[petalas]

Summary

Scoped algebraic proof: classify every forward/inverse multiplicity pair at literal Clifford widths f congruent to 2 modulo 4 under an arbitrary full-space encoder and its exact inverse. The first new identity is

C_2 direct-sum C_2 direct-sum C_2 direct-sum C_2^-1 = E^-1 (-i C_4) E.

For q=a+b and delta=a-b, a child exists exactly when q=2^j |delta| and 4^j divides q for an integer j>=0. Its width, spectator multiplicity and phase are f+2j, q/4^j and sign(delta)i^(-j). The minimum batch for an increase 2j is 4^j lanes. Restored dirty padding cannot enlarge active volume. The written proof constructs rational dyadic encoders on arbitrary inputs.

Before this contribution, Arham Tahir's #1920 covered uniform lanes and a balanced opposite pair. Adam Gall's #1946 covered two unequal-width equal-volume lanes. The new result concerns arbitrary multiplicities of equal-width lanes, including the unbalanced exception. The spectral/Walsh methods and original interface/tariff remain predecessor credit. No literature-priority claim is made for elementary spectral similarity.

Only research/mixed-clifford-dyadic-imbalance/ is added. This contribution is produced by Nick Petalas. Credit Arham Tahir #1920, Adam Gall #1946, ywz65 #2516, Greer Guthrie #679, Ross Richardson's diagonal-child contract audit, ToikY #1924, Douglas Colkitt #665, Jose Manuel and Cinna Davis #1779, Josie Elliston #1817, icekylinx, DreamingOfClouds, Rohan Arun, Jeremy Kothe, Hao Yan and the retained native-kernel/construction lineage. Exact predecessor README provenance is pinned in SOURCES.json; no predecessor code is copied or executed.

Evidence

• Before: the balanced width-two opposite pair has zero first trace and cannot be one phased tensor child, as already proved by Research: Clifford reciprocal-basis obstruction and opposite-direction exception #1920.
• After: the exact 3:1 identity and general integer criterion pass an independently constructed binomial-spectrum comparison. The finite domain has 17,152 targets and 1,170,400 dimension-compatible phase candidates, with 1,360 matches including 336 proper width upgrades. Boundary cases are controls, not new predecessor verification.
• Four complete rational witnesses check 5,870 intertwiner entries and both inverse products. They cover both imbalance signs, spectator multiplicity two, a width increase four, and arbitrary restored dirty complements. Wrong phase, direction, encoder and dirty output fail. Balanced, nondyadic, nonintegral-spectator and invalid-domain controls pass.
• Registered metadata and normal/optimized checks pass locally on CPython 3.12.15, standard library. The two registered commands were individually timed at 0.472s and 0.468s. These are replay durations, not performance claims.
• Committed-byte/source-history preflight passes for all six files at 303bc70074e4fc70abb947012a2dd37caa219ca4. Official export integrity and normal/optimized read-only Git-free replay pass, with all seven exported files unchanged. Generated stdout receipts bind package-file hashes; no receipt was hand-written.

Exact reproduction from the repository root:

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

Other environments may substitute their CPython 3.12 executable. VERIFY.json registers both explicit warning-strict commands, each with a 120-second cap. The standalone verifier needs neither Git nor external inputs.

Limitations and unrun obligations

The quantified theorem is written mathematics; finite checks corroborate it. Independent human and formal proof-kernel review are unclaimed. This is not a complete multiplication theorem or a supplier admission.

The joined child still loses the inherited full-volume upper expression even when encoding is free. No favorable recursive invoice or new kappa follows. Native encoder/decoder realization, frame conversion, scalar reads, physical storage/routing, precision, restoration and complete consumer costs are not supplied. The theorem fixes f=2 modulo 4 and does not provide a uniform operation for all column multiplicities. No predecessor verifier, selected cold replay, Lean or Docker build was run.

The selected source-binding orientation passes. The historical card command research/complex-local-word-ceiling/verify.py raises KeyError: complex_profile against the changed selected schema. It was not repaired and provides no scientific evidence for this result. Canonical main remained 624de9dc66858af0945e5f8a0942488d62280d43 at the final fetch; priority revision 7 and selected frontier are unchanged.

Open next questions

1. Can one actual three-forward/one-inverse width-two caller support a favorable, fully paid operation or volume contract? First pin its charts, independent inputs, outputs and complete encoding/decoding/movement ledger. No physical caller or improved tariff is supplied here.
2. Can a uniform supplier exploit the exception while supporting every required literal width? First identify the caller's column parameter, admissible congruences and legally priced fallback. An actual caller and all-size cost contract remain missing.

Related open work

• Add phase-exact native Clifford compiler and complete contact neighborhoods #4841 supplies phase-exact native Clifford compilation and chart-neighbor searches. This result permits arbitrary rational reciprocal encoders and does not claim that its E is compiled by that interface.

• Add a 72-work literature map with exact native encoder and compiler tests #4792 studies native encoders and compiler tests through a literature map. The present contribution is an exact mixed-lane similarity classification; it gives no cheaper native encoder or complete supplier.

• feat(research): exact 5-array 14-call block-3 unit schedule via 14-neighbor symplectic cascade (κ <= 0.0628) #4813 gives a finite block-three unit schedule. Its engine/observation contract differs from the one-child reciprocal full-space identity here.

• feat(research): exact 3-array 10-call block-4 unit schedule on C^16 (κ = 0.1315 > 0.10) #4818 gives a finite block-four unit schedule. This contribution does not price that schedule or compute its proposed saving.

• feat(research): exact 13-array 41-call block-4 boundary schedule on C^16 (κ = 0.1714) #4819 gives another block-four boundary schedule. No common invoice or numerical saving is recomputed here.

• feat(research): exact 3-array 9-call block-5 unit schedule on C^32 (κ = 0.3174) #4826 gives a finite block-five unit schedule. This theorem fixes directional multiplicities of equal literal widths and supplies no all-width physical completion.

• Count and index shortest binary chart words (no new κ) #4859 counts and indexes shortest words between binary Lagrangian charts. It does not classify full-space mixed-direction tensor-child similarity or compute a competing batch minimum.

• Prove exact cycle-constrained chart neighborhoods (no new κ) #4864 counts cycle-constrained simultaneous chart neighborhoods. Its chart fibers differ from the directional-multiplicity identity here; no common numerical quantity is compared.

• Prove exact four-frame concave subspace pricing (no new κ) #4866 prices three-contact junctions in a schematic full-subspace metric. It supplies no reciprocal Clifford packing criterion or native tariff for this encoder.

The pre-DONE refresh inspected ten open PRs on this account created since the CLAIM, including this PR. None of the nine other submissions studies this exact mixed-lane classification or computes a competing batch value. The nearby geometry results are distinguished above. No extra predecessor computations were run.

Live open PRs, recent handoff claims, targeted merged searches and distinctive all-main README searches were refreshed before committing. No same-question claim was found within that coverage; unpublished work and arbitrary code diffs are not fully covered. Advance CLAIM.

CI notes

Both hosted checks passed on 303bc70074e4fc70abb947012a2dd37caa219ca4 in run 39864461022: planning in 2m19s and registered package replay in 1m28s. Run metadata confirms the exact head, completed status and success. The registered replay executes both warning-strict normal and optimized checks in VERIFY.json. Local registered replay, committed bytes/source history and read-only Git-free export checks also pass. Automated replay does not imply independent human or proof-kernel acceptance or a complete multiplication theorem.

Merge danger

Door: two-way. Blast radius: isolated. Only a new research directory is added; shared code, frozen sources and the selected record are untouched.