[JoyboyBrian]

Change and mathematical scope

pre Y: C^-1 -> backward FULL -> thin Q^-1 C^-1 -> post Y: Q^-1 C Q
                         = [[D,-DQ],[DQ^-1,0]]

Proves every block of one physical stage37/38 pair on arbitrary independent main inputs and the typed all-positive-width lift, retaining all 408 autonomous helpers. The complete emitted field ledger charges 336 extra rank-one jobs, including 224 for the actual exit conjugation; it does not import the scalar p19 invoice. This resolves the changed-operator handoff from #5036, separate from two-stage terminal-code products.

Validation

• Before: scalar A^-1 leaves the signed physical residual D(I-A^-1 C); the old owner1/address1 witness still fails. After: actual source signs pass 57,344 absolute-phase checks, 1,792 basis-column comparisons across width1/2 fibers, eight mixed inputs and every Walsh coefficient of that physical fixture. Six semantic controls reject missing/misordered recoding, signs and helper cleanup.
• Registered normal and optimized checks, committed-byte/source-history checks and a Git-free exported direct replay pass at 65445e68d79d7e9f289d4f97e0aa4a8416f2438f. Direct replay takes about one second. The written operator proof supplies all-width coverage; finite width2 fibers are corroboration, not exhaustive physical-address enumeration.
• Credit Cayden Burkart/SethBurkart123 Consume the #4929 center-free word with coded reverse stages and A⁻¹-mixed query shears (no κ) #4940, Dillon Emit a phase-correct coded p16 caller removing 144 center calls (no κ) #4933, Dugongue Construct a center-free native producer with an explicit rank-8 port code (no new κ) #4929, Cinna Davis Conditional κ = 0.00296 via query-thinned productive controllers (+10.18%) #4440/Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470/Construct a reversible 640-field center interface and exact raw dual (no new K) #4694/Normalize eight center chains and construct a conditional two-call consumer (no new K) #4698, Cao Yifan, zxv, Jacob Alan Sussman, and retained native/Clifford/precision contributors. Prepared for JoyboyBrian with substantial OpenAI Codex assistance, Apache-2.0.

Merge Danger

Door: Two-way.

Blast Radius: Isolated.

Only a new research package. Physical field-map/precision tariffs and complete caller affordability remain open; the family decision is to stop here until that complete budget is supplied. No full native/p19 replay, selected-frontier update, new kappa or full multiplication theorem is claimed.

Task and overlap

Refined task b2_math_physical_inverse_pair on card centre-path-sharing; canonical investigation. Scientific base 4c0a359d88f22f05189d51f48387750b801a45d4, priority revision7; refreshed canonical main is unchanged at the submission checkpoint. Exact dependency heads and retained source/path/hash records are in the scoped package. The claim states predecessor coverage, changed premise, live parallel exclusions and bounded stop rule. Arbitrary diffs and unpublished work are outside overlap coverage.

Credit

Cayden Burkart/SethBurkart123 #4940; Dillon #4933; Dugongue #4929; Cinna Davis #4440/#4470/#4694/#4698; Cao Yifan; zxv, Sussman, retained native/precision lineage; JoyboyBrian #5036/#5037. Substantial OpenAI Codex assistance is disclosed; retained source-specific attribution and licenses are preserved.

Open next questions

See the package questions for the smallest remaining caller/interface obligation and its blocker. This contribution stops at the stated scoped result.

Reproduce with CPython3.12 on macOS arm64: python3.12 scripts/research_package.py --package research/physical-inverse-pair. A separate Astra review of this exact HEAD found no actionable issue and reran normal/optimized targeted checks. Scientific review remains AI-assisted, not human or proof-kernel acceptance.