[petalas]

Summary

Prove the exact fixed-seam contiguous block-copy minimum for the rounded original-coordinate tensor image at arbitrary axis count. Claim type: scoped written proof with exact finite corroboration.

S_D = R_D = 1
d_i = t_i-s_i
b_i = 1 if 3*s_i > 2*t_i, else 0
R_i = d_i*S_(i+1) + b_i*R_(i+1)
S_i = s_i*S_(i+1)

Here t_i=2^ell_i, s_i<t_i<2s_i, and axis zero is fastest. The proof counts exactly which carry boundaries join two selected runs. Copying the maximal runs attains the bound for arbitrary record contents. The contract copies every selected record once in order and copies no hole, with the target field's origin fixed.

At target/source pairs (4,3), (8,7), (8,5), the sharp minimum is 43 blocks for 105 selected records in 256 target records. The theorem supplies a new local layout invariant and closed form. It supplies no new kappa or physical speedup.

Only six files under research/rounded-image-block-minimum/ are added. Credit CubeJerry #6995 for the image and physical extraction boundary, Dugongue / David Ganz #5566 for scalar Beatty runs and the cyclic-seam distinction, and their retained canonical Gaussian, #3851, Tape/Count and normalized-recovery lineage. The new step is the fixed-seam tensor carry count and its interval-readout consequence. Their scientific verifiers and operation totals were not replayed. This is an original local side question, not completion of #6995's production-caller handoff.

Evidence

• Before: Research: Original-q field extraction #6995 supplies a monotone rounded-image extractor; Prove faster periodic Gaussian inversion through Beatty-run blocks #5566 supplies scalar complete Beatty runs after an allowed cyclic relabeling. These are credited inputs.
• After: a fixed-seam tensor formula and matching block-copy lower/upper proof, checked against every ordered dyadic packet with sum(ell_i)<=10.
• Exact enumeration covers 4,180 packets, 3,270,652 target records, 1,831,047 selected records and 657,556 maximal blocks. Literal source-coordinate rounding and independent target half-open interval membership produce the same complete images. The recurrence and finite sum both equal the observed count. Every run-copy readout covers exactly the selected addresses and no hole.
• The named 256-record payload fixture, nine rejected invalid packets and five fixed-seam/trailing-hole/axis-order/large-formula controls pass. The sixteen-axis control evaluates the formula without enumerating its large volume. The universal theorem is the written proof, not finite extrapolation.
• Local CPython 3.12.15 warning-strict normal/optimized checks pass with identical generated hash-bound receipts. The two registered commands took 1.65 and 1.63 seconds locally, below their 120-second limits. Registered metadata/source-pin replay, AST/function bounds and exact committed-byte/mode/source-history checks pass. The maximum function length is 27 lines. These durations describe checks, not physical performance.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/rounded-image-block-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/rounded-image-block-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/rounded-image-block-minimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/rounded-image-block-minimum --check-source-history --check-modes

Expected direct status is PASS_ROUNDED_IMAGE_BLOCK_MINIMUM, with fixture.blocks=43. Another host can substitute its CPython 3.12 path. The verifier imports no predecessor code, writes no files and emits a receipt bound to all six package files and three exact source texts.

Limitations

The interval-copy contract excludes copying holes followed by filtering, a changed producer order, a free cyclic seam change and noncontiguous native reads. No native bulk-copy service or improved scan time is supplied. Source delivery, movement, restoration, stock, precision, compiler closure and the complete paid/productive bill remain separate obligations. All five selected framework assumptions remain. Selected cold replay, full Gaussian/CRT/carry execution, Lean and Docker were unrun. No complete multiplication theorem or independent human/formal-kernel acceptance is claimed.

Python's bounded managed-runtime allocation deviation is documented in README. The executable formula caps axes at sixteen and exponents at 64; the written theorem is unrestricted. Literal enumeration caps target volume at 1,024.

Related open work

• #7251 counts active mixed-width balanced CRT node rows and gives placement extrema. This PR counts contiguous blocks in a rounded original-coordinate tensor image. The endpoints and numerical quantities differ, and no competing value is computed here.

The own-open-PR list was checked again after CI finished, including every returned PR created since the CLAIM. No additional PR on the same rounded-image block count was found. The merged #6995 and #5566 packages remain the credited sources. The earlier object-keyword searches and complete reported 545-open-PR snapshot are bounded coverage; arbitrary unseen diffs and unpublished work remain outside it.

Open next questions

• Optimize the count over axis permutations that a real producer can legally deliver. Start with an adjacent-axis exchange inequality and include the complete physical permutation cost.
• Build one native encoded maximal-block iterator and compare its complete setup, scan, movement, copy, restoration, precision and storage bill. A block count alone is not a runtime improvement.

CI notes

At unchanged head 6fafd9f5983e9d6a573756e73f8b700fe90e4832, run 43023514635, plan job 130532328734, failed in Onboarding helper tests. The log reports 760 tests, one failure and 24 skipped. The sole failed test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record. Its assertion is AssertionError: '`make frontier7000-verify`' not found in the current section of docs/ci-verification.md.

This mismatch is already present in canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while the CI verification document still describes #6030 and make frontier6030-verify. Both base files and the test were inspected directly. This PR changes only its six new package files.

The same exact test, assertion and single-failure summary were independently verified in run 42905094314, plan job 130140894651, head d14f787dc0948188f8f5485115d92f3ac1b0688c, and run 42903313790, plan job 130134720799, head 020d581d9856b6db83e448c3f20418bc63b72c34. Both logs also report 760 tests and 24 skipped. This comparison concerns only the shared CI failure, not those contributions' mathematical results.

The package replay job was skipped because plan failed. CI did not certify this package. The local normal/optimized scientific checks, registered metadata/pin replay, committed-byte/mode/source-history checks, official export integrity and Git-free standalone replay passed as recorded above. Hosted package certification remains unavailable pending the repository-level repair. No fix, retrigger commit or job rerun was made.

Advance CLAIM. The delivered mathematical scope matches it. Canonical main was refreshed before submission and remained e545235; priority revision 8, selected #7000 unchanged.

Merge Danger

Door: two-way.

Blast Radius: research.

Isolated new package; shared scripts, predecessor packages and selected-frontier files are unchanged.