies charged non-nested/phase/common-Pauli searches. Overlapping fixed-ambient sweeps were stopped. Chart-dependent phase-adapter constructions that change actual subspaces remain under investigation, with all paid relatives, broken warps and helper normalizers charged. This research checkpoint supplies no higher kappa or admitted new native profile.

Open work: change the scalar word or helper graph beyond the present contact model, integrate coupled recursive contracts with every restoration/normalization bill, and formalize native phase semantics plus the complete compiler transfer.

Signed: Cao Yifan. Substantial OpenAI Codex assistance. Apache-2.0; inherited notices preserved.

[Matarener]

On open question (3), changing the order of scalar operations inside the contact block: I checked it and found no gain under the frame model of free_warp_milp (warp in/out frames fixed, frames of all-L scalar events re-chosen, climbs priced as d^(1-κ)).

Against the 0.00302152 word (4b4c75b):

• Exact MILP with order variables, one commuting pair per group (3,511 to 5,082 pairs per instance; 8 variants with different offsets, free/fixed G and shuffled role orders): every instance solved to proven optimality (HiGHS, gap 0), gain 0. The solver does reorder 84 to 298 pairs, but only as ties at equal cost.
• Plateau walk (apply those tie swaps, re-solve, repeat; 3 chains, 16 solved steps): gain 0 at every step, profile stays 2201/412/506/135/36.
• Exhaustive window screen: every window of up to K consecutive events of a commuting run on one role (4,040 runs), all acyclic permutations, frames re-chosen exactly with the outer neighbours fixed. K = 3, 5, 6, 7, 8: about 1.6M permutations in total, 0 improvements; the identity-order control never beat the emitted frames.

Against the updated 0.00302176 word (1eb8da6): I re-ran the same window screen for K = 3, 5, 6, 7, 8 (about 1.6M permutations): 0 improvements, control clean. I did not re-run the exact MILP or the plateau walk on this word.

Scope: this only covers reordering commuting scalars with warp frames held fixed. Moving warps, or reordering jointly with new warp maps, is not covered. Scripts are available if useful.

[intqwq]

Addressed Matarener's bounded reordering results in the source-bound review note.

The reported MILP/plateau work applies to the older 4b4c75b word; for the current 1eb8da6 native word the reported evidence is the commuting-window screen through K=8. I credit this as a bounded negative result with fixed warp frames/maps, rather than a global optimum for the current word. This update does not claim an independent replay of the unpublished search scripts.

The next construction work changes paid phase interfaces, owned-centre contracts or grouped recursive job types, with every normalizer, route and restoration charged. Those changes lie outside the comment's fixed-warp search premise. The claimed conditional kappa stays 0.0030217758 until a changed construction passes the full invoice and Lean checks.

Signed: Cao Yifan. Substantial OpenAI Codex assistance.