[adamgall]

Result and exact scope

Adds only research/complex-heterogeneous-child-interface/: a complete unequal-width classification for one inherited Clifford tensor child under arbitrary reciprocal linear encoding, plus exact source and volume accounting.

For two equal-volume lanes with canonical operators C_p and C_q^epsilon, 1 <= p < q, and a candidate child lambda C_d tensor I_t with positive integers d>=1, t>=1, arbitrary restored dirty identity padding does not help. Width-zero scalar children are outside this declared contract:

• Same directions never fit one phased child, even granting arbitrary invertible encoding and its inverse.
• Opposite directions fit exactly when q=p+1 and p=1 or 2 mod4, or q=p+2 and p=1 mod4. The child width is necessarily q+2, with the explicit phase in the proof, and its active volume is the sum of both original lane volumes.
• Three full rational basis witnesses attain the exceptional cases. For example, (C_1 tensor I_4) direct-sum (C_2^-1 tensor I_2) is similar to -C_4 on 16 coordinates.
• Even with both encoders and every other extra charge free, the inherited available child expression 2 A V(q+2)^sigma exceeds A V(p^sigma+q^sigma) for 0<sigma<1. This is an upper-bound-expression comparison, not an optimal-runtime lower bound.

The proof uses Gaussian-prime valuation to force the new width and exact four-trace multiplicities for necessity/sufficiency. It is not extrapolation from the finite sweep.

Concrete source connection and predecessor credit

An independent scan of both 24,767-event selected local orientations reconstructs all 359 higher-rank same-ADD pairs from PR1836, and binds all 215 heterogeneous pairs by source gate, rational coefficient, old/new frame and quotient ranks. None of those 215 same-direction canonical substitutions is realized by the declared reciprocal packing route, at any positive column multiplicity. The 144 equal-rank rows and their prior reciprocal obstruction remain PR1920's credit.

The scope fixes the original per-lane normal-form endpoint charts. It does not infer that the original physical frame charts are reciprocal, or exclude independent nonreciprocal adapters, clean invariant encodings, a different joint primitive, or a new child tariff. PR1836 already labels its supplier unadmitted; this resolves a specific route rather than alleging that its conditional screen was a construction.

Nearest work:

• PR1836 0a62e3f1c0ea493ddb83f25b91c5548c87978bcc: census and hypothetical rank-sum screen (Cinna Davis).
• PR1920 057b214aea45c65926b93ad8b95693c80f0f79c9: equal-width reciprocal obstruction/exceptions and volume comparison (Arham Tahir).
• PR1924 0a7c54dd94bce3784b61c4203adfdda98d13d53d: different clean-invariant-subspace contract (ToikY).
• PR1866 7d22d1d51eb6c0bc040f8181b6c5f83e051fb234: different heterogeneous bilinear tensor, not this Clifford operator.
• PR679 and PR1240: inherited volume tariff and exact ceiling engine; icekylinx and the selected Rohan Arun/Jeremy Kothe/Hao Yan source lineage are retained.

The counterfactual 215-pair-only profile is also screened exactly: baseline cap bracket [0.001690054109,0.001690054110]; hypothetical bracket [0.001733875133,0.001733875134]. It does not clear 5%, and it is not an admitted profile. No supplier, new kappa, selected-record or frozen-source change.

Reproduction and exercised evidence

uv run --python 3.12 --no-project python -I -B research/complex-heterogeneous-child-interface/verify.py
uv run --python 3.12 --no-project python -I -O -B research/complex-heterogeneous-child-interface/verify.py
uv run --python 3.12 --no-project python scripts/research_package.py --package research/complex-heterogeneous-child-interface

Observed on CPython 3.12.12, macOS 27.0.1 arm64, standard library only:

PASS_HETEROGENEOUS_CHILD_INTERFACE spectral_trials=747264 matches=48 heterogeneous_gates=215 fixtures=3 controls=9 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/complex-heterogeneous-child-interface outcome=negative-result

The sweep covers unequal widths through 64, candidate widths through q+3, both directions and all four phases. It cross-checks cyclic/binomial spectra through 67, signed reductions, three complete rational encoding matrices (2304 operator entries), both source orientations, nine named controls and six additional matrix negative witnesses. Normal receipt generation, optimized read-only replay and registered replay all passed. The card's ceiling working example and lever table also passed. Exact commands, hashes, environment and unrun obligations are in VALIDATION.json.

No full selected cold replay, native hosting/adapters, proof assistant, hosted CI success or independent human review is claimed. The all-width theorem is written mathematics.

Independent task provenance

Round 2 instance 05, exact base 0b1cccf86e2f5132ed286a26ef0a681a495740ea; OS-random chooser sampled once with no overrides. Seed c2889b6f0505155ec921e731269f82bf, track complex-mechanisms, selected emitted card complex-rank-shape-decoder. Complete selection is preserved in SELECTION.json. TASK records the explicit scope change from physical synthesis to this prerequisite interface theorem and cost screen.

Claim: #1929 (comment)

Final shared overlap covered 584 open PRs and 565 Discussion records with complete pagination; targeted current heads and local/merged notes were also checked. Metadata search is not a novelty certificate.

Prepared for Adam Gall with substantial OpenAI Codex assistance in mathematics, research, implementation, checking and writing. Apache-2.0. The next useful question is a fully priced nonreciprocal adapter, not another reciprocal basis search.

[Mister-Meeseeks]

Batch review at e55ff15a61286c5bec781db15118fd7a0144b972.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Gaussian-prime valuation and spectral proof classify all unequal-width reciprocal child encodings. Independent scan binds215 selected heterogeneous pairs, none same-direction-admissible; three rational signed exceptions and full spectrum/control replay pass. Positive exceptions lose under full-volume pricing; hypothetical profile is not an admitted supplier.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.