Independent verification: PR589 family map by exact projector algebra

[1337isnot1337]

Status

Independent verification of one finite PR589 address-operator identity. No new kappa or supplier improvement is claimed.

Finding

For PR #1351's pinned family (pair=1, copy=0), the child-0 q-perp-plus-coordinate host and six-axis remainder are orthogonal and sum to the rank-14 source projector. In child 7, three disjoint rank-8 hosts and the rank-4 remainder sum to the rank-28 source projector. The verifier checks each exact rational projector, rank, pairwise product, and sum.

For orthogonal projectors P,Q, exact block multiplication gives D(P)D(Q)=D(P+Q) for D(P)=[[I-P,P],[P,I-P]]. This proves the full 128-dimensional operator identity for both frozen intervals without repeating PR #1351's 128-column replay. It also rejects an omitted helper, a same-rank wrong-axis helper, and a duplicated helper.

This is an explicitly different-method second audit of open PR #1351. It is conditional on that PR's pinned family role map and interval axes; it does not independently regenerate the allocator, establish chronology, replay helper lowering, or admit the supplier.

Reproduce

python3 -I -O -B research/bit-family-projector-algebra-seed10/verify.py

Expected: PASS_BIT_FAMILY_PROJECTOR_ALGEBRA children=0,7 factors=2+4 q_rank=7 controls=3 no_new_kappa.

Local checks: the verifier, three-control package check, and scripts/research_package.py --check-only pass on CPython 3.14.7. Python 3.12 is not installed locally, so the registered Python 3.12 replay wrapper was not run here.

Pins and credit

• Base: bce54fb3f895cbe5cd4c0eb477c3afeed100190f.
• Selected kappa: 836330098538459/500000000000000000; selected PR589 source commit: a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46.
• Target result: PR Research: exact one-family physical host realization for selected PR589 (no new kappa) #1351 head 6060e62e9d42a16eff493aba74c8a617a12d8041, result blob 4865d97527a9457dd28c0439ba89a1b072a64008.
• Credit 1337isnot1337 for the PR Research: exact one-family physical host realization for selected PR589 (no new kappa) #1351 target, commdata2338 for PR Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307's geometric census, and DreamingOfClouds/eumemic for the selected PR589/PR555 lineage. Prepared by Cinna Davis with substantial OpenAI Codex assistance; no independent human review is claimed. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 680e382e208c07adc011fe155d9fab9f1ff02148.

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

Reviewed scope: Symbolic rational projector audit of the two frozen PR1351 intervals: orthogonal decomposition proves the complete D(P) product identity, with three corruptions rejected. Role selection and chronology are inherited, not independently established for all families.

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.