[ywz65]

For the full B_n and C_n root-line configurations with their fixed dense binary orthogonality pairing, prove that the least root-code factorization cost is exactly n², for every n>=1, every coefficient field of characteristic other than two, and any finite number of terms. The cost rank is over F2, separately from the coefficient field.

The new mechanism is an explicit cross-star minor: every nonzero root-code support contains an n-line axis star, and the pairing between any two such stars is invertible even when their anchors differ. Each nonzero rank-one term therefore costs at least n; the mutually orthogonal axis lines force at least n terms. Coordinate Gram factors attain the floor. This extends PR2272/PR2518's fixed-coordinate and bounded basis/one-extra-family evidence to all root-code factors and all sizes, building on the scoped floor method of PR2228/PR2277. PR2536 concerns a different object: fixed-coordinate D punctures with variable pairings.

Validation passed under CPython 3.12.7 using the registered suite:

• Bind both root generators, dense pairing and coordinate prices to byte-exact PR2272 source for dimensions 1–16; verify 2,992 actual ordered stars and 1,360 opposite-sign row equalities.
• Check 131,588 canonical rational support-cell pairs, 18,996 odd-coefficient-field pairs and 140 exact sheared/split Gram factorizations, with six effective assumption controls.
• Independently construct the pairing from graph incidence and check the explicit inverse on 385 matrices and 169,978 binary right-hand sides, without the primary rank engine.
• Git-free export replay; reject source corruption before extraction and a typed bool/int receipt alteration. Separate AI proof/source review and optimized isolated CPython 3.12.14 primary/graph checks passed with no blocking findings.

This is a written all-size theorem with finite exact corroboration. It is not a proof-kernel artifact, new exponent, physical supplier or a floor for arbitrary sparse pairings, punctures, or factors outside the root codes. The card's scope change and remaining questions are explicit. No selected record or frozen evidence changes.

Canonical main refreshed unchanged at 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. The completed overlap snapshot covers 768 open PRs at 15:00 UTC and 1,232 Discussion records at 15:03 UTC; relevant source heads and body hashes are in TASK.json. The PR2518 handoff is complete/released and explicitly leaves unrestricted factorizations outside its one-extra family open. Metadata coverage is not proof that no unpublished work exists.

Credit Aaron Wang for PR2272/PR2518's root generators, tables and screens, and DreamingOfClouds for the root-code floor/model lineage. New proof, checkers and documentation by ywz65 with substantial OpenAI Codex assistance. No independent human review or hosted-CI success is claimed.

Replay: python3 scripts/research_package.py --package research/bc-dense-root-code-floor.