Research: projection-cut frame optimizer and eight costed structural transfers (no new kappa)

[Dugongue]

This adds a constructive frame-optimization theorem, a native nomination tool, exact finite tests, and eight cross-field transfer arguments. It claims no new kappa. The prior checked conditional value remains 0.001591968902833346; the bounded searches in this package found no additional admitted gain.

Main contribution

For a legal finite frame DAG, fix a rank-one projection P(x)=x+lambda(x)v over F2 with lambda(v)=1. Each unfixed scalar-event frame may retain F or become P(F). The package proves exact mixed-edge legality rules and represents the complete paid-objective change as cut capacity minus a constant. One directed minimum cut therefore optimizes ALL legal choices in that family, including dimension-preserving frame rotations coupled to contractions. Shared scalar events, parallel register edges, arbitrary fixed boundary frames and the finite hard-capacity bound are explicit. This is an application of established graph-cut and rank-nullity mathematics, not a claim to have invented those general theories.

The graph search is only a nomination step. A positive candidate would require fresh physical phases/adapters, arbitrary-dirt columns, global owner realization, invoice and exact supplier/outer assembly before changing kappa. No such positive candidate arose in this snapshot, and no check was weakened to manufacture one.

Eight structural transfers retained for future workers

1. Binary domain-wall energies to simultaneous compiler-frame cuts.
2. Rank-nullity and oblique linear projections to coupled frame rotations/contractions.
3. Tensor-network inverse gauges to fully charged cold-carrier conjugation.
4. Linear-matroid circuits to reversible warm entrance elimination, including the two-change causal cross term.
5. Reversible pebbling to dirty-register lifetime reuse with an explicit protected-state reset.
6. Continuation equivalence to multi-carrier response transfer with inverse cleanup.
7. Causal-flow transport potentials to resource shadow prices and integral owner realization.
8. Tropical path composition to controllers carrying complete boundary states and attaining witnesses.

Every note states a sufficient constructive transformation and its actual scope. Theoretical eligibility, finite screens and achieved numerical gain are separated. Consulted OpenAI/Mathlib sources and hashes are recorded; none is represented as newly Lean-certified.

Public lineage and comparisons

The frozen sweep includes PR550@c7337040822978f1611ce9d9d44e7bb96d887e23, PR549@a248674ccdf0ed32f3a122f594b54524c90e4654, updated PR547@eecca20a24258cb6d9f698b1984a95029bac0a1c, PR548, PR541 and PR275. Immutable heads/links are in sources/PUBLIC-PINS.json.

PR550's two proposed rank-five gates already have those frames in the previous wavefront construction. A literal transplant produced no paid-profile change, so the saving was not added twice. PR549's ceiling is scoped to its older fixed provider. PR275's finite-assembly and source-basis ordering work is relevant context but supplies no new provider gain here.

The frozen input comes from the previously checked wavefront construction, using PR543's cited R420 bit profile and the PR531/534/535/544 complex/controller lineage, with our prior collective contractions and compensated source substitutions. PR547 and PR548 supply direct assembly/source comparisons. This toolkit does not reverify unchanged public operators or introduce a new supplier claim.

Validation and bounded results

python -B research/projection-cut-toolkit/verify.py --out /absolute/new/output

Python3.11+, standard library, offline. This checks normalized file bindings, the5,471-node/728-register frame graph, and exact integer projection/cut identities on3,948 legal edge assignments and1,008 graph cases comprising67,600 labelings. Fixed boundaries and parallel edges are included. A full CRLF copy passed the same command. The written universal theorem also received a separate mathematical review. No Lean formalization was performed.

Observed discovery screens:130,816 rank-one projections;1,022 line/hyperplane directions;46 distinct neutral variants followed by another47,012 directional tests;117 single and6,780 coupled representative changes;48 legal carrier-basis candidates;8 recycling seams;586 continuation nominations; and broader two-/three-gate trace searches. No additional paid winner emerged. Floating-point and modular screens are bounded discovery evidence, not global optimality or characteristic-zero impossibility certificates. Separate lane screens are recorded, not replayed by verify.py.

The29-file package contains no build binaries, hidden files or giant data items. The conditional all-size compiler, precision, common-grid and analytic-transfer obligations remain untouched. This is reusable mathematical/search infrastructure, not unconditional faster integer multiplication or a new numerical record.

[Mister-Meeseeks]

Thank you @Dugongue. This contribution at 24321d9347828751adabd86de4a4a7807207afac is incorporated in the reviewed family through #588, published in 9594b61.

Projection P=id+v*lambda is idempotent with rank-one kernel over F2. Mixed inclusions give hard cut arcs; losses (1,1) cost f(r+1)-f(r) and losses (0,1) source arc f(r)-f(r-1). Summed objective equals cut minus constant. Valid for nondecreasing f. Compiler admission remains separate.

Preserved implementation and research:

• research/projection-cut-toolkit

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.