[petalas]

Summary

Scoped all-size proof: the mixed UT4 middle-factor relaxation has exact minimum n=2d+k+[d=k=1], where d counts active diagonal roles and k counts pure-central roles.

one pure-central role + any noncentral partner
separate relaxation blocks: 4 + 2 columns
new joint relaxation block: 5 columns
arbitrary mixed family: combine joint, singleton and credited pure blocks

The new step is the six-partner mixed construction and uniform allocation theorem. ToikY #1662 already solves the pure-central relaxation; Gareth Davidson #2027 supplies mixed-root span unions; Nick Petalas #2175 supplies the mixed lower bound and one z/uz fixture. Their results are cited premises, not audit deliverables. The result identifies a sharp limit of these coefficient equations alone. It supplies no chronological word, physical supplier, new kappa or selected-frontier change.

Only research/ut4-mixed-relaxation-optimum/ is added. The proof and constructor preserve nonzero middle couplings, both active-span requirements, all four coefficient equations and arbitrary padded identity coordinates. This contribution is produced by Nick Petalas.

Evidence

Before, the inherited general mixed lower bound had a special z/uz fixture, while pure-central relaxation attainment was already known. After, explicit five-column blocks cover every noncentral partner and direct sums attain the exact mixed optimum uniformly.

Local normal and optimized checks pass: 5,602 exhaustive mixed/padded fixtures through four active roles, 1,098 larger fixtures through 64 roles, all seven scalar minima, 15 independently multiplied templates with 176 entries, and eight rejection controls. Each direct registered check took about 0.4 seconds locally, below its 120-second limit. These are verification durations, not a performance improvement claim.

From the repository root, using CPython 3.12.15 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/ut4-mixed-relaxation-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/ut4-mixed-relaxation-optimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/ut4-mixed-relaxation-optimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/ut4-mixed-relaxation-optimum --check-source-history

Any CPython 3.12 path can replace the local path. Both scientific commands print PASS_UT4_MIXED_RELAXATION_OPTIMUM; the package runner and committed-byte/source-history checks pass. The generated receipt binds scientific outputs and package/live-input source hashes. Official committed export/integrity and optimized Git-free replay pass, preserving all eleven exported files. Corrupted source and typed-receipt controls reject and preserve their own bytes.

Limitations and unrun obligations

The proof covers every finite diagonal family; the executable supports up to 64 active roles and two dirty coordinates. Necessary coefficient-relaxation attainment does not construct paid prefix/suffix paths or a full formal endpoint. Native/compiler, hosting, physical costs, precision/tape, outer transfer, selected cold replay and proof-kernel verification remain outside scope. Independent human and formal proof-kernel review are unclaimed. Strict Python warnings pass; ruff is unavailable and no external static analyzer was run.

The historical card orientation command scripts/research_baseline.py fails on the current selected record with KeyError: 'bit_lower'. This unrelated orientation failure is recorded in the package. The selected review and source manifest were read directly; no shared code or selected file was changed.

Open next questions

• Find an additional chronological equation that strengthens the mixed bound. Start with one z and one u role, impose actual paid prefix/suffix paths and charge all outer calls.
• Determine complete tariffs for a named larger mixed family outside the existing z/uz weighted region. Retain every formal coefficient and arbitrary dirty restoration.

Related open work

Refreshed after hosted CI: this account has nine other open PRs created since the CLAIM at 2026-10-22T22:41:49Z (#4873, #4874, #4875, #4876, #4878, #4879, #4880, #4881 and #4883). Their titles and bodies contain no work on the same UT4 mixed middle-factor object or quantity, so no same-quantity comparison is made. The scoped live search before submission likewise found no other open UT4 or mixed middle-factor PR.

The merged #1662, #2027 and #2175 premises are credited above. #2314 studies a complete z/uz weighted-word region, whereas this contribution solves only the necessary coefficient relaxation; no same-quantity numerical comparison is claimed.

CI notes

Both hosted checks passed at exact head 7d94b58da38d73d570814a4fe97e8818690594ae: plan in 3m22s and the registered package replay in 1m12s. Successful run 39867516380 is bound to that head. This certifies the registered automated checks, not independent mathematical acceptance.

Both registered checks, metadata/source pins, direct normal/optimized runs, committed bytes/source history and Git-free export replay also pass locally. No proof obligation was deferred solely to CI.

Merge danger

Door: two-way. The isolated package can be reverted.

Blast radius: package. No shared verifier, predecessor source, frozen evidence or selected result changes.