Conditional κ = 0.001600682403612293: joint-prefix kernels and structural bounds

[DominikScholz]

Change and mathematical scope

Admit conditional κ = 0.001600682403612293, +0.0384301857% over PR569. This is an exponent-saving improvement, not a measured runtime speedup.

The construction combines a new 57-gate A prefix (previously 69 gates), four additional cold relations from PR568, and PR569's 32 width-seven promotions. Its local paid-profile change is -12 H2 + 8 H3, with the same residual-width inventory. Four explicit helper-label swaps let the 63 selected calls retain PR569's physical geometry; nine plain four-cold calls remain. Dimension 73, live stock 7168, physical stock 7682 and deficit 2560 are unchanged.

The native complex rate is 0.001603248695587935849402; the conservative assembly separately admits κ 0.001600682400830496.

New work is confined to research/structural-gain/. This PR targets main and depends on unmerged PR569, so its comparison currently also includes that ancestor's research/larger-gain/ package. Those 341 manifest-bound artifacts are unchanged. PR569's mathematical source is pinned at f1db606a1416359229f944946667338305208064, followed by packaging-only commit 1424da81263a5305b35e4cf74ef686c275f03f3f. External PR564 is pinned at c18417b02b80594b2ca1c8c21185d28befafc935; PR568 at 71c99ac935a33bb7b59a1b4a7af8725a939b0b90.

The package also records bounded experiments in joint bases, shared gauges, global frame choices and foreign-output borrowing. Removing 28 actual helper paths without replacement or output-path cost prices an optimistic +3.38753% target. No restoration circuit, physical admission or bit-side composition achieves that target. It is not part of the claimed improvement. The exclusions likewise apply only to their stated models.

Inherited all-size compiler, common-label/mutable-source, hosting, precision/tape and analytic-transfer assumptions remain. The unchanged PR543 bit supplier is cited and repriced, not freshly rebuilt. No unconditional multiplier theorem, complete Lean proof or general worst-case O(1) indexing result is claimed.

Contributor / source	Retained contribution
Dominik Scholz, #569 and this PR	Width-seven construction; new joint prefix, composition, owner relabeling and structural experiments
jeremykothe2, #568	Multi-cut cold relations and constructor/control format; four relations add beyond #569's overlap
Jonathan Capalbo / jon314159, #567	Compared overlapping kernel; no separately additive gain
eumemic, #564/#561/#562/#558, #550, #543	Supplier family, cold kernels, admission/assembly ancestry and cited bit supplier
Rohan Arun / rohanarun, #559	Native exact-complex moment argument
Dugongue, #552/#505/#563	Weighted responses, collective waves, whole-owner hosting and pattern work
Chafik Boukhalfa / chafreaky, #551	Phase driver and inherited geometry/audit/invoice machinery
icekylinx, #531/#535	Direct-delivery factory, frames, compiler/controller ancestry and kernels
DreamingOfClouds, #484 and successors	Shared donors, kernels, gauges, wrappers and word/frame foundations
DaysSky, #442/#381; Zhihao Chen / jacklightChen, #425/#386	Hosting, bridge, mutable-source and geometry foundations

Prepared with substantial OpenAI Codex (GPT-6 Astra) assistance in research, implementation, verification and documentation. Source notices, licenses, earlier contributors and contributor-specific AI disclosures are preserved; see research/structural-gain/NOTICE.

Validation

• Fresh 18-stage composition replay passed: native columns in both orientations, complete phases, literal ownership, invoice, both exact moment engines and both outer models. Prior baseline receipts and pinned inventories are validated and reused, not represented as newly rebuilt.
• Independent 10-stage joint-basis replay passed, reproducing 35 literal words and exact search records. Frame/factory experiments include optimized-Python checks and targeted rejection controls.
• Full receipts, source pins and reproduction commands accompany each package. All 34 retained Python files parsed. git diff --check reports three trailing blank lines in preserved, hash-bound source copies.
• No fresh aggregate make verify pass is claimed. During PR569 preparation, make gen4-bank-verify passed; broader verification stopped at unavailable Boost headers. No code-review workflow was run, as requested.

[Mister-Meeseeks]

Thank you @DominikScholz. This contribution at b7bdc5a8de18fcf418cfb9c7d83a2b1097a5e4a3 is incorporated in the reviewed family through #588, published in 9594b61.

A combined prefix replaces the literal old prefix after an exact shared response/frame match. The reusable frontier driver verifies source, baseline, provider and prior-global receipts including every stage/log before reuse. Ten joint-basis bounded searches have replayed; those negative findings are scoped to their declared catalogs.

Preserved implementation and research:

• research/larger-gain
• research/structural-gain

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.

[higherbeing]

On "does not cover a different packet decomposition": the natural alternative, #493's D4 partition, has no sharing lattice at all (exact check).

Your factory note fixes the face-packet decomposition and lists other decompositions as uncovered. I checked the obvious candidate on the complex side, the 10 D4 blocks of 12 lines from #493's bit skeleton, with an exact inventory over the 112 code lines (packet = signed neighbour set of a target inside a block, readers = targets with the same set and the same signs up to a global sign). Scripts and numbers are in my notes; summary:

	Hamming 14 × 8 (current)	D4 10 × 12
foreign packets	168 four-sets, port span 3	120 six-/five-/four-sets, span 4
readers per packet, sign-consistent	8	6–8
own-block correction	4-set	8-set
pairs of packets sharing ≥ 2 readers with the same relative sign	168 pairs with 4 readers at join dim 5 (the 8-sums), 2688 with 2 readers	none (216 pairs share 2 readers, all sign-mixed)
per target: joins of two packet spans	dim 4 or 5	dim 6–8
readers of a packet that can still read it after it is raised to the join with another packet of one reader	2–4	1–2

So under D4 there is no shared intermediate sum at any level: about 7 of a target's 8 foreign packets would arrive at private frames, against 3 cap reads today. Fewer packets do not help when nothing can be bundled. A sweep over the other natural partitions (Hamming complementary 7 × 16, D4 complementary 5 × 24, the 30 D4 4-frames, random 14 × 8) finds a nested sharing lattice only in refinements of Hamming (the 28 half-faces: 2-sums with 16 readers inside the 4-sums with 8). The lattice 16 → 8 → 4 → 2 → 1 readers at dims 2 → 3 → 5 → 6 → 8 appears to be unique to the Hamming algebra, which makes your plane bound the relevant one.

A census of the delivery frames in that algebra (joins of a target's 12 face planes inside its cap) may also be useful for the borrowing design: per target there are 12 planes (dim 3, shared by all 8 readers of the face), 6 frames of dim 4 and 54 of dim 5 shared by exactly 2 targets, 6 of dim 5 shared by 4 (the 8-sum frames), 103 of dim 6 (3–4 faces, at most 2 owners), 27 of dim 7 (6–8 faces, mostly private) and the cap. Any design in this family delivers at most 1 + 1 + 2 + 4 faces below the cap and at least 8 of 12 at frames owned by ≤ 2 targets, which is why the two-copy interface and the 16-sums exist.

Prepared by higherbeing with Claude Code assistance. Happy to share the scripts if useful.