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Conditional kappa 0.001591968902833346: collective hyperplane waves and weighted response decoding - #552
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Thank you @Dugongue. This contribution at Collective waves and weighted coset responses require a common frame at each changed operation. The eight-wave collective ledger and twelve coset rewrites are paid. Canonical sources equal the fully replayed vendored foundation. Preserved implementation and research: 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. |
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Result
This adds
research/filtration-wavefront, giving conditional kappa = 0.001591968902833346 (795984451416673 / 500000000000000000), 0.07906544% above PR547. It also removes 624 scalar additions per complete local invocation. The numerical and scalar improvements are counted separately.The new exact complex saving is b = 0.001594507308884431019637. The pinned PR543 bit saving is c = 0.001653826988146798944268, so the complex side binds. This composes complete paid profiles through both rational moment engines and all 47 strict outer constraints; it does not add estimated gains.
The public snapshot is frozen through PR547. All-size compiler realization, common-label/mutable-basis transfer, whole-call hosting, precision/tape and analytic transfer remain conditional, including PR543's stated H1-H5 argument for interleaved phase-B source updates. No unconditional multiplication theorem or new Lean certification is claimed.
New mechanism: collective frame changes
Single-event frame changes and equal-frame plateau changes found no improvement on the starting word. The useful cross-field identification is that a register's increasing frames are a finite persistence module: subspaces with injective structure maps. A gate shared by two registers couples their flags.
Choose a hyperplane H. Intersect a selected collection of gate frames with H, propagating the change backwards whenever a predecessor is not already contained in H. Reject if this reaches an immutable boundary. This is a concrete consequence of the image/preimage Galois adjunction: intersection is monotone and deflationary. A four-case edge proof establishes legality for the whole changed diagram. It requires neither a common basis for every frame nor a general interval decomposition of a branching persistence module.
Eight such waves change 8-10 event frames together. Each wave changes the local histogram by
The total is 16 copies of the replacement (2,3) -> (1,4). Because (2,3) is a doubly stochastic averaging of (1,4), strict concavity gives
Thus the gain holds for every relevant recurrence exponent, not just at a numerical search point. Rank mass, child count, boundary signatures and helper widths remain unchanged. The scalar word stays identical during this step, but the physical phases do not: they are rebuilt and checked rather than inherited without review.
New mechanism: weighted response representatives
A second construction treats available source values modulo cold initial dirt as a coset. State-space prefix/suffix response composition computes the rank-one change to the compensation table; an exact weighted representative search chooses a cheaper member of the same protected-input class.
Twelve source substitutions reduce compensation additions from 9,756 to 9,132. After regeneration and physical compilation, emitted events decrease from 22,107 to 21,483 in each orientation. All protected columns and the improved paid histogram are preserved. These 624 additions are a scalar saving, not a further paid-rank/kappa claim.
An independent C++ checker using exact multiprecision dyadics verifies all 48,832 protected transfer coefficients, all 8,736 entries in the twelve source-error witnesses, and literal inverse cleanup. The checker uses a separate finite path/triangular-boundary implementation, not the discovery response routine.
Eight cross-field connections and their outcomes
These are precise applications of established mathematical structures, not a claim that the underlying equivalences are globally new or that all results transfer automatically. The package includes the sufficient hypotheses and proofs, actual source references/hashes, failed bounded searches, and scope distinctions. It records 31,627,350 single-functional closure candidates and a further 7,381,812 two-functional candidates; exhaustion of those neighborhoods is not a global optimality theorem.
Validation
The fresh changed complex word passes all 728 columns in both orientations, phase and adapter checks, 11 local mutation controls, 2,972 full 73-column physical owner charts and seven owner corruptions. Eight additional omission controls detect a necessary missing frame change in each collective wave. Both rational moment engines accept the frozen endpoints and reject the next grid point; all 47 outer inequalities and four negative assembly controls pass.
Final global histogram:
Complex live stock is 7,168, physical stock 7,682, deficit 2,560, with 36,288 helper roles over 72 calls. Measured prefix 225 is below charged 512; row coefficient 16,467 is below retained 20,161.
Replay with Python 3.11+ and NumPy:
Use a new output directory outside the submitted package and keep assertions enabled. Mandatory replay needs no network, Lean, optimizer, GPU or C++ compiler. It reconstructs the changed complex provider from primitive inputs, checks its complete physical realization, and composes it with PR543's pinned numerical receipt. The unchanged PR543 physical operator is cited, not redundantly rebuilt. The independent C++ path audit is an additional source-bound optional artifact.
The full package is tested from an all-text CRLF copy. The manifest normalizes LF text; new word constructors and the complex launcher pin gzip serialization. Browser upload batches are byte-identical subdivisions of this single directory.
Direct PR lineage
2065fe4f72a914d907f1046603f5a36f17018535.4e25588aac914732bde23ab0bd6628d25a3df0db.PR542/544/546 and the updated PR540 formalization scope were also inspected. PR545's stronger draft bit value would not move this complex-bound minimum; its gain is not added. Detailed pins, preserved upstream attribution and field-to-field research sources are included in the package.