[petalas]

Summary

Scoped written mathematical proof with exact finite corroboration, produced by Nick Petalas. For two heterogeneous dyadic tensor-cut types with disjoint odd-prime defect sets, the new rational crossing rule returns the exact best unit-channel density at every positive length and every attaining multiplicity.

r(K)=r, c(K)=c, r(L)=s, c(L)=d
0<c<r, 0<d<s, disjoint odd-prime defects

find k with (r/c)^k <= (s/d)^(n-k)
        and (r/c)^(k+1) > (s/d)^(n-k-1)
optimum = max((d/s)^(n-k), (c/r)^(k+1))
only k and k+1 first-type factors can attain it

For K=diag(1,1,3), L=diag(1,5) and length four, the sharp density is 8/27, attained by a 3:1 mix, with capacity 16 at rational rank 54. The general catalogue bound gives 4/9. Equal multiplicities give density 1/4 at the different rank 36, so balanced counts do not optimize this objective.

Credit zxv / PR6054 for the fixed defective-catalogue question, full saturated tensor-module theorem and general switching bound. Nick Petalas / PR5882 supplies primewise tensor capacity; Ali Romman / PR5615 supplies the single-cut criterion, with PR5602/5611/5599 and the retained helper/native/controller lineage. These are cited premises, not audit or replay targets. Smith theory, tensor ranks and monotone crossing optimization are classical. The new contribution is the finite-length schedule optimum, exact bounded optimizer and standalone schedule certificate. No global literature-priority claim, new kappa or selected-frontier change.

Advance CLAIM. Canonical main was refreshed before commitment and remains 7c49fe3, priority revision 8. The card's scope change to an algebraic side result is explicit. Only research/heterogeneous-tensor-schedules is added.

Evidence

Before, the general defective-catalogue bound does not give the optimal heterogeneous mix. After, PROOF.md gives the all-length theorem and a strict attaining refinement; schedule.py computes its two candidate counts without logarithms.

Local checks on CPython 3.12.15, macOS 27.0.1 arm64:

• All 14,112 bounded profile/length contracts, 246,960 individual count observations and 562 tied-optimum cases agree with a separate exhaustive-count oracle.
• All 248 ordered binary tensor words in four diagonal catalogues are materialized, including 18,504 diagonal coefficients. Independent integer gcd/lcm Smith reduction establishes all 20 length/catalogue optima and their complete attaining count sets, without using prime profiles or the optimizer.
• Eleven invalid-profile rejections and nine semantic controls pass, including a shared-prime counterexample, exact crossing equality, maximum accepted sizes and the sharp heterogeneous example.
• Warning-strict normal and optimized direct runs pass with identical generated hash-bound receipts. The final commands took 0.374 and 0.438 seconds locally, below their separate 120-second limits. These are replay-budget observations, not a performance claim.
• Both registered commands and metadata/source validation pass: PASS_REGISTERED_RESEARCH_PACKAGE. In-memory compilation, function bounds, attribution scan and whitespace checks pass.
• Committed-byte, mode and source-history checks pass at 09d4b6b. The official committed export and declared-input coverage check pass. Both direct modes pass with empty PATH in a read-only Git-free export; all eight exported files retain their hashes. Receipts are generated from execution, never handwritten.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -S -W error -B research/heterogeneous-tensor-schedules/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -S -O -W error -B research/heterogeneous-tensor-schedules/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/heterogeneous-tensor-schedules
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/heterogeneous-tensor-schedules --check-source-history --check-modes

Another CPython 3.12 executable may replace the absolute path. Expected markers are PASS_HETEROGENEOUS_TENSOR_SCHEDULES, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Standard library only; the scientific verifier writes nothing and imports no predecessor executable.

Limitations and unrun obligations

The theorem fixes tensor length and maximizes capacity divided by rational rank. It does not fix total rank, maximize absolute capacity, or price per-type physical work. Disjoint prime defects are a mathematical premise and cannot be authenticated from capacities alone. The theorem covers all positive lengths; the executable caps ranks and length at 4096, and the exhaustive corroboration has the smaller domains listed above.

Physical basis preparation and inverses, native common charts, chronology, carriers, delivery, restoration, precision/growth, stock, tape, independent productive targets, compiler and outer obligations remain unpaid. No actual selected product cut, complete signed invoice, supplier admission, complete rounded multiplier or discharge of the five framework assumptions is supplied. Selected scientific replay, Lean and Docker are unrun. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Determine exact optima for at least three heterogeneous types or overlapping defects. Start with a finite prime-rank table and fixed length, derive its integer minimax objective, and supply a complete optimality certificate. A source-specific paid/target objective is still missing.
2. Extract two actual native product cuts with full saturated modules, common physical timing and disjoint prime defects. Price basis preparation, delivery, restoration, native charts and the complete signed invoice before crediting productive targets.

Related open work

• Prove sharp one-channel capacity change for paired cut edits (no κ) #6462 studies dyadic unit-capacity change under paired prefix/suffix ADD edits. This fixes the two cut types and optimizes their tensor multiplicities; it computes no common edited-cut instance.
• Prove sharp dyadic channel loss under frozen snapshots (no κ) #6465 studies the same unit-capacity quantity under frozen snapshot restrictions. This assumes fixed unconstrained prime profiles and supplies no snapshot update or comparison instance.

No inspected open PR states this two-type scheduling optimum. Parallel direct-sum activation, Smith-axis ADD-precision ordering and physical helper edits have different objectives. No extra computations checked another contribution's reported values.

Completion refresh inspected the ten other account open PRs created since the CLAIM: #6648 and #6650 through #6658. None studies the same two-type dyadic unit-channel scheduling optimum or computes a conflicting common value. The tensor-root COPY construction #6650 and target-refinement optimizer #6651 have different objects and objectives; their rank or tariff numbers are not compared with unit-channel density. No additional computations checked those contributions.

CI notes

Both hosted checks passed at exact head 09d4b6bb8a30d6c616fa2263d6ec99ee8c47a483 in run 41660512907: plan in 4m31s and replay (research/heterogeneous-tensor-schedules) in 1m34s. The completed check list was read once, and the PR still has that head.

Hosted replay certifies execution of the registered normal and optimized finite checks and metadata validation. Local publication/source-history and export checks above also passed. This is not independent human or formal proof-kernel acceptance of the universal written proof, and the physical/supplier obligations remain open. The delivered mathematical scope matches the advance CLAIM. No failed or waived check remains.

Merge Danger

Door: two-way.

Blast radius: package. Seven isolated new files and opt-in replay only. Shared implementations, pinned predecessors and the selected frontier retain their bytes.