[ChaloupeMusic]

Stacking note: this PR stacks on #6398, which is open and not yet in main. The diff against main therefore also shows #6398's files; the diff against #6398's head dee2663ba0918be86993fd9d63c8d64b020f4c83 is research/mixed-root-third-anchor-composition/ only (19 files). The branch is a direct child of that commit.

Change and result

This replaces the PR6375 native word inside the PR6398 mixed-root composition by the same word with one more helper anchor (helper 234, final frame 127 -> 246) in group 1091. Only the native word changes: the six-field caller, the compiler and the outer ledger are PR6398's. The complete conditional claim is kappa = 0.00328637739144124222748802, i.e. +1.626e-7 (+0.00495 percent) over PR6398's 0.00328621476572836015146308. The exact raw crossing with the unchanged caller lies in (0.00328637739144124222748803, 0.00328637739144124222748804).

Honest standing: this does not exceed the open draft PR6417 (0.003286849), which composes the PR6399 donor-transport merge with the PR6398 root. This package is built on PR6398 alone and is independent of PR6399. The two native edits touch disjoint registers and frames (group 1091 / frames 245, 246 against role 400 / frames 1090, 477), so they should stack; that stack is a separate recompute, in progress on my side, and I will say so here if it is published.

Mathematical idea

On the transposed PR6353 word, group 1091 has joint capacity exactly 3 across the entrance frames 245 and 246 (response-free input directions U_245 and U_246 each have dimension 2, the helper-only departure functionals have dimension 7, joint pairing rank 3). PR6372 used two anchors (224, 227 at frame 245); its particular basis left no room at frame 246. A joint re-choice, with dyadic coefficients, places a third anchor (234) at frame 246. It is stored in SPEC.json and rebuilt by construct3.py, a k-anchor generalization of the PR6372 contact-block constructor that reproduces the PR6372 word byte for byte with no spec. The PR6375 lift rounds are then re-applied unchanged.

Per native call the paid profile changes by +1 at rank 4 and -1 at rank 5, and the productive profile by +1 at rank 8 and -1 at rank 9 (paid call count unchanged). The local gain (H5-H4)-(H9-H8) is positive for every strictly concave increasing tariff. The word keeps PR6375's scope for its six input lifts (positive only for power exponents of about 0.873 to 1; zero at linear cost). The claim is made at the stated exponent only.

Costs: 44 owned lifetimes (PR6375: 43), real scalar allowance 131,052 (+1,382), one new native chart (1091, 246), 2,942 distinct native maps (PR6375: 2,941). Paid mass 5,925, helper mass 4,205.

Evidence

All commands run directly on CPython 3.12.10, Windows, exit 0:

Run	Result	Time
python -I -B .../verify.py --write-result	all stages PASS, wrote RESULT.json	58m43s (heavy load)
python -I -B .../verify.py (exact comparison with RESULT.json)	PASS_MIXED_ROOT_THIRD_ANCHOR_KAPPA_00328637739144124222748802	16m05s
python -I -O -B .../verify.py	same marker	12m50s

Stages: exact source reconstruction, root, normalization and complete caller; source inputs, dense transpose (799 columns, 638,401 coefficients), native phases, physical caller (7,488 contacts) and owned controls; both prices, complete compiler and outer ledger, source controls and the exact raw family limit. Seven third-anchor rejection controls plus all inherited controls reject. An old-word control confirms the PR6375 word with this caller fails the new claim.

Independent, partial checks by separate agents (own exact-rational code, not scientific acceptance): the capacity and dyadic solution (dims 2, 2, 7, pairing rank 3, host determinants), the histograms and margin with a rigorous interval enclosure, the owned-label lifetimes (779-781 disjoint, all zero rows at CLEAR in the transposed preparation), a 1,255-mutation red-team of the word (no soundness gap; 39 accepted mutants are value-redundant events, also present in the PR6375 word), the stale-pin and diff review (no weakened check), and that the physical check reads the native word only at its ports. Not checked independently: native phases and caller contacts beyond the verifier, the compiler and outer ledger beyond the verifier.

Reproduce

python3.12 -B scripts/research_package.py --package research/mixed-root-third-anchor-composition
python3.12 -I -B research/mixed-root-third-anchor-composition/verify.py
python3.12 -I -O -B research/mixed-root-third-anchor-composition/verify.py

On slow hardware a run can exceed the 590 s per-command cap declared in VERIFY.json (a timeout, not a failed check). The six live inputs of mixed-root-source-composition are pinned with head: commits that are ancestors of PR6398's head; this branch is a direct child of PR6398 head dee2663ba0918be86993fd9d63c8d64b020f4c83 so they are reachable.

Scope, assumptions and limits

• The five inherited framework assumptions remain: native_all_columns, owned_erasure, compiler, precision_tape, analytic_recovery. This is a conditional finite certificate, not a proof of the multiplication theorem and not a runtime speedup. Author checks only; no independent scientific review, no formal-kernel acceptance, selected frontier unchanged.
• Inherited all-width native claims (Q^2 = X etc.) are not discharged.
• The shadow rows for owned labels 779-781 are zero in the transposed preparation but not at the forward CLEAR (the inherited owned_erasure obligation, as in PR6372).
• RESULT.json field price.refinement_kind is a label inherited from PR6398 and does not describe this word.
• Only group 1091 with frames 245/246 was examined for additional capacity. A track of broader exact tests (other groups, carrier-paying clusters, input-lift variants, caller-shape variants) found nothing further; negative results are in my notes and can be shared on request.

Credit and AI disclosure

Builds on Ali Romman (aliromman) (PR6398 mixed root, PR6353, PR6372, PR6348, PR6349), DreamingOfClouds (PR6375 lifts, PR6314), Cinna Davis, Mason Hale, Pierre-Loic Doulcet, Chris R / Clear1nsight, Jacob Alan Sussman, zxv, Cao Yifan and the other contributors named in NOTICE, whose notices are retained byte for byte. Apache-2.0.

Prepared by Claude Sonnet 5.5 via Claude Code, orchestrating subagents, for Erwan L., with substantial AI assistance; the user pre-authorized publication of a certified result.

🤖 Generated with Claude Code

[ChaloupeMusic]

Cross-reference: the same third group-1091 anchor is stacked on the clean mixed root of PR6417 (PR6399 merge + PR6398 root) in PR6453 (kappa 0.003287014, +1.65e-7 over PR6417), which is the stronger statement. This PR (on the PR6398 root alone, 0.00328637739144124222748802) remains as the minimal, independently certified step; if PR6453 is preferred, this one can be closed by the maintainers.