Research note (no κ): #2435's entrance raise is a width exchange paid by 72 partner splits; further raises saturate at ≈ +3e-7 (split-free raises worth up to +4.3e-6)

[thakrarsagar]

Research note (structural pricing screen); no κ claim. #2435 (Jeremy Kothe) gains +1.3164e-6 by raising 24 copy-buffer entrance frames per word from dimension 1 to 2. The change is frame-only (#2446). This note decomposes it per register and prices further raises of the same structural kind.

1. It is a width exchange paid for by the partners. In both providers, 96 registers' climbs change.
   • The 24 raised helpers each go from a rank-3 child to rank 2, and from width 8 to 7.
   • 72 gate partners (6 X registers, 66 helpers) each split a rank-3 child into rank 2 + rank 1, which is locally worse.
   • The net gain is mainly packing: width-7 helpers fill the rank-16 bins as [7, 9].
2. Calibration. The width-aware bin-pattern LP (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/Conditional κ = 0.001674097969905390: an attaining T37 construction and fixed R480 owner optimum #1129) moves the ceiling by +1.3166e-6 from Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322 to Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435. The realized κ gain is +1.3164e-6.
3. Further raises at Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435's point (k raises, each width 8 → 7):

k	average #2435 pattern, partners included	own climb only, no partner splits
1	+3.05e-8	+3.84e-7
10	+3.05e-7 (peak)	+3.85e-6
12	+9.6e-8	+4.35e-6 (peak)
16	−1.40e-6	+4.26e-6
26	−6.02e-6	+3.15e-6

• With average partner costs the lever saturates after about 10 more raises, at about +3e-7, and then packing turns it negative.
• Partner splits consume about 92% of each raise's value. Raises without partner splits are about 12× more valuable.

4. Census of remaining sites.
   • 112 dimension-1 entrances remain. Their (width, first climb) shapes are {(7,2): 30, (8,2): 26, (8,3): 52, (8,4): 4}, so 26 share the shape of Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435's 24.
   • Every one of them has its first gate at the entrance frame, and only 16 have those gates in A alone. Further raises therefore need a joint frame re-solve that treats entrances as variables.

Checks.

• The recounts of the four pinned words equal their reports, and the decomposition agrees across providers.
• The LP calibration reproduces the realized gain within 0.1%.
• --check-only, the 3.10 grammar check and the research tests pass. Run on CPython 3.14.4; the registered 3.12 replay was not run.

python3 -I -B research/t74-entrance-raise-prices/entrance_raise.py --fetch   # ~25 s
# PASS_T74_ENTRANCE_RAISE lp_gain=1.3166e-06 realized=1.3164e-06 best_further=+3.051e-07@k=10 own_only_k1=+3.839e-07 no_new_kappa

Scope. These are LP-ceiling prices. The average ladder assumes #2435's average partner cost. No raise is constructed; entrance delivery and Lemma A wrappers remain the open obligation, as in #2446. Base: main 0fdd7d82. Assignment: structural-decoder track, seed 41; card complex-rank-shape-decoder, scope changed.

Credits.

• Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435 / Conditional κ = 0.001697194950901770: exact owner fill on #2322's handoff construction #2374 / Conditional κ = 0.001696891067241618: helper width exchange on the borrowed T74 words #2312 (Jeremy Kothe).
• Improve conditional κ to 0.001697060296258836 with coherent handoffs and retimed hosting #2322 (Vlado Vojdanovski).
• Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458 (chafreaky, Lemma A).
• 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).
• 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 / Independent verification (no κ): #2435's words, 289,044-role owner map and endpoint reproduce; its change is frame-only (24 raised entrances), entrance delivery unchecked #2446 (this author): the tools, vendored with pins.

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

[philippeVV]

FINDING (no κ): on #2474's words there are no split-free entrance raises, and every legal single raise loses.

Artifact: research/t74-entrance-raise-census @ 14bc0925 (fork). Standard library only.

I took the minimal forward frame closure of a raise S→S+x. Gate frames are shared, so x must enter every partner's later frames until those frames already contain x. I enumerated it over all 141 entering helpers × every one-dimension raise on #2474's borrowed words, 30,851 candidates per word.

• 30,833 reach an end frame that does not contain x: a target's port hyperplane or a cold end, directly or through warps. 10 more fail at non-target ends.
• 8 are legal (width 5→4 helpers 606, 635, 638, 639, 667, 669, 670, 671). All 8 need partner splits, e.g. 606 is {1:+9, 2:−5} and 669 is {1:+20, 2:−11, 3:−1, 4:+1}. Your vendored PR2387 LP prices every one of them, and their joint stacks, as negative.
• The forced requirements of a joint raise contain the union of the singles' requirements, so stacking cannot rescue an infeasible single raise. Lifting a whole frame class fails for all 52 seed classes.
• Control: the same closure code reproduces Conditional κ = 0.001698511343697629: entrance raise of 24 copy buffers on the handoff T74 words #2435's 24 raises as exactly its 72 changed events, and it recounts Conditional κ = 0.001703676387993676: a fifth #2436 alias on the entrance-raised T74 words #2474's profile exactly.

So the +4.3e-6 "split-free" figure is not available on the current frontier words. Its 12 hypothetical {3:−1, 2:+1} raises would all flood into target port hyperplanes. Not explored: turning non-entering helpers into entering ones, which changes the contract rather than only the frames.

Prepared with substantial Claude Code assistance (a sub-agent did the census). I re-ran the enumeration (byte-identical) and the breakdown, but not the LP pricing.