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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. diff --git a/research/native-quotient-interface/LICENSE b/research/native-quotient-interface/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/native-quotient-interface/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. 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We also recommend that a + file or class name and description of purpose be included on the + same "printed page" as the copyright notice for easier + identification within third-party archives. + + Copyright [yyyy] [name of copyright owner] + + Licensed under the Apache License, Version 2.0 (the "License"); + you may not use this file except in compliance with the License. + You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, software + distributed under the License is distributed on an "AS IS" BASIS, + WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + See the License for the specific language governing permissions and + limitations under the License. diff --git a/research/native-quotient-interface/OVERLAP.json b/research/native-quotient-interface/OVERLAP.json new file mode 100644 index 0000000000..973d568f3a --- /dev/null +++ b/research/native-quotient-interface/OVERLAP.json @@ -0,0 +1,108 @@ +{ + "checkpoint_utc": "2026-10-22T14:05:00Z", + "canonical_main": "fbc9ea0249746acca4f504d4fda2b84e0e36cf0d", + "coverage": "Latest 30 updated PR metadata at first refresh, latest 15 updated at second; detailed relevant sources #4596/#4604/#4610/#4629/#4633/#4635/#4636. No claim to have reviewed every historical PR or proved novelty.", + "pulls": [ + { + "number": 4638, + "head": "385a4436a47ea8e80cc1503948fe7a36e1b1c5af", + "state": "open", + "title": "Sharpen chronological width to d+1 and extend batching to adaptive heights (no new kappa)" + }, + { + "number": 4637, + "head": "60d0b09f13a9f9ebdd93d98d83d16ce6907d70b6", + "state": "open", + "title": "Join the CRT scanner-owned mask to conditional sparse repair (no new kappa)" + }, + { + "number": 4610, + "head": "1eb8da6cfb35512c0cc2533413e5ca42f48e13c6", + "state": "open", + "title": "Conditional κ = 0.00302176 (+0.00000024): joint helper interfaces and Lean certificates" + }, + { + "number": 4643, + "head": "0c084991a7d3bcd18ef4426a5940516c0171ae74", + "state": "open", + "title": "Independently check #4628's m ≥ 4 projection bound: 2.99172, 2.97290, 2.9447 reproduce; tail proved for all k; two uncertified block-4 inputs isolated (no new κ)" + }, + { + "number": 4642, + "head": "d8c59e36a8efb301f0298871218cfe00f4e043f7", + "state": "open", + "title": "Independently check #4629's re-padding ledger and price: κ 0.015637149091 reproduces; re-pad is +1×187/+5×26, block-7 surplus −256, ninth axis reached only by padding (no new κ)" + }, + { + "number": 4641, + "head": "99007ce1bb6a3d2889ddeb763bbc7546724e1f52", + "state": "open", + "title": "Independently check #4635's native κ step: the +1.6099e-6 is #4610's paid profile, not the Word RAM transfer (no new κ)" + }, + { + "number": 4640, + "head": "2d64f3521bcfad893abf0aa8022914c7429abfbc", + "state": "open", + "title": "Independently check #4626's helper621 chart move: ΔP=H1+2H2−H3−H4, rank 9→7 and crossing reproduce; +2.9e-8 of the gain is the compiler-gap retune (no new κ)" + }, + { + "number": 4639, + "head": "ade9e6ab6ff3cb2c3328ef9601976eb599eb654c", + "state": "open", + "title": "Independently check #4623's monotone extension, decoder rank and idle-free controller; sharpen ceiling to C≥.005687/.006512 (no new κ)" + }, + { + "number": 4636, + "head": "b0a325a3e312b7c5ebceb420de4b8cb13181f43e", + "state": "open", + "title": "Audit #4633: full replay, native contract failures and verifier false positives" + }, + { + "number": 4596, + "head": "477232ced13ec75fa40e0451ad1377db54c9d385", + "state": "closed", + "title": "Conditional κ = 0.0034 (BIT-limited, +12.58%): complex supplier to 0.011417749 via the ambient quotient by span{511}" + }, + { + "number": 4629, + "head": "2cf7d081118f6f702aa317816d231b41ce073579", + "state": "closed", + "title": "No new kappa: complex supplier to 0.015637149 via the best ambient quotient (kernel 153 of 32 replay-valid directions)" + }, + { + "number": 4633, + "head": "b390d64abfcdba1a3816b86fc763cbc95d43c12c", + "state": "closed", + "title": "Conditional κ = 0.012401086416 (BIT-limited, +310.6%): BIT supplier to 0.012401086416 via the ambient quotient (kernel 150)" + }, + { + "number": 4634, + "head": "9f053f7051e5fc3327d4337bb4bd8ac52cf5fcd0", + "state": "open", + "title": "Prove optimal binary Schur sharpness dimension n = 3d for d >= 2" + }, + { + "number": 4635, + "head": "898b0c8e2ca23897c312230988515c2553154f30", + "state": "open", + "title": "Conditional Word RAM κ = 0.0030217754 (+0.0000016099 over #4630)" + }, + { + "number": 4631, + "head": "48aef2349dc761f6b3b6bf3e59c3ee8080dbbe7a", + "state": "open", + "title": "Fix sparse baseline preflight for selected explicit inventories" + } + ], + "credit_and_overlap": { + "4636": "Independent negative audit of quotient admission, native contract and verifier controls. Our package constructs replacement shared maps and typed positive interfaces; it does not repeat a full negative audit.", + "4604": "Predecessor main-endpoint objection remains valid; our output is a changed typed interface, not the old canonical main interface.", + "4629": "DaysSky kernel-153 supplier frames and profile retained, with native maps repaired here.", + "4610": "Cao Yifan joint helper/native-map search and supplier lineage.", + "4635": "Dugongue conditional native Word RAM consequence; no superior exponent established here.", + "4641": "lm1909 independent arithmetic audit confirms that #4635 numerical change comes from #4610 paid profile; native transfer is lossless and adds no numerical gain. Native/provider admission remains outside that audit.", + "4639": "Independent monotone/decoder/idle-free-controller audit of #4623, no new kappa; distinct from this repaired typed local supplier." + }, + "final_canonical_main": "b58ad9ef4d4e861b0eaa5dba41742bdc072dca51", + "main_delta": "Only docs/ci-verification.md, scripts/check_selected_ci_inputs.py and its tests changed; AGENTS.md, autoresearch.md, CONTRIBUTING.md, priority/card and selected record unchanged. Frozen mathematical source inputs retained." +} diff --git a/research/native-quotient-interface/PHASE-CUT.md b/research/native-quotient-interface/PHASE-CUT.md new file mode 100644 index 0000000000..cebae17d8b --- /dev/null +++ b/research/native-quotient-interface/PHASE-CUT.md @@ -0,0 +1,80 @@ +# Rank-one phase-flow optimization by a minimum cut + +Prepared for Dugongue with substantial OpenAI Codex assistance. Apache-2.0. + +This is a constructive extension of PHASE-FORM-GRAMMAR.md. It preserves the old subspaces, scalar chronological word, fixed logical native maps and exact main ABI. It permits and pays additional Gaussian children when a formerly monomial physical warp becomes nonmonomial. + +## Binary gauges + +Fix a nonzero ambient functional l on F2^n. At a contact node v with subspace U_v choose s_v in {0,1} and form + + b_v(x,u)=s_v (l·x)(l·u), x,u in U_v. + +An actual gauge is F_v=C_(U_v) D_(s_v ll^T), with the same inverse-image convention as the phase-form grammar. A scalar contact has one common gauge for all roles it touches. Fix all main endpoints to s=0. Initially also fix owned-center/non-G adapter boundaries to s=0, so those frozen instructions remain unchanged. Fix one endpoint of every helper to zero; allow its opposite endpoint to vary. Both choices, all helper entrances fixed or all helper exits fixed, are valid separate search problems. + +## Every physical edge has an exact rank invoice + +For an original paid edge U→V with I=U intersection V, its old rank is r=dimU+dimV-2dimI. The two rank-one forms restricted to I agree when both switches agree, so the added rank is one exactly when switches differ and l is nonzero on I. Its complete child cost at exponent t is (r+added_rank)^t, not an unpriced phase correction. + +For a fixed logical native warp K_G, the retained subspaces satisfy GU=V. The transported source form is the rank-one functional G^-T l restricted to V; the target form is l restricted to V. Write these restricted functionals as a and b. The physical operator is + + F_target K_G F_source^-1. + +Its Gaussian rank is exactly rank(s_source a a^T + s_target b b^T), hence one of 0,1,2. If a and b are equal nonzero, the rank table for switches (00,10,01,11) is (0,1,1,0). If they are distinct nonzero, the table is (0,1,1,2). If one vanishes, the table is (0,0,1,1) or (0,1,0,1); if both vanish every entry is zero. These cases are decided by exact binary linear algebra. + +The physical warp is emitted as its actual exact Clifford circuit. The retained Clifford normal form factors it into two explicit monomial adapters and one rank-one or rank-two Gaussian call, or only a monomial when the rank is zero. Thus a switch cannot make a formerly free native warp expensive without that expense appearing in the paid profile. All adapters, owned work and absolute phases remain covered by their explicit native construction and replay, rather than treated as cost-free semantic labels. + +## The binary edge costs are submodular + +Let f(r)=r^t for fixed 0= 0, + A=c10-c00-C, B=c01-c00-C. + +Thus C is a nonnegative cut-edge capacity, and A,B are unary contributions. Negative unaries are represented by a nonnegative terminal capacity plus a constant. There is no sign ambiguity or appeal to a floating optimization status in this reduction. + +## Productive rewards remain unary + +Suppose a helper has nested source/end subspaces U⊂V and original productive rank g=dimV-dimU. One endpoint switch is fixed zero. The relative helper's extra rank is s times rank((l|U)⊗²), hence zero or one. Its productive-moment reward is the unary quantity + + s * ((g+1)^t-g^t) + +when l is nonzero on U, and zero otherwise. The reward must be multiplied by every actual controller occurrence of that independent helper. Every helper remains a distinct arbitrary input and its true new Gaussian job is normalized explicitly. + +Consequently total added paid cost minus total added productive reward is a binary submodular energy and is reducible to an s–t minimum cut. The all-zero assignment is feasible with energy zero. Any negative-energy cut gives a strictly improved complete moment at exponent t. This conclusion requires the actual graph multiplicities, owned-center charges and controller invoice; it is not a local gain that can be added across overlapping windows. + +## Exact arithmetic and discovery + +At rational t the weights are algebraic powers, not generally rational numbers. A portable discovery implementation may solve min-cut problems for rational enclosures/approximations of these coefficients, then evaluate every selected cut with the retained outward-rational power engine. Only a strictly certified negative true energy is a mathematical improvement. Approximate optimality alone is not an exact ceiling. Because the graph has finitely many cuts, any strictly profitable true cut is eventually detected by sufficiently fine rational approximations; a bounded search that finds none gives no universal obstruction. + +All n-bit nonzero functionals l can be enumerated at the fixed local dimension n=9. This is 511 distinct rank-one phase families, not a claim to search arbitrary symmetric phases, simultaneous moving subspaces, or all native algorithms. + +## All-input semantic assembly and finite compiler + +The new physical edge for every old logical operation M is F_next M F_prev^-1. Adjacent gauges cancel exactly. At scalar contacts the same F acts on each operand, so the original scalar opcode and coefficient are retained. Main gauges are frozen to their original values. In the mode with all helper entrances frozen, the initial gauge is unchanged on every field, so every main endpoint is immediately unchanged on every arbitrary input. In the mode with varying helper entrances, additionally use the retained independent-helper direct-sum contract: the old global endpoint has zero blocks from helper inputs to main outputs. The original all-column certificate must check those zero blocks explicitly. Right-multiplying independent helper input blocks then cannot affect a main output. Thus both declared modes preserve the actual main ABI; freezing main gauges alone would not suffice for an arbitrary globally coupled operator. Helper endpoints change to the actual relative Gaussian/Clifford operators indicated by their gauges; they are not validated against an automatically redefined main contract. + +The original chronological word order is unchanged. Each new Gaussian child finishes before the suspended parent resumes. The exact normal form provides whole-job monomial adapters, so the retained direct mixed-target compiler can treat the additional rank-one/rank-two paid jobs and increased helper ranks as actual canonical Gaussian jobs. Owned erasure is not inferred from the moment: frozen owned instructions and their original all-column proof remain explicit. Actual all-column and absolute-phase replay must validate the emitted witness and helper normalizers. + +For every digit-column lift width f, tensor the actual one-column Clifford factorization independently over all f columns. Operator identities tensorize, active paid ranks multiply by f, and the helper rank gain becomes f. Identity spectators add no rank. This supplies an all-fiber algebraic extension of an admitted candidate; it is not a new unknown adapter hypothesis. + +After a negative cut is emitted and verified, the entire changed target/paid histograms must pass the critical-root mirror, funded majorant, finite uniform-word absorption, precision and native outer assembly checks. No new kappa follows from a min-cut objective alone. This theorem gives a sufficient exact candidate-generation and lowering mechanism; it does not assert a useful cut has been observed in the current word. + +Credit the classical submodular graph-cut reduction and symplectic/Clifford normal form; Cinna Davis's supplier/controller interfaces; Cao Yifan's actual round-three word; Dugongue's native phase-flow route and Word RAM assembly; zxv, Jacob Alan Sussman and all retained compiler/normalizer sources. The new role here is the source-convention-aware phase-flow energy that permits paid native warp changes while preserving the actual main ABI. + +## Extension: an arbitrary fixed symmetric ambient form + +The minimum-cut reduction is not restricted to rank-one Q=ll^T. Fix ANY symmetric ambient Q and choose b_v=s_v Q restricted to U_v. On a paid edge I=U intersection V, the extra rank is zero for equal switches and rank(Q|I) for unequal switches. Its incremental paid cost is a nonnegative Potts cut weight f(r+rank(Q|I))-f(r). + +For a formerly monomial logical warp G, pull both forms back to its source U. Put A=Q|U, B=(G^T Q G)|U, a=rankA, b=rankB and c=rank(A+B). The four paid costs are (0,f(a),f(b),f(c)). Rank subadditivity gives c<=a+b, and the increasing subadditive function f(k)=k^t gives f(c)<=f(a+b)<=f(a)+f(b). Thus this table is submodular too. A helper with one endpoint fixed has unary reward f(g+rank(Q|I))-f(g). All earlier source/normalizer/owned/main-ABI obligations remain unchanged. The actual diagonal Clifford D_Q is explicit and its phases must be carried literally. + +Consequently any chosen symmetric Q, including alternating rank-two and high-rank forms, supplies another exact min-cut discovery problem. Searching finitely many Q is bounded evidence; it does not exclude a useful form among the other 2^(n(n+1)/2) possibilities. diff --git a/research/native-quotient-interface/PHASE-FORM-GRAMMAR.md b/research/native-quotient-interface/PHASE-FORM-GRAMMAR.md new file mode 100644 index 0000000000..3e91b0b7e2 --- /dev/null +++ b/research/native-quotient-interface/PHASE-FORM-GRAMMAR.md @@ -0,0 +1,79 @@ +# Symmetric phase forms enlarge the native contact grammar + +Prepared for Dugongue with substantial OpenAI Codex assistance. Apache-2.0. + +## Exact model and conventions + +Let V=F2^n with the ordinary nondegenerate dot product. Write a Pauli exponent as (x,z), with symplectic pairing x·z'+z·x', and put y=x+z. Let L0={(0,z)}. The retained canonical frame C_U has involutive symplectic action and sends L0 to + + L_U={(x,z): x in U, x+z in U^perp}. + +A frame gauge F is assigned its INVERSE image chart L(F)=F^-1(L0). This orientation matters: a physical paid edge is F_next F_prev^-1. Its Fourier rank is n-dim(L(F_prev) intersection L(F_next)). For canonical C_U the inverse and forward symplectic images coincide; this coincidence must not be assumed for an arbitrary new gauge. + +For a subspace U and a symmetric bilinear form b:U×U→F2 define + + L(U,b)={(x,z): x in U, (x+z)·u=b(x,u) for every u in U}. + +The old canonical-frame grammar is precisely b=0. + +## Theorem 1: explicit gauges for every such chart + +L(U,b) is an n-dimensional Lagrangian. Every Lagrangian whose x-projection is U has this form for a unique symmetric b. + +Proof. For each x in U, the allowed y=x+z form an affine translate of U^perp, so total dimension is dimU+(n-dimU)=n. For two members (x,y),(u,t), the symplectic pairing is x·t+y·u=b(u,x)+b(x,u)=0. Conversely the vertical kernel of a Lagrangian with x-projection U is exactly U^perp; hence y·u is independent of the chosen y above x. It defines b, and isotropy proves symmetry. + +The construction is an exact Clifford circuit. Choose basis columns B of U, extend them to an invertible n×n matrix H, write S for the symmetric matrix of b on B, and put + + Q = H^-T blockdiag(S,0) H^-1. + +Then Q is symmetric and x^T Q u=b(x,u) on U. The retained diagonal Clifford D_Q acts symplectically as (x,z)→(x,z+Qx), equivalently (x,y)→(x,y+Qx). Choose + + F(U,b)=C_U D_Q. + +Because D_Q has involutive symplectic action and C_U^-1(L0)=L_U, its inverse-image chart is D_Q^-1 L_U=L(U,b). In the retained Op.then convention this circuit is `diag(Q).then(canonical_frame(U))`: D_Q acts first. The diagonal's Z4 linear coefficients have parity equal to diag(Q), as in the retained exact implementation. This fixes an actual circuit and absolute phase, not just a projective tableau. Different extensions of b may give different gauges with the same chart; their relative operator is monomial and is not silently identified as an exact circuit. + +## Theorem 2: complete paid rank formula + +For charts (U,b),(W,c), put I=U intersection W and let D be the symmetric form (b+c) restricted to I. The exact Fourier rank of F(W,c) F(U,b)^-1 is + + r = dimU+dimW-2dimI + rank(D). + +Proof. A vector in the intersection of the two Lagrangians has x in I. Its prescribed y-functional on U+W exists precisely when b(x,v)=c(x,v) for every v in I, that is, x lies in the radical of D. This radical has dimension dimI-rank(D). For each such x, allowed y form an affine space of dimension n-dim(U+W). Thus the intersection dimension is n-dimU-dimW+2dimI-rank(D). The B-block rank equals n minus this dimension, by applying F(W,c) to the inverse-image intersection. This proves the formula. + +In particular, every old paid rank is preserved exactly if the two forms agree on I. These are linear equations over F2, not nonlinear rank constraints. Once a candidate is emitted, the retained exact Clifford normal form constructs and checks its two monomial adapters and rank-r Gaussian child, with absolute phase. Every adapter has the native volume-linear implementation already charged by the source-bound provider theorem. This is a sufficient physical realization of the proposed rank, not free nonmonomial motion. + +## Theorem 3: native warp constraints are linear + +For the retained monomial K_G, Q_G=I+G^-T G^-1 gives + + (x,y)→(Gx,G^-T y). + +Therefore K_G sends L(U,b) to L(GU,b'), where + + b'(Gx,Gu)=b(x,u). + +The physical fixed-logical-warp edge F_next K_G F_prev^-1 is monomial exactly when K_G L_prev=L_next. With the retained subspaces already satisfying GU=W, this is exactly the displayed bilinear transport equality. It is linear in the local form coefficients. Its exact monomial and absolute phase are emitted from the full retained Clifford circuit, rather than inferred from the chart alone. + +## A sufficient positive profile mechanism + +Keep the actual scalar chronological word, original subspaces and logical warp maps fixed. Assign one form to each shared scalar-contact gauge and to all role endpoints. Impose: + +1. zero forms at all frozen main input/output endpoints; +2. equality of forms on every paid-edge intersection; +3. the bilinear transport equations for every fixed logical native warp; +4. one common actual gauge at each scalar contact; +5. zero forms at frozen owned-center and non-G adapter boundaries, unless their exact symplectic constraints are separately implemented. + +Every equation is homogeneous over F2. Solve it exactly. For a helper entrance (U,b) and exit (W,c), its productive Gaussian rank becomes its old rank plus rank((b+c)|I). It remains an arbitrary independent input; no initialized storage or redundant productive credit has been introduced. If ANY helper difference is nonzero on I, its productive rank strictly increases while every paid rank is unchanged. All other helper ranks weakly increase. Hence the complete productive moment increases for every positive exponent, with the entire paid invoice unchanged. The controller's multiplicity must be retained when rebuilding its whole moment. + +This is a genuine sufficient route to a changed supplier invoice. A numerical exponent additionally requires all-column endpoint reconstruction, absolute-phase lowering, actual whole-job adapters, exact critical-root/majorant and finite-compiler admission. The homogeneous equations alone are not a new multiplication theorem, and existence of a useful nonzero solution is not assumed. + +The all-column semantic proof is ordinary gauge telescoping: replacing each logical operation M_j by F_j M_j F_(j-1)^-1 gives total F_final(M_s…M_1)F_initial^-1. Main gauges are frozen, scalar contacts use a common gauge, and helper relative endpoints are updated honestly. If helper entrance gauges vary, the retained independent-helper direct-sum endpoint contract is also used: helper input blocks have zero transfer to main outputs. This must be checked in the old all-column certificate. Under that explicit source fact, the old main ABI is preserved exactly; frozen main endpoint gauges alone do not imply it for an arbitrary globally coupled operator. No quotiented or missing main chart is substituted. + +## Minimal exact coordinates for the solver + +If U has basis B=(u1,…,ud), use d(d+1)/2 variables S_ij with i<=j. Express x=B a and u=B c. The coefficient of S_ii in b(x,u) is ai*ci; that of S_ij for i