[petalas]

Summary

A scoped all-size proof settles the intermediate-axis punctures left open by ywz65's #2582. Keep every signed edge line and any k coordinate-axis lines in dimension n>=4. Over any coefficient field of characteristic other than two, every finite exact root-code Gram factorization has dense binary support-rank cost at least n(n-1)+k. Coordinate factors attain it, and every optimum has exactly n nonzero terms.

all signed edges + k retained axes
  exact coefficient identity on the retained k coordinates
  at least k terms contain two complete axis stars
  signed-edge tariff + retained-axis charge
  exact minimum n(n-1)+k = retained line count

The new case is 0<k<n. ywz65's #2582/#2719 already settle the endpoints and supply the cited cross-star and Dn lemmas. DreamingOfClouds retains the #2228/#2277 root-code and one-extra coupling credit; Aaron Wang retains #2272/#2518's root generators/screens, and ywz65 retains #2536's different coordinate/pairing result. This contribution was produced by Nick Petalas.

Evidence

• Before: the arbitrary-axis puncture family had no all-term floor in the cited packages. After: research/axis-punctured-root-code-floor/PROOF.md supplies the retained-axis rank argument and all finite term-count cases.
• Standalone exact replay reports PASS_AXIS_PUNCTURED_ROOT_CODE_FLOOR: 802,334 support pairs, 314 coordinate attainments, 125 rational expansions, 58,625 count inequalities and eight controls. The complete finite support domains are all intermediate axis masks and projective-vector pairs over F3 at n=4,5 and F5 at n=4. The written proof supplies the universal quantifiers.
• Warning-strict normal and optimized registered commands pass locally in about five seconds each, 10.38 seconds together. The direct verifier, metadata/pins, seven-file committed-byte/source-history preflight and committed export integrity pass. The generated receipt binds six local source/metadata hashes. Git-free optimized replay and exported receipt/source corruption controls also pass.

From the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/axis-punctured-root-code-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/axis-punctured-root-code-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/axis-punctured-root-code-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/axis-punctured-root-code-floor --check-source-history

Limitations and unrun obligations

Fixed dense pairing, exact Gram target, root-code factors, all signed edges and 2 invertible are essential hypotheses. The predecessor lemmas are cited premises, not audit deliverables. Physical frames/access/restoration, hosting, precision, compiler and a complete recursive supplier remain outside the result. Independent human and formal proof-kernel review are unclaimed. No native, Lean, Docker, historical full replay, new kappa or selected-frontier change is claimed.

Open next questions

• Which specified signed-edge deletion admits a strict exact root-code gain? Start with one n=4 mixed-axis configuration and a complete factorization with all literal support ranks. Its replacement tariff and physical consumer remain open.
• Can one shared sparse fitting pairing lower the coordinate bill at n=4,k=1? Enumerate compatible pairings and charge every term; deleting entries is not rank-monotone.

Related open work

• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies sparse fitting pairings for full D4 and exceptional units. This proof fixes the dense pairing and adds an arbitrary retained-axis subset to the full signed-edge set.
• Decoder floors: universality, projection transfer and batching reductions; C(P112) ≥ 64 reduced to named conjectures, C(P120) ≥ 72 restated on N₀ (no κ claim) #2832 studies E8/P112 decoder reductions and outside-code factors. This result uses a different root set and keeps both factors inside its root code.
• Copied-information floor of the selected centre path #3120 studies the selected E8 centre-path demand floor. This obstruction does not change that unit or its selected charge.

The post-claim related-work refresh found no additional open PR addressing the same intermediate-axis Gram floor. Broader cached overlap covered 379 open PRs and 3,855 Discussion records, with incomplete sparse local notes; distinct all-main README/proof and merged-term searches supplement it. These searches do not certify novelty.

CI notes

Both hosted checks passed at head 8dbf33d72f181af5986b3f703f9a827d64d18688 in run 39864600668: plan and registered package replay. The run is completed with conclusion success. These checks certify the registered automated replay, not independent human or formal proof-kernel acceptance.

Both registered commands are bounded at 180 seconds and pass locally in about five seconds each. The selected source-binding orientation control passed; scientific predecessor suites were not replayed.

Claim: #1166 (comment). The delivered scope matches the claim.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Seven new files in one opt-in package. Archive intake would preserve this scoped result without selecting a frontier or admitting a supplier.