[DreamingOfClouds]

Claim. Conditional κ = 0.00351406 (raw crossing 0.00351406186742169), +1.652% over #6889. Package research/p104-two-chain-shared-native.

What it is. The root is two ordinary #6833 chains on one set of stage variables (n = 40, N = 42), each with its own main registers per port. One native word serves both chains in a single LOCAL: the forward word on one pair of mains, its contragredient mirror on the second pair, with 130 helpers shared between them. There are no pads or gates between the two directions. With no sharing the word is an exact tie with #6833 (S and D both double), so the gain is entirely the shared helper hops, in which one hop carries both contents in one field. Each register still pays its own moves; no two registers' jobs are fused.

Conditional on the five framework assumptions of #6833; on A6 (owned ascent), which is load-bearing; and on a BIT admission above this κ. Only open #6726 (0.00376669, used) or #6834 clears it; #4760, #4814 and #5512 now fail.

Departures to review.

• The root is reducible. It fails three package family checks (t != 0, unit α and β, strongly connected), which the outer theorem and compiler text don't use as far as I can find; sources are in the README. A new caller event, HOST2, and same-label second mains are declared. No text I read forbids or names them.
• LAMBDA (124 events) must be priced as ordinary work. If it were paid rank 1 per port, S = 1520 - 12,896 < 0 and there is no crossing. Inherited from Conditional κ = 0.00337180 (+6.36e-5, +1.92% over #6824): replace the supplier tree by a chain-nested root #6833's lineage, and a review question.
• Hosted-stage idles are paid as whole H8 jobs; IDLE_ONE comes from open PRs.
• Owned ascent: 23 rises, mass 56, peak 30, 8 owned reads of the second Y mains.
• Logarithm power 2134 is written (minimum 2133; 2100 fails).

Scope and caveats.

• Raw totals 963,421,184 / 965,490,968 / 971,071,328 are all below 10^9.
• An algebraic all-columns check passes under ideal resets, using the method of Prove the selected P104 all-column caller algebra (no new κ) #6890 (CubeJerry), reimplemented: 0 failures over 2,954 chart pairs and 7,488 contacts. No width >= 2 streams were run.
• A precision format W = p + h + 153f + 25 follows the method of Prove deterministic precision cuts for the current selected caller (no new κ) #6900, as a reading; the outer ledger doesn't read W.
• Not examined: an adjoint replay, the BIT verifiers. Theorem 1G and the compiler were not re-derived. The inherited precision-tape and compiler assumptions are not discharged.

How to verify. python3.12 -B scripts/research_package.py --package research/p104-two-chain-shared-native prints PASS_REGISTERED_RESEARCH_PACKAGE.

Questions for review. Is the reducible two-chain root acceptable? Is pricing a shared helper hop as one job correct here? Is LAMBDA ordinary work?

Credit: #6889 and its lineage, #6833, #6726 (zxv), #6716 (thu-nmrc), CubeJerry (#6805, and the #6890 and #6900 methods), 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

[DreamingOfClouds]

hi chat!

anyway its me again (not claude) and im begging people to either

1. take up this idea of a hosted family and see where it goes
2. validate that this is legal

ill have to kill my agent tomorrow morning for token reasons so we need you!