Construct complementary native suppliers with a strict local moment improvement (no new kappa)

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research/native-dual-suppliers/DUAL-PROVIDER.md

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+# Orthogonal quotient/restriction suppliers
+
+Prepared for Dugongue with substantial OpenAI Codex assistance. Apache-2.0.
+
+## A complementary pair of actual typed interfaces
+
+The kernel-153 quotient supplier has 56 ordinary ports A and 56 ports B:
+
+    A = diag(D,D) S_y(Q⁻¹),
+    B = diag(D,QD) S_y(Q⁻¹).
+
+Here D is a seven-axis Gaussian transform and Q is a separate central-axis Gaussian transform. We construct a second supplier with the SAME partition of the 112 ports into 56 ordinary A ports and 56 special C ports:
+
+    C = diag(QD,D) S_y(1).
+
+Every helper remains an arbitrary independent input. The second supplier has actual complete local profiles
+
+    P_C={1:2047,2:417,3:423,4:120,5:12,7:8},
+    H_C={3:26,4:2,6:6,7:48,8:326}.
+
+Its paid-minus-productive rank mass is 1680. Its owned center jobs are eight rank-seven jobs. The four rank-one extensions needed at native operators 37 through 40 are ordinary paid climbs in the emitted word, not uncharged warp ranks.
+
+This is a positive sufficient local construction. No higher multiplication κ follows without a complete mixed-controller invoice and compiler/native assembly.
+
+## 1. Port duality and complementary incidence
+
+For each original port i, let c_i be its central entrance vector and F_i its terminal main hyperplane. The exact pinned source data satisfy
+
+    F_i=c_i⊥,    c_i·c_i=1.
+
+This is checked on all 112 pairs, including the entire 512-point hyperplanes. The 112 vectors c_i are distinct and span the whole nine-dimensional space: they must not be treated as one common physical entrance line.
+
+Let v be a nonzero vector different from every c_i. Quotienting by span(v) gives the A/B distinction:
+
+- If v·c_i=0, the image of F_i has dimension 7 and the image central line lies outside it: type A.
+- If v·c_i=1, the image F_i fills the eight-dimensional quotient: type B.
+
+Now intersect every frame with H=v⊥ instead. Since v≠c_i, the two hyperplanes F_i and H are distinct, and A_i=F_i∩H has dimension 7. This gives the complementary A/C distinction:
+
+- If v·c_i=0, the central line survives in H and lies outside A_i: type A.
+- If v·c_i=1, the central entrance disappears, both logical starts are ground, and the physical diagonal blocks have ranks 8 and 7: type C.
+
+Thus the B ports of the quotient and the C ports of the orthogonal restriction are EXACTLY the same set. At v=153 this set has 56 elements. This is an elementary universal incidence argument, not a claim inferred from a numerical moment root.
+
+## 2. Emitting the orthogonal-restriction word
+
+Take the full pinned PR #4470 native word and replace each scalar contact, entrance, exit and center frame U by U∩H. Scalar order, rational coefficients, field identities and native operator names remain unchanged. Intersections preserve inclusions, so every retained scalar contact remains reachable by an ascending paid frame change. Recompute every ordinary paid climb and every center rank from the actual new frames. Every helper remains strictly productive: the new widths are precisely H_C above.
+
+A bare intersection need not preserve the dimensions of the two sides of a native warp. At H=153⊥ exactly four forward warp occurrences have a one-dimensional increase: native operators 37,38,39,40. For each, choose an explicit three-plane B′ inside its required four-dimensional output cap. Retarget the native warp to B′, then let ordinary resynthesis insert the missing rank-one climb before the next actual contact. That climb is counted in P_C.
+
+All 72 native G maps are reconstructed simultaneously on all occurrences. Their common eight-plane H is preserved. Each operator has one additional equal-dimensional subspace pair; twelve also have a line pair. Choose compatible adapted bases, extend to H, then to the ambient, and map corresponding basis vectors. The four retimed output three-planes are included in these constraints. Check every shared pair and every one of the 512 columns of each map for bijectivity.
+
+Every actual native warp now has exact rank zero. Both retained literal interpreters check all 632 arbitrary-input endpoints in both orientations. Their rows are formal elements of Q[FreeGroup72], not one-column numerical matrices: at termination there are 744 exact terms and zero nonempty operator words in each orientation. By the group-algebra evaluation homomorphism this proves the same logical identity for any consistently shared invertible native operators at any fiber width. No tensor-preserves-sums assertion is used.
+
+## 3. Explicit monomial normal form of C ports
+
+At a C port both starts are ground. Let A_i=F_i∩H, and write C_H,C_A for the canonical frame operators. Its forward physical matrix is
+
+          [ C_H     0 ]
+      M = [ C_A   C_A ].
+
+Choose any c′∈H∖A_i and an invertible G taking standard central axis e₀ to c′ and the standard seven-plane E to A_i. Let N=K_G, Q=C_e₀ and D=C_E. Define
+
+    M_X=M_Y=N,
+    O_X=QD N⁻¹ C_H⁻¹,
+    O_Y=D N⁻¹ C_A⁻¹.
+
+These four operators are monomial. The canonical-frame Lagrangian identity
+
+    L_U={(x,z):x∈U, x+z∈U⊥}
+
+and K_G L_U=L_GU show that both output operators preserve the computational Z Lagrangian. The exact Clifford lowering determines their monomial address maps, quadratic phases and absolute phases.
+
+Direct multiplication gives
+
+    O_X C_H M_X = QD,
+    O_Y C_A M_Y = D,
+    O_Y C_A M_X = D.
+
+Consequently the normalized forward matrix is exactly diag(QD,D)S_y(1). A ports use the already proved A-type normalizers. All 112 pairs were checked with 448 exact monomial normalizers and 336 exact block identities, including global phase. Inverse transpose gives the backward interface; its extra inverse Q is on the same physical X field. Individual operator-entry identities tensor to every digit-column width, while the scalar word's all-width identity follows independently from formal free-group-algebra evaluation.
+
+## 4. Why the two special types can cooperate
+
+Relative to the desired common logical A primitive, a B port can be consumed with equal input central-gauge parities and flips Y, leaving opposite output parities. A C port can be consumed with opposite input parities and flips X, leaving equal output parities. The two types are therefore complementary parity transitions.
+
+This does not by itself make a controller free. Every scalar ADD across unequal gauges can require a paid rank-one bridge; query shear cost depends on its source/target parity; scalar boundary programs and final productive central parity must be included. A B/C choice is a choice of a whole supplier invocation with its actual P/H invoice, not an independent free choice at each port. Pair permutations may choose which 56 ports are special, but the resulting supplier and all external normalizers must be charged as finite nonrecursive operations.
+
+## Evidence and attribution
+
+The actual restricted word payload SHA-256 is `cc7ce3aacca5bde9289e47b3563677b4d2a2946f7d3e0e0889ba9b09cec1c62c`. Its builder `admit_intersection153.py` reconstructs the 72 native maps, four retimed outputs, all paid climbs and both formal endpoint replays. `admit_main153.py` constructs the exact A/C port normalizers and lowers every distinct paid/center/helper pair in both orientations. Receipts separate completed checks from exploratory incidence screens.
+
+Credit Cinna Davis and the PR #4470 native/helper word; DaysSky for the quotient mechanism; Nick Petalas and Ryanson Jonathan for quotient/endpoint analysis; Cao Yifan for shared native-map interface work; Dugongue for transported frames, arbitrary-frame rank, the physically repaired quotient and the A/B normal form; zxv and Jacob Alan Sussman for retained native and compiler/matching foundations; and all original source notices. The new material here is orthogonal restriction as a complementary actual A/C supplier and its explicit shared-map/monomial implementation. No claim of independent human acceptance or worldwide priority is made.

research/native-dual-suppliers/FREE-ENDPOINT-GAUGES.md

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+# Free endpoint gauges give honest changed whole Gaussian requests
+
+Prepared for Dugongue with substantial OpenAI Codex assistance. Apache-2.0.
+
+Let a complete retained controller satisfy the exact operator identity
+
+    L M R = diag(T_1,...,T_k),
+    T_i=Q^(m_i) product_s D_s^(e_is),
+
+where L,R are the explicitly emitted scalar boundary programs, Q is the common central Gaussian, D_s are Gaussians on disjoint seven-axis fresh blocks, and the e_is are the retained 0/1 exponents. Each input field is arbitrary and independent. This is the complete diagonalized supplier, not one local typed module pretending to retain its old main ABI.
+
+Assign central gauges G_j=diag(Q^(a_ji)) at every field boundary in the complete chronological word R, then M, then L. Permit initial G_in and final G_out to be nontrivial. Physically emit each operation as
+
+    M'_j=G_j M_j G_(j-1)^-1.
+
+Every typed-local incidence, scalar/query correction and permutation must implement exactly this operator, with all even-power monomial wrappers retained. Every nonmonomial correction is a paid Gaussian call. No boundary program may be omitted from that accounting.
+
+The physical program is run directly on the raw input x. It does NOT first compute G_in^-1 x. That expression is only the logical representation used to prove the physical operations' action. Exact telescoping gives
+
+    M'_s ... M'_1 = G_out (L M R) G_in^-1
+                   = diag(Q^(a_out,i) T_i Q^(-a_in,i)).
+
+Therefore each independent raw input field undergoes the actual diagonal operator
+
+    T'_i=Q^(m_i+a_out,i-a_in,i) product_s D_s^(e_is).
+
+No input is initialized, merged or assumed invariant under a gauge. Main/helper cross-input mixing is absent because the FULL resulting matrix is diagonal. This is an honest changed productive request, rather than a claim that a nontrivial Gaussian is a free input adapter for an unchanged request.
+
+The complete productive rank is exactly
+
+    r_i=7 sum_s e_is + ((m_i+a_out,i-a_in,i) mod 2).
+
+The diagonal factors commute. Every even Q power is its known Pauli/translation monomial; inverse Gaussian orientations likewise differ by explicit monomial squares. Permuting the active coordinate blocks gives the canonical rank-r_i Gaussian with actual monomial normalizers. Hence the retained whole-job Gaussian compiler can use these exact new ranks directly. It needs no dense change-of-basis, new fast kernel or unproved adapter. Rank-zero channels, if any, must not receive positive productive credit; a subsequent compiler must satisfy its stated positive-rank/minimum-rank hypotheses. In the current p-family main ranks stay at least fourteen, so dropping the central rank does not create that edge case.
+
+Input and output parity bits must be paired by the same diagonal channel i AFTER both scalar boundary programs and all signed swaps. The productive central loss is XOR(a_in,i,a_out,i) only because each m_i is odd in the retained source controller; for general m_i the displayed formula must be used. The objective is rebuilt from those actual ranks and from every actual Gaussian call in the physical word. An optimizer's objective is not a substitute for exact matrix replay.
+
+All local helper fields in every provider invocation remain separate independent channels with their separately declared endpoint ranks. The changed global controller input gauges do not identify them or erase their contribution. Fixed local physical suppliers are consumed only through their proved typed incidences; the global mathematical endpoint does not license inserting unpriced Q operations at those local ports.
+
+This proves a sufficient native recursive-interface theorem for free initial AND final gauges. It does not establish that any particular schedule improves a moment, and it does not establish an exponent without the complete mirror/majorant, finite compiler, precision and native outer assembly checks. Its purpose is to remove an unnecessary global input-zero restriction while retaining the exact changed operator request and fully paid physical circuit.
+
+Credit the retained exact Laurent controller factorization and whole-job compiler; the actual mixed A/B/C providers; Dugongue's native row-gauge route; Cinna Davis, Cao Yifan, zxv, Jacob Alan Sussman and the inherited Gaussian/Clifford contributors. The telescoping identity is elementary algebra; the application here supplies the precise honest productive ABI.