[DreamingOfClouds]

Claim. Conditional κ = 0.003452786, tested price 0.0034527867 (raw crossing 0.00345278688457740), up from 0.00342928953721633 of #6886 by +0.685%, and from #6883 (the base) by +1.555%. Package research/p104-hosted-joint-merge. Root, caller and hosted pair are those of #6883 (t = -1/2, S = 1520). Not claimed as a frontier change.

Conditional on the five framework assumptions of #6833, on A6 (owned ascent) as declared in #6883, and on a BIT admission above this κ. The pin used is 0.00376669 from the open #6726 (margin 3.139e-4). That's a choice, not a necessity: the ledger passes at the merged archive values #4760 0.0034597, #4814 0.003475458 and #5512 0.003483969976, none of which is admitted, and fails at 0.0034, 0.00343, 0.00343635 and #5421's 0.00344412.

What changes against #6886.

• X-side merges are added and chosen jointly with the Y-side merges and the shared helpers, rather than in sequence. dD per call 7451.30 to 7284.61, 166.69 R0 better.
• Helpers read Y directly on 42 changed ports, and writes into X change on 51. The moved endpoint reads and writes are ascending and paid. 8 fewer registers, 116 fewer MOVEs. No new frames; main paths are the same.
• Correctness rests only on the exact replay of the final word and the physical and caller replays. Value and frame controls are rejected.
• Owned ascent has the same scope: 23 rises, mass 56, 54 lifetimes, peak 25. One owned read of Y (event 98540).

Scope and caveats.

• Raw totals: 948,952,288 (literal) and 982,697,752 (actual per-call work) are below 10^9. The review's conservative count is 1,022,800,096, 2.28% over and above Conditional κ = 0.00339993 (+0.834% over #6833): host a pair of units at every root call #6883's. No outer-ledger term reads 10^9; the growth bound is written out, as a reading rather than a proof, and the precision and tape contract stays inherited.
• Coefficient growth: sum log2(1+|q|) = 246,857 against the predecessor's 416,000; largest |q| 62,224; 31 adjacent clusters of ADDs split to fit the 2^16 alphabet, largest true coefficient -2,689,024 in 83 pieces, each event legal.
• Logarithm power 2100 suffices (minimum 2096); the wall at that power is about 0.0034607.
• κ rests on pricing hosted-stage idles as whole H8 jobs, and IDLE_ONE and LAMBDA come from open PRs, as in Conditional κ = 0.00339993 (+0.834% over #6833): host a pair of units at every root call #6883.
• Not independently examined: an adjoint replay, streams at tensor width >= 2, the BIT verifiers and the content of the BIT admissions, and which ADD events differ from Conditional κ = 0.003429289 (+0.388% over #6884): remove 24 Y pads from the mirror #6886's deletions. The inherited precision-tape and compiler assumptions are not discharged.

How to verify. python3.12 -B scripts/research_package.py --package research/p104-hosted-joint-merge prints PASS_REGISTERED_RESEARCH_PACKAGE (about 2 min). Comparison controls rebuild #6883, #6884, #6886 and #6833 by the same code.

Question for review. Which BIT admission above 0.0034 should the outer statement use?

Credit: builds on #6883, #6884 and #6886 and their lineage: #6833, #6726 (zxv), #6716 (thu-nmrc), CubeJerry (#6805), Ali Romman (#6349, #6353, #6372), Jacob Alan Sussman (compiler), #6702 and #6824.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

🤖 Generated with Claude Code