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The original overlapping-rank flow could accumulate leaf exits too quickly to justify near-root convergence. This changes its stopping height and supplies a uniformly funded recursive majorant. The resulting direct compiler accepts three raw native profiles without chronological tensor padding, paid completion or splitting.

For p16, the sufficient uniform rank falls from 2^558100005581 to 2^2800000000, a 199.32× smaller exponent, while retaining the same uniform saving, outer parameter point and conditional κ = 0.00302015 as PR #4479. The initial native supplier uses tensor power 1 instead of 682. This is a compiler improvement; it does not claim a new κ.

What changes

The new package, research/rank-convergence/, contains an arbitrary-height integer forest, two complementary Chernoff bounds, an exact mirror-window boundary certificate, sharp source/exit cancellation, and a concave shifted-logarithmic majorant with its entire base range and final rank funded. Exact rational gates cover every local exponent and all larger heights. This supplies a convergence argument for profiles satisfying the gates; no universal gate-existence theorem is claimed.

Profile	Retained conditional κ	Prior H	New H	Reduction in H
P7, same saving as PR #4480's unchanged-C witness	.00268647	130,200,000,000	3,040,000,000	42.83×
PR #4453 native profile, same saving as PR #4480's main witness	.00300120	125,790,000,000	4,300,000,000	29.25×
PR #4470 p16 profile, same saving as PR #4479	.00302015	558,100,005,581	2,800,000,000	199.32×

Here N=2^H. A fourth certificate gives H=1,300,000,000 for the native4453 profile using its original consumer's weaker saving requirement; the 29.25× comparison above holds that requirement fixed.

Predecessors and related PRs

• My PR #4359 κ=.0023 construction and cumulative native/outer lineage, PR #4436 integer overlapping flow, PR #4453 native retiming, and PR #4480 net-tax/log-moment compiler.
• Ryanson Jonathan's corrected PR #4459 mirror identity. Its withdrawn earlier reduction is not used.
• The finite-flow, leaf and width analyses in PR #4456, PR #4465, PR #4467, and PR #4476.
• Cinna Davis's PR #4470 joint-helper native construction and Dugongue's PR #4479 compact compiler and comparison point.

Credit Jacob Alan Sussman's integer compiler and all retained BIT, scalar, Clifford, geometry, helper/controller and outer sources. SOURCES.json authenticates exact inputs, including original Sussman license and notice. The diff includes byte-for-byte predecessor packages absent from main so their pending inputs remain available; the new mathematics and code are confined to research/rank-convergence/. Shared documentation matches canonical main. Retention is not scientific admission of those pending packages.

Validation and limits

• Registered replay: python3.12 -B scripts/research_package.py --package research/rank-convergence passed.
• Four exact certificates, 13 test methods, 80 literal fixtures, 24 independent ordered-word flow comparisons and eight rejected numerical witnesses.
• Committed-byte preflight and checked 26-file export passed. The exported package reproduced with no Git, PATH=/nonexistent and python3.12 -I -O -B; source and receipt mutation controls rejected corruption.
• Native and outer producer receipts are inherited and authenticated, not freshly replayed. Physical, precision, analytic, finite-program and exact-recovery premises remain inherited. Winning words are not materialized; the smaller ambient rank is not a practical width/runtime or eventual multiplication-input threshold claim. The selected frontier is unchanged.

Known baseline CI issue: four existing current-status/navigation tests fail on unchanged canonical main 2a13eb1827784206202db90e5dc8327cf8a250db, as reproduced during PR #4480. No frozen verifier or historical source is changed to bypass them. Hosted validation for this PR must be assessed on its actual run.

Open next questions cover changed native profiles for an actual κ gain, broader compiler admission gates, and complete physical constants. Substantial OpenAI Codex assistance, including three Sol 6.1 medium subagents and separate author-side audits; no formal or independent human verification is claimed. Apache-2.0.