[petalas]

Summary

Scoped mathematical proof with finite exact corroboration. In the four-register abelian bit model, the 48-pool noncoordinate-kernel orbit cannot realize three invertible rational 3-by-3 targets in eight rank-one calls at frame span six, even when the fourth arbitrarily dirty role may end at any diagonal monomial.

- unresolved exceptional flagless pools: 72
+ unresolved after this obstruction:    24

The new argument retains the coordinate-diagonal endpoint. Common left/right kernel supports force zero noncoordinate contributions on two roles. One data target consumes three coordinate calls, leaving a five-call subword for two invertible targets, contradicting the credited six-call theorem.

woklerx #1846 retains credit for the scalar catalogue and 24/48 orbit split. unpaidmoon #4119 supplies the relation-support reduction and explicit open question; JoyboyBrian #5075 leaves this branch open; waynenilsen #1882 supplies the two-target dirty-auxiliary theorem; Ali Romman #468 and the retained eumemic/huxint lineage supply the model and low-span reasoning. Their scientific suites are cited premises, not the deliverable. This contribution was produced by Nick Petalas.

Only six new files in research/flagless-kernel-obstruction are added. No new kappa or selected-frontier change.

Evidence

Before, scalar classification did not exclude realization of the 48-pool orbit. After, PROOF.md excludes it for every rational frame choice satisfying the stated span-six premises.

Local CPython 3.12.15 normal and optimized verification pass. Evidence includes 2,500 independent literal matrix products, 1,920 formal words with 7,632 coefficient matrices, 3,888 diagonal assignments and all coordinate-call allocations, leaving 96 critical allocations excluded by the cited two-target theorem. Nine named boundary controls pass. Three disposable optimized implementation mutations reject.

From the repository root, using the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/flagless-kernel-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/flagless-kernel-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/flagless-kernel-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/flagless-kernel-obstruction --check-source-history

Any CPython 3.12 executable can replace the local interpreter path. Direct modes print PASS_FLAGLESS_KERNEL_OBSTRUCTION and a generated receipt bound to the six package-file hashes. Both registered commands are timed locally at about 0.35 seconds, below their 120-second limits.

Metadata/provenance, committed bytes and source history pass at head 90e9334. Official export/integrity and read-only Git-free normal/optimized replay pass, with all seven exported files unchanged. Canonical main remains ec2780e. The advance CLAIM and source pins record overlap coverage and its limits.

Limitations and unrun obligations

The universal proof depends on #4119's relation-support reduction and #1882's two-target theorem. The finite checks corroborate the new argument; they do not establish arbitrary rational-frame feasibility or independently accept the predecessor premises.

The remaining 24 coordinate-kernel pools and span five are unresolved here. Native legality, physical caller/frame/access/restoration prices, stock, precision/tape, compiler, supplier admission and the complete multiplication theorem remain outside scope. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker, predecessor scientific suite or selected operator replay was run. Every registered package command ran locally.

Open next questions

• Determine eight-call feasibility of the remaining 24-pool orbit, starting with the literal pool (0x0003,0x0022,0x0550) and preserving the target coordinate projectors and both cube relations. Its kernel support union has size one, so this obstruction does not decide it.
• Identify one literal native caller for a changed register mechanism and charge all independent targets, arbitrary dirty inputs, frame changes, access and restoration. No complete physical caller or invoice is supplied.

Related open work

Live open-PR searches for span-six, flagless, two-pattern and three-target dirty-auxiliary work found no other open PR on this object. The related #1846, #4119, #5075 and #1882 are merged and credited above. No extra computations were run to audit them. Sparse local notes, arbitrary diffs and unpublished work remain coverage limits.

CI notes

CI pending at head 90e9334. All registered checks passed locally. Hosted CI has not yet certified this head.

Merge Danger

Door: two-way. Blast radius: isolated package. No shared implementation, frozen evidence, predecessor credit or selected frontier is changed.