[aliromman]

This proves two stronger loss bounds for the paired-stage, n+2-query P104 caller model and supplies the missing mixed query/stage coefficient test. With the complete PR #5599 bill, no such caller can reach kappa 0.003 through 92 fields, or through 176 fields when the stage-pair count is odd. The predecessor excluded 50 fields. This is a scoped obstruction and search filter; complete kappa is unchanged.

Proof, reusable chronology test, exact receipt, verifier, external source pins. Base: canonical main 6cf7a68, priority 7.

A two-call stage has rank 0, 2 or 4. Thus the previous positive loss is at least two. For odd stage-pair counts, the rank-two complement of the query Gram matrix forces at least two deficient stages and a loss of at least four. Exact power intervals give the stated complete-price bounds. The original P104 bill gives 98 fields and odd-pair counts through 184 fields. These are bounds for separate ports, full calls, disjoint eight-axis blocks, binary scalar bases, autonomous helpers and no idles.

The result also resolves the proposed growing-loss strategy under the existing cuts: an explicit compatible family has constant loss eight at arbitrarily large odd sizes. It passes all previous rank and intersection cuts. At 1024 fields, both complete relaxed bills are positive. It nevertheless fails the new mixed coefficient test in every order. This identifies a concrete false lead that the next controller search can reject before native work.

PR #5617 supplies the earlier capacity/intersection cuts, full-capacity obstruction and complete source profiles; its proof and receipt are pinned and retained byte-for-byte. Cinna Davis supplies the private-projector lemma in the source proof. Retain the P104, Douglas Colkitt and source contributor notices. The new results are the parity bounds, constant-loss family, chronology test and complete-price consequences.

Normal and optimized registered checks pass: all 560 binary rank-two projectors in dimension four and all their idempotent pair sums; six constant-loss instances through 1024 fields; 131328 predecessor intersection checks on the largest instance; a separate literal mixed-coefficient interpreter; all 120 relative orders of the smallest restricted family; a positive complete six-query word; a deleted-query control; exact price endpoints and hinge sums. Committed-byte, export-integrity and Git-free optimized replay checks pass.

python3 -B scripts/research_package.py --package research/paired-stage-chronology-cut

Python 3.12 and the standard library suffice. Expected markers: PASS_PAIRED_STAGE_CHRONOLOGY_CUT and PASS_REGISTERED_RESEARCH_PACKAGE.

No native source replay, selected cold replay, full source-history ancestry, independent review or proof-kernel check is claimed. The checkout is shallow. Native admission and source complete profiles are inherited evidence. Coupled ports in this longer-word case, paid idles, partial block overlap, joint main/helper codes and characteristic-zero reductions remain open. Passing the new necessary cut still does not prove the whole word. Coordination. Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.