Conditional κ ≈ 0.0017228163198632: four paired extractions at normal 45 on #2917's words (+3.16e-7 over #2930)

[mdumitrean]

#2930 showed that the fill layer of #2917's words is exhausted (its allocation attains their exact packing cap), so a further gain must change the words. This package changes them: four further paired input/output extractions on #2917's final words, all at one hyperplane normal (45), with a new literal T74 allocation of the changed inventory, through the complete admission:

κ = 0.001722816319863214327996 − 10⁻⁴⁶ − 10⁻⁷⁰, exactly 316413419939763639/10²⁴ ≈ +3.1641e-7 over #2930 at 81f3b5d12977eba09e06fbff729167d59faabe90 (≈ +1.2277e-6 over #2917).

What changes

Helpers 249, 267, 251, 265 are restored at E_45 = ⟨256,128,64,33,16,9,5,2⟩ (width 9 → 8), each with its own private zero buffer (86 in total, physical stock 66,808), using #2908's paired construction exactly as #2917 did: literal input gates at frame zero after the prelude, output gates at E_45 appended to restore, compensation C·S, cleanup re-derived, plus ten declared inherited-frame edits. data/TRANSFORM-<h>.json holds every literal gate and frame edit (indices relative to each step's parent); producer.py (#2917's model) composes them on #2917's words.

	local profile per call	width inventory	buffers	κ
#2917 / #2930 words	{1:2416, 2:640, 3:280, 4:110, 5:26, 8:8}	{8:96, 9:222} (+{4:26,5:2,7:62})	82	#2930: 0.001722499906443274564357 − …
this PR	{1:2434, 2:632, 3:286, 4:104, 5:26, 8:8}	{8:100, 9:218}	86	0.001722816319863214327996 − …

Why four and not one

On #2917's final word no single further extraction reaches break-even (about 2,091,300 local units per freed width unit): on existing frames only normals 486/504 pair (29,645 short); with exact frame lowerings the best direction is 4,579 short. At normal 45 the first extraction is expensive (seven registers brought into E_45 at once, 20,801 short), but the next three reuse that lowering and each needs at most three more registers: cumulative −8,378,128 units = 4 × break-even − 12,928. Only after step 4 does the fixed-capacity cap of the changed inventory exceed #2930's (0.0017228172707 vs 0.0017224999064). Exactly, the certified cost of a composition at one hyperplane depends only on the touched set U and the pivots P: total(U,P) = Δ_end(U ∪ P) + c1·|U \ P| (here −25,311,496 + 8·c1 = −8,378,128). Every complete rank-4 composition at normal 45 touches the same 12 registers, whose only helpers are the four pivots, so no order or first step does better (three other orders end at exactly −8,378,128), and no other normal has a positive margin before or after: within this cost model the composition is the optimum of the paired-extraction lever on these words. The step table is in the README.

Validation

• Registered (run_registered.py, stdlib, ~20 s): pins; producer reconstruction of both words; Conditional κ ≈ 0.001720398337: two-sided helper extraction and charged T74 packing #2908's unchanged structure.py two-sided certificate at every step; Conditional κ = 0.001718657437166855: dirt-entry rewrite of helpers 530 and 542 on #2802's construction #2833's auditor forward and dual over all 632 inputs on both words (86 restored endpoints); frame accounting; 40 semantic rejection controls (drop an input gate, flip an output coefficient, wrong hyperplane, stale cleanup, omitted frame edits); owner engine with five controls; the moment auditor re-pinned to Conditional κ ≈ 0.0017224999064433: cap-attaining allocation of #2917's unchanged words (+9.11e-7) #2930 (strict gain); the Fractions-only cap certificate of the changed inventory (its basis is fractional at T74; the literal fill is 9.5e-10 below it, recorded exactly); stored receipt binding.
• Complete replay (reproduce.py): exact Git exports of Conditional κ ≈ 0.001720398337: two-sided helper extraction and charged T74 packing #2908/Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917/Conditional κ ≈ 0.0017224999064433: cap-attaining allocation of #2917's unchanged words (+9.11e-7) #2930 (and through them Conditional κ = 0.001720184528377469: two more staged quartets on #2863's shared-kernel words #2894/Conditional κ = 0.001718657437166855: dirt-entry rewrite of helpers 530 and 542 on #2802's construction #2833), fresh retained bootstrap, Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917's derived source (98d57c82…), then the new overlay stage normal45-paired-extractions (exact edit spec replayed by the source's own rebuild chain after Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917's stage; manifest 79956f19…, 637 files), 86-buffer adapters with pinned Conditional κ ≈ 0.001721588647: nine additional paired helper extractions #2917/Conditional κ ≈ 0.0017224999064433: cap-attaining allocation of #2917's unchanged words (+9.11e-7) #2930 comparators, fresh native chains for both providers, complete admission: PASS_REPLICA_ORDER_FILL_DERIVATIVE_COMPLETE_RELEASE, PASS_RELEASE, strict gate κ > Conditional κ ≈ 0.0017224999064433: cap-attaining allocation of #2917's unchanged words (+9.11e-7) #2930 passed.

git fetch https://github.com/CrocSwap/integer-mult-bounds.git 81f3b5d12977eba09e06fbff729167d59faabe90 \
  224bd5e56041c40078ff6b5fa7393733a215de5c 51309592971e3b5be935599eed2a95404dc7781e \
  18d03c447886895ef9abf888398178e0428533ca e9307b48d7d1b690939935ecef0f8e524e6d92f4
python3.12 scripts/research_package.py --package research/paired-extension-normal45
python3.12 -B research/paired-extension-normal45/reproduce.py --repo . --work /tmp/normal45-complete --jobs 2

Scope and next

Conditional finite supplier/outer certificate with all inherited hypotheses; no selected-frontier change, no unconditional theorem, no independent human review. The search and optimality statements are relative to the stated cost model and census. Next: edits outside the lever (gate re-timing, output gates before the end of restore, entering-helper pivots) to lower the 88,556-unit overhead of the 12-register universe, and integral packing closer to the fractional cap (9.5e-10).

Credit: Jeremy Diviney (#2908) for the paired extraction method and structure.py; Mason Hale (#2917) for the parent words, producer model, chronology and admission tooling; Jeremy Kothe (#2894/#2833); Yiyang Lu (#2857/#2848) for the flow model; Dugongue (#2863/#2839); Tekalign; icekylinx. Marius-Paul Dumitrean, prepared with substantial AI assistance (Claude Code). Apache-2.0.