[petalas]

Summary

Prove the sharp active-row count extrema for CubeJerry's #7152 mixed-width balanced CRT schedule, at every dyadic axis count d=2^h, ell>=2 and number k of long axes. Claim type: scoped written combinatorial proof with exact finite corroboration.

A(S) = (2ell-3)(d-1) + 2 F(S)
min F = max(0,k-1)
max F = sum_{j=1}^h min(k,2^(h-j))

Here F counts occupied aligned right subtrees. Prefix placement attains the minimum. Complemented bit reversal attains every levelwise maximum with one placement. The proof also characterizes equality at the minimum. This contribution was produced by Nick Petalas.

Only six files under research/crt-active-row-extrema/ are added. Credit Jerry Chen / CubeJerry #7152 for the activity predicate and whole-tree interface, and Ali Romman #5353/#5625, JoyboyBrian #5053/#5219/#5092/#5496/#6100, ToikY #4195, Cinna Davis #4100, commdata2338 #5519 and their retained numerical, row-entry, workspace and CRT lineage. The exact predecessor head and observed file hash are recorded. Its scientific executable and numerical providers are cited, not imported or rechecked as this contribution.

Evidence

• Before, Compose the complete balanced CRT state tree across changing precision with 127 stable owned roles (no new kappa) #7152 supplies a mixed-width activity predicate and compiled schedule. After, the new proof gives exact placement extrema and explicit attaining orders at every size. This is a scheduling count theorem, not completion of its topology-controller implementation.
• Complete enumeration covers 65,812 placements and 197,436 literal schedules at d=2,4,8,16 and ell=2,3,5, totaling 14,722,192 active commands. A separate bottom-up occupied-tree engine checks the identity, and independent literal loops determine the actual active command lists.
• Every k through 1,024 axes gives 2,056 constructor cases and 18,489 levelwise checks. Twelve domain rejections and three disposable semantic mutations are rejected. Normal and optimized receipts agree and bind all six package files by SHA256.
• Both registered commands pass locally on CPython 3.12.15, macOS, standard library only, timed separately at 2.789 and 2.783 seconds. Metadata/pins, in-memory AST compilation, exact committed bytes/modes/source history, official export integrity and both read-only Git-free exported modes pass.

From the worktree root, reproduce with:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/crt-active-row-extrema/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/crt-active-row-extrema/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/crt-active-row-extrema
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/crt-active-row-extrema --check-source-history --check-modes

Any CPython 3.12 installation can replace the local interpreter path. Expected markers are PASS_CRT_ACTIVE_ROW_EXTREMA, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The all-size result rests on the written proof. Independent human and formal proof-kernel review are unclaimed. Actual row costs are unequal, and changing width order changes node precisions and coefficient geometry. No legal physical axis/modulus permutation or complete nonuniform tariff optimum is supplied. Numerical/ROM providers, source preservation, restoration, binary lowering, precision and the five selected framework assumptions remain separate obligations. No selected scientific cold replay, whole CRT execution, Lean or Docker run was performed. No new kappa, physical speedup or selected-frontier change is claimed.

The predecessor is an external exact-head citation, not a reachable Git pin or live replay input. Canonical main was refreshed unchanged at e54523517d0f03e58352e647c403a5af3e525a57. Live claims, current open and merged searches, general-family main searches and the task overlap report were inspected before committing. The cache reported complete pagination over 545 open PRs, 7,209 Discussion records and 10,178 HEAD README/TASK notes. That coverage does not include arbitrary historical diffs or unpublished work. Advance CLAIM.

Open next questions

• Can a physical caller choose a minimizing width placement with a favorable complete bill? Bind one actual axis/modulus permutation and pay its preparation and inverse.
• Which placement minimizes the full nonuniform numerical/ROM bill? Begin with actual node precisions and row_bound, and specify legal axis orders before optimizing.

Related open work

• Compose the complete balanced CRT state tree across changing precision with 127 stable owned roles (no new kappa) #7152 supplies the exact mixed-width schedule and whole-tree physical-state interface used as the premise here. This PR adds sharp all-placement activity counts without replaying its providers.
• Derive balanced CRT node selection and precision from the physical source before invoking the numerical producer (no new kappa) #7154 derives one encoded node's geometry and precision from the physical source. This theorem neither implements that frontend nor removes its remaining traversal/selection obligations.
• Audit uniform Beneš compiler with charged sequential arena scans #4192 studies the uniform Beneš compiler and sequential arena tariff inside a CRT row. This PR counts active row invocations and does not compute its compiler tariff.
• Audit conditional CRT last-K two-slice streaming tariff #4187 studies a last-K two-slice payload schedule. This PR concerns mixed-width whole-tree row activity and computes no competing payload cost.
• Prove the exact rounded tensor-image block minimum (no κ) #7247 studies contiguous block copies of a rounded tensor image. Its extraction endpoint and block count differ from the active CRT node-row count here; neither numerical quantity is cross-checked.

The completion check inspected all ten returned account open PRs created since the advance CLAIM, including this PR. No additional PR studies this active mixed-width CRT node-row count. The related rounded-image block-count PR #7247 is distinguished above. No extra mathematical computations were run to compare other contributions. The delivered scope matches the CLAIM.

CI notes

At unchanged head 6555bd480b70f63bfdcc9ce63f203b0061b7f242, run 43024009248, plan job 130533991471, failed in Onboarding helper tests. The downloaded job 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.

The mismatch is present in canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both base files were inspected directly, and this PR changes neither. Its diff consists only of the six new package files.

The same sole failing test, missing-command assertion and 760-test/24-skipped summary were confirmed directly in the failed-job logs of all seven comparison runs:

• Other contributors: 42905094314 at d14f787dc0948188f8f5485115d92f3ac1b0688c, and 42903313790 at 020d581d9856b6db83e448c3f20418bc63b72c34.
• Prove the exact rounded tensor-image block minimum (no κ) #7247: 43023514635.
• Prove four independent full guards beat their product over every field #7248: 43023633785.
• Prove the sharp rectangular native-fusion approximation floor (no κ) #7249: 43023693429.
• Prove exact sparse tensor-annihilator coefficient valuations (no κ) #7250: 43023829856.
• Prove exact square-zero tensor determinant images (no κ) #7252: 43024030495.

These comparisons concern only the shared CI failure, not those contributions' mathematical results. Run metadata confirms failed planning and skipped package replay in every comparison.

The package replay job was skipped because plan failed. CI did not certify this package. All local checks recorded above passed. Hosted package certification remains unavailable pending the repository-level repair. No fix, retrigger commit or job rerun was made.

Merge Danger

Door: two-way. Blast radius: isolated.

One opt-in research package. No shared implementation, frozen evidence or selected-result file is changed.