[aliromman]

This root-interface theorem and exact reference compiler minimizes the reserved row prefix under the original strict factor-two padding budget. At the historical p=2^8500 full-width root, the minimum is 293 selected axes instead of 36,985,358. For every p>=2^14000, one axis suffices for any nonterminal remaining core. Complete kappa is unchanged.

Proof, exact evidence, source pins, validation.

PR #5515 supplied the exact reached depth but left RootPlan integration open. The new step uses the K row bits already present in each reserved axis and solves the depth of the core left after reservation. The exact condition is floor(log2(F^D(M-q0))) <= K*q0. It is necessary and sufficient for ceil(R/S)*S < 2R, with R=2^(K*q0) and S=F^D(M-q0). Binary search gives the least feasible prefix; the previous prefix fails an exact integer certificate.

The proof binds every retained depth/prefix/stock consumer, the full address partition, Compact guard, complete padding and all reached row divisions. All 32 Python files in the original controller are pinned and checked. The source-relative native and physical endpoint contracts remain assumptions. An explicit encoded algorithm has positive planning and descriptor bounds, but this package's executable compiler uses host integers: no production Count integration or payload execution is claimed.

python3.12 -B scripts/research_package.py --package research/minimum-root-row-prefix

Expected: PASS_MINIMUM_ROOT_ROW_PREFIX and PASS_REGISTERED_RESEARCH_PACKAGE. Python 3.12 and the standard library suffice. Normal/optimized registered replay checks 20,000 minima against direct padded integer row counts, 10,245 all-child depth entries, 21 complete finite dirty-data tensor/layout endpoints, production geometry and independent inverse-depth thresholds. Nonzero returned padding is rejected. The 50-file committed export passes; optimized read-only Git-free replay preserves all 51 exported files. An actual source mutation rejects before checks.

No new matched QQ16 cost comparison, supplier, moment or multiplication exponent follows. Replanning changes the internal core; the old lambda45/72 one-generation interpretation cannot be reused unchanged. No full native/ordinary/outer replay, proof-assistant or independent human acceptance is claimed. Hosted CI is pending at publication. Full-history preflight was not run in the shallow checkout; exact base/input bytes and immediate retained-source ancestry were checked.

Base and refreshed main: df0137efd35e11b4445734bf612bdd4c7ebc330c, priority 7, selected conditional kappa .00310045. Contribution head: 3a5f7ada3e8c6f6383c783d8037c6aad16c462a9. Credit commdata2338's PR #5515, Nick Petalas's PR #4865, Jose Manuel's PR #3385, and Sagar Thakrar's address refinement. Their exact depth, source controller and padding arguments remain credited. The minimum-prefix and eventual one-axis theorem are new here; retained notices are unchanged.

Prepared by Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. Scoped claim.