Research note (no κ): T74 #2322 words pay 525 rank units/call of end-of-call restoration climbs — each more cold-restored helper worth +0.6–1.5e-6 (LP ceiling); buffers are free in κ

[thakrarsagar]

Research note (pricing screen); no κ claim. This note finds and prices the largest remaining item of pure restoration work on the live T74 line: the end-of-call climbs to FULL that FULL-cleanup helpers make only so the inverse-reverse cleanup can run at FULL. The object is the #2322 words, which the open #2349 / #2363 / #2374 line builds on.

item (both providers, per call)	value
FULL-cleanup / cold-restored (hyperplane-ended) helpers	356 / 78
end climbs paid by FULL-cleanup helpers	{1:249, 2:42, 3:64}: 525 rank units, 9.8% of local paid mass (5,371)
end climbs paid by cold-restored helpers	none
one more cold-restored helper, end climb 1 / 2 / 3	+5.77e-7 / +1.04e-6 / +1.46e-6 in the LP ceiling b
all 249 / 42 / 64 of one class	+6.31% / +2.33% / +4.00%
ideal, all 355	LP ceiling 0.0018806 (+10.8%; an upper bound for the lever, not achievable as stated)

For comparison (#2402): on the same words a #2322-type handoff is worth +4.27e-8, and all remaining owner-fill headroom is at most 8.1e-10.

Buffers are free; legality is the obstacle. These facts are read from #2374's pinned SPEC, OUTER-CLAIM and invoice:

• κ = C − 1/F − B with F = 1e3046 and B = 1e-3022. Zero buffers, CLEARs and adapters therefore enter only through admission inequalities, which have huge slack.
• Physical stock would go 66,826 → 67,181, against a parent capacity of 112,178.
• Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322's namespace bound NZ ≤ R becomes 433 ≤ 434.

So the open obligation is whether each helper's cleanup can legally move before its end climb. The existing hyperplane-ended helpers come from #601's sufficient commutation test and #604/#617-style cold tails. Cold ends are a well-known move class on older words (#636, #640, #676, #685, credited). This note prices it on the live line; the prices are what a stronger exact commutation test, or a changed cleanup contract, would be worth. #1995/#2013 hold the cleanup contract fixed, so this lies outside their scope.

Checks.

• restoration_prices.py recounts both pinned Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322 words, using the vendored Negative result (no κ): legal 400-helper owner fill of #2339's 80-alias T74 words reaches b = 0.00168654, 99.994% of a certified width-aware cap b < 0.0016867 — below #2269 #2387 frame-climb rule patched only to label which register and phase pays each child. Each recount matches the word's reported profile.
• Prices come from the width-aware bin-pattern LP of Negative result (no κ): legal 400-helper owner fill of #2339's 80-alias T74 words reaches b = 0.00168654, 99.994% of a certified width-aware cap b < 0.0016867 — below #2269 #2387 / Certify PR2339's whole-helper T74 packing cap below 0.0016870 #2360 (pattern dual of Conditional κ = 0.001674097969905390: an attaining T37 construction and fixed R480 owner optimum #1129), extended to widths 3–9. As a control, the extended LP reproduces the Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322 ceiling.
• A separate AI subagent wrote its own GF(2) replay from a plain rule statement and agreed on every census number in both providers.
• --check-only, the 3.10 grammar check and the research tests pass. Everything ran on CPython 3.14.4; the registered 3.12 replay was not run.

python3 -I -B research/t74-restoration-climb-prices/restoration_prices.py --fetch   # ~2.5 min
# PASS_T74_RESTORATION_PRICES full_cleanup=356 end_climb_mass=525 convertible=355 one_d1=+5.767e-07 one_d3=+1.463e-06 ideal=+10.80% no_new_kappa

Scope.

• No conversion is constructed or shown legal.
• Prices are LP-ceiling differences on the fixed bin stock.
• They assume an earlier cleanup changes no other register's climbs. Conditional κ = 0.001614202339036103: 46 hyperplane-ended helpers per call via commuting-prefix cleanup #601 records that intervening partner climbs stay paid; Conditional κ = 0.00162128508944984: cold factory-scratch helpers end at 4-dimensional frames on the PR601 witness (+0.439%) #604-type tails would have to be priced on the actual word.
• Base: main 0fdd7d82. Assignment: recursive-interfaces track (seed 55, "eliminate restoration work"). Card: complex-new-helper-inventory, scope changed, with the reason in TASK.json.

Open next questions.

• An exact commutation test over the 355 helpers, starting with the 64 end-climb-3 helpers.
• A Conditional κ = 0.00162128508944984: cold factory-scratch helpers end at 4-dimensional frames on the PR601 witness (+0.439%) #604-style cold tail on one helper, priced on the recounted word.
• The same census on the selected k8 word's 402 FULL-cleanup helpers.

Credits.

• Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322 Vlado Vojdanovski (the words).
• Conditional κ = 0.001697194950901770: exact owner fill on #2322's handoff construction #2374 / Conditional κ = 0.001697065994108880: owner relocations on #2322's handoff construction #2349 Jeremy Kothe and Improve conditional κ to 0.001697147064845350 with whole-role augmenting paths #2363 mason-hale (outer, invoice and buffer accounting).
• Conditional κ = 0.001614202339036103: 46 hyperplane-ended helpers per call via commuting-prefix cleanup #601 / Conditional κ = 0.001626010676080313: E1 entering-pair hyperplane ends on the PR604 cold-ended witness #615 / Conditional κ = 0.001635183729255778: second dim-4 cold end per factory block (28 per word) on the PR604 cold-ended witness #617 / Conditional κ = 0.001659319032684569: six dim-8 cold ends and three entrance promotions on the PR659 no-triangle word #676 Jeremy Kothe, Conditional κ = 0.00162128508944984: cold factory-scratch helpers end at 4-dimensional frames on the PR601 witness (+0.439%) #604 DreamingOfClouds, Independent construction: 46 hyperplane-ended helpers per call via the exact commutation condition, conditional κ = 0.001613569610580215 (below #601) #608 higherbeing, Conditional κ = 0.001636269762440408: combined cold ends and PR635 owner repair #636 / Conditional κ = 0.001636464689763295: four CK cold ends and concentrated packing #640 vectorsss, Conditional κ = 0.001660339298410333: four entrance promotions and 72 cold ends #685 chafreaky, Research: 80-helper anticipatory borrowing component and exact price rejection (no new kappa) #2226 Cinna Davis (cold and hyperplane-ended helper mechanisms).
• Research: bound joint helper-end and frame contractions of the selected k8 word #1995 / Research: bound arbitrary helper terminal frames of the selected k8 word #2013 (fixed-contract frame bounds).
• Negative result (no κ): legal 400-helper owner fill of #2339's 80-alias T74 words reaches b = 0.00168654, 99.994% of a certified width-aware cap b < 0.0016867 — below #2269 #2387 / Research note (no κ): T74 #2322 words — owner fills capped at b < 0.001697195759 (#2374 within 8.1e-10); exact prices of handoffs, aliases and rank-1 climb merges #2402 (this author; tools vendored with pins).
• Certify PR2339's whole-helper T74 packing cap below 0.0016870 #2360 Jeremy Diviney and Conditional κ = 0.001674097969905390: an attaining T37 construction and fixed R480 owner optimum #1129 Ali Romman (pattern-dual relaxation).

Prepared with substantial AI assistance (Claude Code, Anthropic, Claude Opus 5.5). No independent human review is claimed. Apache-2.0.

🤖 Generated with Claude Code

[thakrarsagar]

Follow-up in #2451 on this note's first open question. On #2322's words, only 8 of the 355 FULL-cleanup helpers admit a single commuting restoring gate. With reader clearing, their converted words pass the exact replay, but the readers' frames join to FULL, so the saving is 0. A loose frame estimate caps the lever at +8.86%, not the +10.8% ideal here; an unreviewed span check suggests far less. Any realization needs low-frame readers or earlier restoration.