Prove a complete four-width criterion for affine-quadratic native identities

docs/research/native-tensor-cutoff.md

@@ -0,0 +1,53 @@
+# Certify identities with actual affine and phase relations
+
+Use [native-tensor-cutoff](../../research/native-tensor-cutoff/README.md) when
+every payload operator is a binary affine permutation with a quadratic
+Gaussian phase. This is the fixed monomial recipe grammar in [PR #2250](https://github.com/CrocSwap/integer-mult-bounds/pull/2250).
+The checker respects actual inverse, finite-order, commutation and phase
+relations. It checks all dirty inputs and all returned roles.
+
+The proof gives a finite criterion for a fixed n-axis program: equality at
+widths **2n+4 through 2n+7** is equivalent to equality at every positive
+width. The tool checks canonical native coefficients and constructs a short
+basis witness on failure. It does not materialize those four tensor spaces.
+The number of stored terms can still grow exponentially.
+
+```sh
+python3 -I -B research/native-tensor-cutoff/native_identity.py \
+  /path/to/problem.json --output /tmp/native-certificate.json
+python3 -I -B research/native-tensor-cutoff/check_certificate.py \
+  /path/to/problem.json /tmp/native-certificate.json
+```
+
+Copy the input format from the package's source-conjugation example. Bind
+each generator to exact source recipe bytes. Use a second complete program
+as the reference. Keep a borrowed role's identity output when it must return
+its dirt. Save both input and certificate files with the source pins.
+
+The actual nine-axis example is a relation-dependent identity. Its deliberate
+constant-phase error passes even widths but fails the separate checker's
+width-25 witness. Another nine-axis example vanishes at every width below 9.
+Small-matrix tests alone therefore do not establish the all-width claim.
+
+Keep three boundaries explicit:
+
+- Scalars stay outside tensor powers. A recipe phase c contributes f*c at
+  width f. A scalar multiplying the complete register operation contributes
+  that same scalar, not its f-th power.
+- The four-width criterion requires a single fixed program and fixed scalar
+  coefficients. Arbitrary width-dependent choices need the proof's separate
+  phase-residue statement.
+- This grammar excludes Hadamard gates, rectangular operators, nonlinear
+  leaves and external aliases. Physical costs, precision, tape movement and
+  supplier admission need their own proofs.
+
+For independent operator symbols, use the
+[universal dirty-circuit guide](universal-dirty-circuit-guide.md). Its smaller
+grammar supports compact polynomial-size certificates. Named inverses alone
+have their own [inverse certificate package](../../research/inverse-dirty-certificates/README.md).
+Choose the grammar that the actual source satisfies. A failed independent
+operator check does not refute an identity which uses native relations.
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance. The package
+retains the source and classical mathematical attribution. This is an
+author-checked research interface, not a selected-frontier update.

docs/research/universal-dirty-circuit-guide.md

@@ -74,6 +74,12 @@ not tensorize sums or admit physical implementations. See the
 
 For free-group native programs, use the existing
 [free-word tensor audit](../../research/free-word-tensor-lift-audit/README.md).
+For fixed binary affine maps with quadratic Gaussian phases, the
+[native tensor-cutoff guide](native-tensor-cutoff.md) supplies a relation-aware
+endpoint checker and a four-width completeness theorem. It preserves actual
+global phases and dirty roles; nonmonomial Cliffords and physical admission
+remain outside that interface.
+
 For rectangular scratch and one-sided invariants, the existing
 [dirty-phase audit](../../research/universal-dirty-phase-identities/README.md)
 has a different contract. Nonlinear leaves, aliasing, rounding, native frames,

research/native-tensor-cutoff/MUTATIONS.json

@@ -0,0 +1,53 @@
+{
+  "status": "PASS_NATIVE_CUTOFF_MUTATIONS",
+  "cases": [
+    {
+      "mutation": "one-copy-phase-repair",
+      "optimized": false,
+      "exit_code": 1,
+      "rejection": "complete phase-character statistics"
+    },
+    {
+      "mutation": "one-copy-phase-repair",
+      "optimized": true,
+      "exit_code": 1,
+      "rejection": "complete phase-character statistics"
+    },
+    {
+      "mutation": "ignore-central-residue",
+      "optimized": false,
+      "exit_code": 1,
+      "rejection": "complete phase-character statistics"
+    },
+    {
+      "mutation": "ignore-central-residue",
+      "optimized": true,
+      "exit_code": 1,
+      "rejection": "complete phase-character statistics"
+    },
+    {
+      "mutation": "drop-native-constant",
+      "optimized": false,
+      "exit_code": 1,
+      "rejection": "separate composition formulas"
+    },
+    {
+      "mutation": "drop-native-constant",
+      "optimized": true,
+      "exit_code": 1,
+      "rejection": "separate composition formulas"
+    },
+    {
+      "mutation": "omit-last-dirty-return",
+      "optimized": false,
+      "exit_code": 1,
+      "rejection": "negative control was accepted"
+    },
+    {
+      "mutation": "omit-last-dirty-return",
+      "optimized": true,
+      "exit_code": 1,
+      "rejection": "negative control was accepted"
+    }
+  ]
+}