[DreamingOfClouds]

No new κ. The package states and proves when one independent input of an equal-target class can be retired at a cut: over the dyadic ring, a retirement (basis change on the class at the start and end, basis change on the combinable registers at the cut, pivot register emitted and reloaded with a fresh input) exists exactly when there is a pure dual pair P c = s u, a P = s d, a u = 1. The completion to ordinary events is explicit over Z[1/2]; d c = 1 already splits the class, so no separate primitive-vector condition is needed.

Also proved: ordinary events leave the pairing unchanged (only cuts after non-ordinary events matter); the number of simultaneous retirements of one job is the number of unit elementary divisors of the pairing, not its rank over Q (an explicit rank-one pairing with divisor 3 admits none); inside native calls, helper classes see only the native block, caller classes reduce to the caller cut before the call, and consecutive calls never share a pair (the statement stops at interleaved calls).

Exact checks on the literal words, with the retired word replayed from a freshly computed completion each time:

• [codex] Conditional κ = 0.00330420 (+6.60e-6 over #6702) #6716 unretired caller: the only pair is at cut 353 (class {4,20,21}); Conditional κ = 0.00330720 (+3.000e-6, +0.09079% over #6716): two paid idle fragments unlock independent outputs #6726 unretired caller: pairs at cuts 158, 383, 392; Conditional κ = 0.00330720 (+3.000e-6, +0.09079% over #6716): two paid idle fragments unlock independent outputs #6726 final caller (the caller of Conditional κ = 0.00330814 (+9.4e-7 over #6726): retimed native helper split #6805): no pair at any cut.
• Conditional κ = 0.00330814 (+9.4e-7 over #6726): retimed native helper split #6805 native word with the retimed move: cut 12372, frame 243, helper 368; pure rows exist only with the retimed role in the frame; Gram [4,4,4,4], a unit of Z[1/2] but not of Z; retired endpoint checked on all 771 public inputs.
• Controls: pivot outside the class, non-pure row, non-pure column, zero pairing, frame without the retimed role, word without the retiming, rank one without a unit.

Base: 7c49fe3. Inputs are unmodified copies pinned in retained/SOURCE-PINS.json and SOURCES.json to #6716 at 4a40713, #6726 at 435bbe9 and #6805 at 0ecd4fd.

Reproduce (CPython 3.12, standard library):

python3 -B scripts/research_package.py --package research/p104-dual-pair-retirement-criterion

Expected: PASS_DUAL_PAIR_RETIREMENT_CRITERION (normal and -O) and PASS_REGISTERED_RESEARCH_PACKAGE.

Scope: retirements of the conjugation type defined in PROOF.md section 1; equal-target classes with block-diagonal endpoint; scalars Z[1/2] (the same proof covers Z[i][1/2], not executed); the level section uses an explicit block model; classes mixing caller roles with helper inputs and interleaved calls are not covered. Pricing, native phases and the five framework assumptions of the cited words are not re-derived and no κ changes.

Overlap: #6764 states the pair conditions as a necessary interface for the binary full-bank model; Ali Romman's helper-basis completion supplies the a b = 1 completion step. This package gives the general equivalence with the converse, the unit-divisor count and the level propositions.

Credit Ali Romman (helper-basis criterion), thu-nmrc (#6716), zxv (#6726), CubeJerry (#6805) and DreamingOfClouds (#6702). Apache-2.0.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

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