Construct an exact typed seven-axis native quotient supplier (no new kappa)

research/native-quotient-interface/.gitattributes

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research/native-quotient-interface/FREE-GROUP-LIFT-REVIEW.md

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+# Why the repaired local word holds at every fiber width
+
+Prepared for Dugongue with substantial OpenAI Codex assistance. Apache-2.0.
+
+## A universal finite certificate
+
+The exact row evaluator `verify_buffers.py` represents a row by a finite dictionary
+
+    (independent_input_role, reduced_signed_generator_word) -> rational_coefficient.
+
+Its `warp` operation left-prepends the actual shared operator symbol, cancelling only an immediately adjacent inverse symbol. Its `add`, SCALE, COPY and owned CLEAR rules perform exact rational linear algebra on these dictionaries. Distinct backward operator symbols are also represented formally. It assumes no finite matrix identities among different generators. The second evaluator `native_extensions.literal_replay` invokes those same free-word arithmetic primitives while separately checking every physical frame incidence and all owned-buffer conventions.
+
+Thus a successful endpoint comparison is a finite coefficient-wise identity over the rational free group algebra Q[F_72], with distinct independent field labels. It is stronger than numerical agreement of 512-dimensional matrices. In particular the expected helper rows contain only their own input label and the empty word, while main rows contain the stated elementary shear and no unwanted helper input labels. The owned C row is identically zero. Those exact zero coefficients must remain explicit acceptance checks.
+
+## Evaluation theorem
+
+Assign any invertible operator K_j on a vector space A to each positive generator j and K_j^-1 to its negative occurrence. There is a unique rational algebra homomorphism
+
+    ev : Q[F_72] -> End_Q(A)
+
+sending a reduced word to the ordered product of its assigned operators. Adjacent inverse cancellation is valid because each K_j is invertible. Exact addition, scalar multiplication and multiplication are preserved by linear extension. Consequently every formal endpoint identity remains true under this assignment, on every arbitrary independent input field.
+
+This handles sums correctly. It does NOT assert that tensor powers preserve addition. Instead it first proves a free-group-algebra identity, then evaluates each generator by K_j^(tensor f). Evaluation is an algebra homomorphism for every f separately, including f>1.
+
+## Physical frame restoration
+
+At a field in frame C_U, represent its physical data as C_U times its logical row. A paid frame transition applies exactly C_new C_old^-1, leaving the logical row unchanged. At a native warp, the physical operator C_new K_j C_old^-1 realizes logical multiplication by K_j. Scalar ADD/COPY operations use a common frame for their operands and therefore enact the exact field-level rational operations. Owned CLEAR produces zero at any declared frame. Exact monomial/Clifford normalizers must implement those stated physical operators, with their absolute phases, rather than only their projective tableaux.
+
+This is a direct induction over the chronological emitted program. Therefore the successful formal endpoint identity yields the actual physical endpoint C_final S C_start^-1, where S is the stated elementary main shear and independent helper identity. It also proves zero transfer from helper inputs to main outputs and zero final owned state.
+
+## Application to the repaired quotient
+
+The repaired construction supplies one invertible G_j for each of all 72 shared native operators and uses its inverse consistently at every negative occurrence. Its exact physical K_(G_j) are invertible monomials. Fresh literal free-word replay checks the complete emitted program in both orientations and every borrowed scalar-frame incidence. Hence the evaluation theorem applies to the repaired maps without retaining a new unknown all-fiber identity as an assumption.
+
+For f digit columns, choose K_j^(tensor f) and each canonical frame C_U^(tensor f), with the corresponding fixed coordinate ordering. The same physical-frame induction proves the endpoint with its actual tensorized address-operator blocks. Independent coefficient-block identities for the explicit main normalizers then give the stated type-A/type-B interface. The 2×2 FIELD matrix is never tensorized; only the ADDRESS operators assigned to its coefficients are lifted. Identity spectators add no Gaussian rank.
+
+This proves the all-fiber algebraic lift of the concrete repaired local word from an exact formal certificate and explicit Clifford/frame constructions. It does not establish a complete controller, recursive stock, finite compiler majorant, native precision/caller, or improved kappa. Those remain separate obligations. Credit the retained free-word row evaluator and its original authors, Cinna Davis's native/helper framework, Cao Yifan's shared-map interface, the repaired quotient construction, and all original notices. No Lean/kernel verification or general novelty priority for free group algebra is claimed.