Conditional kappa 0.001590591090759927: observable pruning, Steinberg cancellation and response-owner composition

[Dugongue]

Result and scope

This adds research/pruned-trace-response, a reproducible finite witness for conditional kappa = 0.001590591090759927 (1590591090759927 / 10^18).

It improves PR539's 0.001584931113695072 by 0.35711187%, and the previous local preload60 result by 0.21286603%. The snapshot is frozen through PR540. These comparisons concern this retained conditional framework; this is not an unconditional multiplication theorem, a new all-size compiler proof, or a claim against later or differently conditioned proposals.

Both changed suppliers are reconstructed and physically checked. Their exact savings are:

• bit: c = 0.001593125101352604961534;
• complex: b = 0.001593519419282306763502.

The bit side binds. Gains are not summed: both complete paid profiles are fed through two rational moment engines and the 47-constraint outer assembly.

Mechanism of the improvement

1. Observable pruning changes the legal optimization space. Exact backward propagation of every output response, including retained-copy scatter at its actual time, finds 227 zero-response preparation gates. For a grouped shear T, the identity Q(T-I)=(Qe_t)u^T certifies deletion when Qe_t=0. Rebuilding literal inverse cleanup preserves arbitrary dirty inputs. This first step removes 454 grouped body/cleanup instructions while leaving the paid histogram unchanged, but it removes dependency blockers.
2. Transport plus paid commutator cancellation exploits that space. Twenty frame-labelled transports and seven Steinberg three-to-two cancellations are checked on the evolving word. The latter replace i+=j; i+=k; j+=k by j+=k; i+=j after certifying the intervening commutations. Relative to preload60, the exact histogram delta is 17(e1-e3-e4+e6) + 10(e2-2e1), a strict concavity gain. The final local histogram is {1:1257,2:1330,3:43,4:163,6:137,8:9}.
3. The complex provider is recomposed, not treated as a fixed ceiling. The static44 response presentation accepts 477 compatible frame lifts from PR534. A finite register-version expansion then proves an exact cold-dirt source relation, allowing one source substitution with regenerated compensation. This saves four scalar operations; it does not claim a paid-rank gain of its own.
4. A new whole-owner pattern allocation makes the complex gain usable. Width inventories and legal call times are realized by a constructive matching decomposition into 36,288 literal helper roles. Exact full-column charts and all physical ownership checks pass. Concave remainder costs guide the search; the checker receives the explicit witness and does not assume an optimizer is correct or globally optimal.
5. PR539's explicit fallback charge is recomputed for the new bit word. The charge uses bad fraction 1/2^61 and per-edge cost 98,682, with the actual changed child count. The inherited bad-class transport premise is stated explicitly.

Two OpenAI structures per worker

The package contains sufficient mathematical arguments, concrete witnesses and source-file hashes for all eight applications:

Lane	Structures tried and applied	Actual contribution
Root	QuantumCircuit zero-action pruning; FourierCircuit rectangular-shear factorization	Exact grouped-gate pruning, enabling the next lane. A separate rank9 copy-factor screen found no compression.
Trace	Chromatic ordered-product noninteraction; BooneHigman Steinberg chain relation	Twenty legal transports and seven compensated cancellations; both have strict paid-moment improvements.
Response	OddKaplansky finite nilpotent inverse; AuslanderReiten dual-kernel factorization	Exact version expansion and cold-dirt quotient substitution, saving four scalar operations after full compensation.
Resource	SwitchChain fiber-preserving switches; MatchingCount cut algebra and concave potentials	Compatible owner allocation and constructive matching lift, admitted by the physical checker. No unconstrained-min-cut optimality assertion.

These are applications of source structures from openai/math, not claims that the underlying identities are globally new. The sources were inspected and hashed; no new Lean certificate is asserted. PR534's numerical frame improvement is explicitly credited to PR534 rather than relabelled as an OpenAI-derived gain.

Bounded failures are retained: compensated moves on the unpruned bit word, the full-rank copy-factor screen, neutral/blocked response packets, and the earlier weaker owner-exchange witness. The final construction shows why the pruning and transport order matters; the notes do not call exhausted restricted searches impossibility proofs.

Validation and replay

Run:

python -B research/pruned-trace-response/verify.py \
  --work ../fresh-pruned-trace-run --cxx g++ --boost-include /usr/include

Requirements: Python3.11+ with NumPy, a C++17 compiler and genuine Boost multiprecision headers. No network, Lean, optimizer or GPU is required. Use a fresh output directory outside the submitted package and keep Python assertions enabled.

The replay reconstructs both suppliers from pinned primitive inputs, then runs native local, complete physical controller, finite invoice and exact assembly checks. It covers all constructor columns, 960/1440 native bit columns, 18 bridge, 6 structural and 3 mutable-source controls, 17 parent and 16 hosted bit controls, 728 complex columns in both orientations, 44 response omission controls, phase/adapters checks, 11 complex local controls, 2,972 full physical charts and seven owner corruptions. Both moment engines certify the endpoints and reject the next grid point; all 47 outer constraints and their negative controls pass. Decimal arithmetic is used only to nominate endpoints.

The root manifest normalizes LF text, including headers and SHA files, before source binding. New word constructors and the complex replay wrapper explicitly fix gzip compression level, timestamp and OS byte. Full replay from an all-text CRLF copy is included in the release validation. Upload batches are content-identical subdivisions of one directory, with no large or generated build artifacts.

Direct upstream lineage

• PR539: sharp fallback fraction and scalar charge, applied to this changed bit profile.
• PR535: post-center complex construction, controller and assembly.
• PR534: source of the 477 transported frame lifts.
• PR531: transport, phase, local emission, whole-owner audits and assembly implementations.
• PR510, PR503, PR493: nine-stage bit controller, r480 source word and retained frame machinery.
• PR484, PR463, PR444, PR442, PR386: retained realization, local phase and complete-owner interface lineage.

Changed-head commit pins and preserved attribution notices are included. PR533/536/537/538 and PR540 were also inspected; their claims are not added to this result. PR540's narrower literal-invocation verification is not used as a proof of the all-size root.

All-size compiler realization, common-label/mutable-basis transfer, whole-call hosting, precision/tape and analytic-transfer interfaces remain conditional. A helper's complete endpoint includes its residual Q action; scalar-body dirt restoration is a separate fact, not a license to erase that residual.

[Mister-Meeseeks]

Thank you @Dugongue. This contribution at 66dbc2bffc1bfa8d50b3f9a2ac7ccba8fd9f6f36 is incorporated in the reviewed family through #588, published in 9594b61.

Pruning changes the selected response implementation; actual scalar and frame replay is required, not a subtraction from the predecessor histogram. Fresh native replay passed.

Preserved implementation and research:

• research/pruned-trace-response

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.