[ToikY]

Change and mathematical scope

NO VERIFIED GAIN. The κ change is exactly 0%. The strongest independently reproduced comparable result is #7161 at κ = 0.003644890067940, head e3f8d39. The fixed campaign target is 0.0040075385047236.

Two independent rank-one full-fibre BIT address reflections cannot save the paid tariff by packing into other full-fibre reflection children when the no-child map preserves ambient addresses. The proof allows linear mixing across roles and cover rows, overlapping banks, unequal volumes, any number of children and clean padding. Over the odd local ring the retained split involution is diag(I, I−2P) in sum and difference coordinates. A triangular minor gives the same rank bound for address-preserving cover routes as for an identity no-child map. A single rank-two child for two equal input fields needs at least 2s/(s+1) times one field’s volume, where s=q^f. The charged tariff exceeds the two original calls throughout 0 ≤ κ ≤ 1/2.

Task and overlap

Changed bit-new-register-model scope under Discussion #7111. Base e545235. The pinned #7001 native word comes from the #6726 line. The exact reframe permutes all 111 port interfaces and gives 660 independent rank-one pairs, including native events 19152 and 19167. #673 supplies the rank-defect argument for one fixed permutation. This proof extends that argument to different literal reflections, full-fibre child ranks, cover routes and volume-weighted tariffs. #1920 and #1924 concern related complex encodings. #7162 bounds changes to the caller schedule at fixed native invoice.

A separate, previously validated prerequisite commit 6c8c9a0 reuses the exact selected-replay navigation from #7161, following #7147/#7156. It lets hosted onboarding reach the scientific package. No selected record or historical source pin changes.

Validation

From a clean checkout on CPython 3.12 (tested 3.12.14):

python3.12 -I -B research/bit-paired-reflection-bound/verify.py
python3.12 -B scripts/research_package.py --package research/bit-paired-reflection-bound
python3.12 -B scripts/check_research_commit.py --package research/bit-paired-reflection-bound --check-modes

The verifier authenticates the native word, checks all 111 port correspondences and 1,386 rank-one MOVE sites, and verifies 53,001 split-address cases for q = 5, 7 and 11. The GL₂(F₅) cover fixture retains all 9,600 pivots with address-preserving routes and cancels them in an address-changing control. It also checks 28 rational tariff inequalities. The all-width inequality and the variable-bank telescoping argument were independently audited. Pinned proof and replay

Credit and open next questions

Credit Androo Lloyd #673 for the fixed-permutation rank method, zxv #6726 for the word, DreamingOfClouds and CubeJerry for the #6834/#6980/#7001 paired lineage, and Clear1nsight for the comparator and navigation fix. OpenAI Codex and subagents materially assisted the derivation, audit and replay.

The first changed-source pair is event 19167, role 225, charts 3→213. A follow-up can test whether an already-required address-changing map supplies the missing defect, with the full adapter cost charged. The retained cover right routes alone cannot supply it. Incomplete fibres, other child primitives, and a changed external contract remain open. No new physical provider or completed multiplier is claimed.