Conditional Word RAM κ = 0.0030217754 (+0.0000016099 over #4630)

research/native-provider-lift/.gitattributes

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+* -text

research/native-provider-lift/ADDITIONAL-SOURCE-PINS.json

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+[
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+    "source": "PR4483 head fb0c208864e6b24e3473318a058aa5b2dbd5764b:research/direct-mixed-flow/README.md"
+  },
+  {
+    "local_file": "sources/PR4630-4535-BRIDGE-PINS.json",
+    "sha256": "82b86fbc8a09bb7c72044bdfda1f7b45d694ef47f0bcbb4c4f5b38fc4134f248",
+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699; retained source maps supply earlier proof commits"
+  },
+  {
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+    "sha256": "40d160b96402054c7cac05d21e383eda449445eceba2520a2a6a90c17b117c55",
+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699; retained source maps supply earlier proof commits"
+  },
+  {
+    "local_file": "sources/PR4630-MODERN-SOURCE-MAP.json",
+    "sha256": "cc48b51a9b2ae46cef437e0a56c4e68b613f8c3bd8b03c3386bc6cf4745c6c57",
+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699; retained source maps supply earlier proof commits"
+  },
+  {
+    "local_file": "sources/PR4630-NATIVE-ASSEMBLY.md",
+    "sha256": "a1f200292ea13d72bd64cb91a4daa28261bc220a80fe9c94983f3b9a1dffaa42",
+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699; retained source maps supply earlier proof commits"
+  },
+  {
+    "local_file": "sources/PR4630-NATIVE-PROVIDER-4535.md",
+    "sha256": "242af59962fe466dc0bfe5826bd33da1b9b7d2430a2f8dfceb71e4282ecc410c",
+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699; retained source maps supply earlier proof commits"
+  },
+  {
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+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699; retained source maps supply earlier proof commits"
+  },
+  {
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+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699; retained source maps supply earlier proof commits"
+  },
+  {
+    "local_file": "root_seed.py",
+    "sha256": "9b323f84ade7decfa6152dd4385e90ec43b56087927ac23ad4ef73ae4d5d3042",
+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699:research/native-kappa-30201655/root_seed.py"
+  },
+  {
+    "local_file": "native_convolution.py",
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+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699:research/native-kappa-30201655/native_convolution.py"
+  },
+  {
+    "local_file": "record_product.py",
+    "sha256": "6967feb00fbcb13640504f26c5e20bcea249e1dc89c41713aa8e8f161def873e",
+    "source": "PR4630 head f4b974db1f651250a252721cf127d0491a802699:research/native-kappa-30201655/record_product.py"
+  },
+  {
+    "local_file": "critical_engine.py",
+    "sha256": "e656ef12f73a8dd3bdb256eca95a712de1269facb18a652fc4068ba718619b4d",
+    "source": "PR4535 a6d7237f697185e857090e174c23c1cb22c6563c:research/adaptive-gate-existence/root_application.py"
+  },
+  {
+    "local_file": "majorant-benchmarks.json",
+    "sha256": "513efe2fe74af82891c1918e462ba8338b1bbd06403cbfcec284bcf3b4f8f850",
+    "source": "Canonical main f626db5c3a3d781d8f7695038776c621651f75e1:research/rank-convergence/BENCHMARKS.json"
+  },
+  {
+    "local_file": "CONTROLLER.json",
+    "sha256": "1f5d6aa267e4e66048818714b5d3b5eae17a58abe91d15deeaa34574f7391a5f",
+    "source": "Canonical main fbc9ea0249746acca4f504d4fda2b84e0e36cf0d:research/joint-helper-lattice/CONTROLLER.json"
+  },
+  {
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+    "source": "PR4610 1eb8da6cfb35512c0cc2533413e5ca42f48e13c6:research/kappa-round3-warp-20261022/ARITHMETIC.json"
+  },
+  {
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+    "source": "PR4610 1eb8da6cfb35512c0cc2533413e5ca42f48e13c6:research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz"
+  }
+]

research/native-provider-lift/ARBITRARY-FRAME-BRIDGES.md

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+# Exact arbitrary-frame bridges
+
+Prepared for Dugongue with substantial OpenAI Codex assistance. Apache-2.0.
+
+Let U,V be arbitrary subspaces of F₂ⁿ, with the retained source's canonical frame operators C_U and C_V. They need not be nested. Then the operator C_U.adjoint().then(C_V) has exact Fourier rank
+
+    r(U,V)=dim U+dim V−2 dim(U∩V).
+
+It can therefore be implemented by one rank-r recursive Gaussian transform between two explicit monomial adapters, with all absolute phases retained. When r=0, only a monomial adapter is required. The finite exact Clifford lowering constructs and checks these adapters; this is a sufficient realization, not an assumption that arbitrary frame changes are free.
+
+## Proof
+
+Write Pauli exponents as (x,z)∈F₂ⁿ×F₂ⁿ and let L₀={(0,z)} denote the Z Lagrangian. We claim
+
+    C_U(L₀)=L_U={(x,z):x∈U and x+z∈U⊥}.
+
+For the coordinate r-plane E, this follows directly from each retained C gate sending a Z basis generator to the corresponding XZ generator on selected axes and fixing unselected Z generators. For a general basis of U, the canonical frame uses an invertible matrix G with GE=U and the monomial K=D_Q P_G, where Q=I+G⁻ᵀG⁻¹. Its symplectic action is
+
+    (x,z) ↦ (Gx,G⁻ᵀz+QGx).
+
+Consequently the new x coordinate is Gx and the sum of new x and z coordinates is G⁻ᵀ(x+z). This sends L_E precisely to L_U. The canonical-frame conjugation and the fact that K preserves L₀ prove the claim. The special zero and full-frame definitions satisfy the same formula.
+
+Now L_U∩L_V is parametrized independently by x∈U∩V and y=x+z∈(U+V)⊥. Its dimension is
+
+    dim(U∩V)+n−dim(U+V)
+    =n−dim U−dim V+2 dim(U∩V).
+
+The Fourier rank of a symplectic operator is the rank of its B block, equivalently n minus the intersection dimension of its image of L₀ with L₀. Each canonical frame is a conjugate of commuting selected-axis C gates, whose symplectic actions are involutions; hence C_U and C_U⁻¹ have the same symplectic action. Applying an invertible canonical-frame action transports the intersection calculation above to the relative operator, in either composition convention. Therefore its rank is exactly r(U,V).
+
+The source's exact Clifford normal form factors any such relative operator into a right monomial, the r-axis Gaussian transform, and a left monomial. Its tableau and exact zero-state vector determine and check the absolute phase. Tensoring with identity spectator coordinates preserves the factorization and its active rank r. Taking f independent digit-column tensor copies gives active rank rf, as required by the retained all-fiber interface.
+
+## Finite check
+
+A separate checker evaluated 1,024 deterministic random pairs among the actual round-three frame registry. The Fourier rank read directly from each composed exact Clifford tableau matched the formula above in every case. The sample result is `RANK-FORMULA-SAMPLES.json`; it corroborates the proof and does not replace it.
+
+This primitive permits nonnested contact proposals in the native model, provided every actual bridge and both its adapters are charged and its endpoint is checked. It does not by itself establish any improved paid invoice or κ. Productive helper credit must use the same actual relative-operator rank; a helper's input remains an arbitrary independent field.
+
+## Attribution
+
+The canonical-frame and exact Clifford implementations are retained repository sources, including Cinna Davis's native/helper interface, the transported warp-frame work, zxv's native framework, and all original compiler notices. This note gives an explicit arbitrary-frame rank proof and its sufficient native realization. No Lean verification or novelty priority for the general Clifford normal form is claimed.

research/native-provider-lift/ARITHMETIC-INDEPENDENT-REVIEW.md

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+# Independent review of the new native target
+
+The source reviewed is Cao Yifan's PR #4610 at head 1eb8da6cfb35512c0cc2533413e5ca42f48e13c6, including round-three paid profile
+
+    {1:2201,2:412,3:510,4:127,5:40,8:8}
+
+and unchanged productive census {4:28,7:7,8:47,9:326}. This is a genuinely changed paid invoice relative to the old #4470/#4535 supplier. Its acceptance still requires the exact endpoint/phase/whole-job conditions reviewed separately; a histogram alone cannot establish them.
+
+I read critical_provider_check.py and its frozen majorant engine. The exact interval root certification and the majorant checker both pass on the complete 64,404-paid, 10,928-productive profile with rank surplus 752. The retained witness a=1,b=1,height_base=1024,ell0=650,s=1/20,lambda=1/10,p=2/5,c=133 and upper_exponent=.997067 is applied to the changed profile, not to a stale old root. The bracket is theta in (.9969782241,.9969782242).
+
+For q_C=.0030217755 and N=2^(10^12), the gap 1-theta_hi-q_C is exactly 3/10^10. The source majorant constant obeys log A<20 from (1-theta)log n0 and n0=145*1024^650. The logarithmic overhead is below16 because 133+log N<2^40 and log2<1. Thus total logarithmic overhead is below36. Since log2>2/3, absorption has logarithmic margin greater than
+
+    (3/10^10)*10^12*(2/3)-36 = 164.
+
+No enormous integer N is materialized. The derivative check gap > (2/5)/(133+(2/3)*10^12) ensures continuation for larger ranks. The finite compiler theorem still requires its credited chronological all-column construction and whole-job normalizers; this review does not manufacture them from the root.
+
+I independently recomputed the crossing using a separate 70-digit Decimal exponential/logarithm sum and 180 bisections (provider_review.py), obtaining a saving near .0030217758 in the same bracket. This is corroboration, not the exact interval certificate. The source engine's wrong-root and unfunded-mirror controls reject as intended.
+
+Finally, native_lift.choose('.0030217755','.0030217754') gives positive values for every interface guard and every nine-row excess. NEW-TARGET-PARAMETERS.json records the exact rational choices. In particular q=60435509/20000000000 and delta=1/80241742036, so the full-coordinate group-key exponent is 1-2delta<1, cap xi<epsilon, chirp guard and global gamma guard are strict. The minimum paid-row excess is
+
+    80181306527/1604834840720000000000 > 0.
+
+This establishes the numerical assembly of conditional native κ=.0030217754 from the admitted stronger source provider and the retained analytic caller. It does not assert unconditional mathematical acceptance, a practical algorithm implementation, a new TM frontier or a further paid-profile improvement over the current #4610 round-three head. Prepared for Dugongue with substantial OpenAI Codex assistance; Apache-2.0.