[petalas]

Summary

Prove an exact storage obstruction for eight fixed half-open helper lifetimes. For every positive real scale q:

peak live width                 4q
minimum with split fixed sets   4q
minimum with contiguous blocks  5q

Claim type: scoped mathematical proof with finite corroboration. The explicit family forces a 25% storage surcharge even when all lifetimes are known in advance. A real-to-integer compression lemma and a four-cell contradiction prove the lower bound; displayed layouts attain both optima.

This takes the contiguous-block boundary explicitly excluded by Ali Romman's PR #5763 at cee2336bc3a20c586b1dbb55152d97a114c657a6. The predecessor is cited as the question source and split-range context, not audited or replayed. Classical dynamic storage allocation is credited; novelty of the general fragmentation phenomenon is not claimed. This result was produced by Nick Petalas.

Evidence

• Before: peak live width alone provides no contiguous layout certificate under the changed contract.
• After: the explicit family has exact optima 4q and 5q. The verifier checks all 40,320 total address orders, independently excludes all 9,216 capacity-four integer placements, validates both layouts at six exact rational scales, and rejects four invalid layouts.
• CPython 3.12.15 warning-strict normal and optimized runs pass with PASS_CONTIGUOUS_HELPER_STORAGE_GAP, timed locally at 0.52s and 0.53s. Both registered VERIFY commands and package metadata checks pass.
• Committed bytes and declared source history pass at d46194d7f001210e49ef155409b76182c773e6e7. A generated local receipt binds all five package hashes. Read-only review found no mathematical error in the new proof or layouts.

From the checkout root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/contiguous-helper-storage-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/contiguous-helper-storage-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/contiguous-helper-storage-gap
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/contiguous-helper-storage-gap --check-source-history

Elsewhere, substitute a CPython 3.12 interpreter for the absolute interpreter path. Standard library only.

Limitations

The proof fixes lifetimes, widths and addresses throughout each lifetime. It does not permit movement, change chronology, prove a smallest obstruction, or establish a worst-case approximation ratio. The all-positive-real statement has a written proof; finite enumeration corroborates its unit instance.

No actual supplier is asserted to require contiguous blocks. Source-specific cleanup, routing, ownership, precision, tape movement and all-size compiler obligations are unrun and outside this claim. No native construction, new kappa, selected-frontier change or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed.

Related open work

• Allocate optimal weighted helper pools with independent replay certificates #5763 supplies the split-range allocator and explicitly excludes contiguity. This contribution proves an exact obstruction for that changed interface, without testing its predecessor implementation.

• Materialize P104 in 12 private slots while preserving every native query #5822 allocates actual P104 private lifetimes into unit slots. The present family has unequal widths and does not recompute or challenge its source-specific slot count.

• Prove sharp migration tariffs for two reusable resources (no new κ) #5893 charges migration across two eligible reusable resources. The present contract fixes addresses during each lifetime and prices capacity instead of transfers.

• Construct optimal reversible scratch schedules with replay certificates #5780 and Prove exact weighted five-node clean scratch (no new κ) #5895 optimize reversible scratch with recomputation and chronology choices. The present obstruction uses supplied immutable lifetimes.

• Reduce bounded NTT temporary storage from 17,920 to 1,282 (no new κ) #5886 renames bounded NTT temporaries into reusable cells. The present family is a storage-contract example and does not change that program or its claimed count.

• Prove sharp clean storage for reversible Gaussian overwrite (no κ) #5880 proves clean information storage for reversible Gaussian overwrite. That information-theoretic contract is separate from fixed weighted lifetime placement.

• Prove the six-ADD bank-exchange floor with one clean helper #5898 studies one initially clean, restored helper in scalar bank exchange. Its six-ADD gate-count floor does not price contiguous weighted lifetime storage; no common numerical fixture is evaluated here.

• Prove sharp one-pass child batching storage (no new κ) #5905 proves retained payload storage for one-pass child batching. Its streaming-information model permits encoding and has a different fixed input/output order; this result prices fixed contiguous placement of already specified lifetimes. No common numerical instance is computed.

Open next questions

• Can an actual native caller use split fixed ranges with its complete route and address-description bill? Start with a source-pinned access trace and a declared movement contract.
• Which source-derived lifetime restrictions guarantee contiguous storage at peak width? First identify a restriction that excludes this eight-helper pattern.

CI notes

Head: d46194d7f001210e49ef155409b76182c773e6e7. Run 40500640359 completed with plan failed and replay skipped.

The plan job failed only in Onboarding helper tests, in the unchanged tests/test_research_current_assignment.py: FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failures concern current/historical first-step metadata and sparse-reference expectations. The same step, test file and eight-failure/two-error result were also checked in PR #5896's plan job, based on the same canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a.

The package replay job was skipped. Hosted replay has not certified this package. Only the local mathematical and package checks listed above have run. This PR changes none of those shared tests or helpers; their repair is left to maintainers.

Merge Danger

Door: two-way. Five new files are isolated in research/contiguous-helper-storage-gap/.

Blast radius: research. No shared code, selected record or predecessor files change.