EXTRACTED TEXT: equations and tables can be transcribed incorrectly. Consult the unchanged PDF and exact TeX. PAGE 1 Guarded Boolean Operators: Exact Dispatch and a Polynomial Guard-Degree Tradeoff Brian Theory Abstract A finite family of continuous scores determines a Boolean operator wherever all scores are nonzero. We study exact identification of that operator under an explicit guard language, using the real polynomialF (x,y ) = x +y +xy on signed positive-magnitude representatives as a complete case study. Its parameter domain has exactly four operator chambers, labelled by AND, the two projections, and XNOR. In the original magnitude coordinates, letmd be the minimum number of binary polynomial-sign tests of total degree at mostd needed for exact dispatch on the zero-free domain. We provem1 =∞, m2 =m3 = 3, andmd = 2 for d≥ 4; moreover, both polynomials in any two-test solution must have degree at least four. Reciprocal coordinates linearize the chamber boundaries, making the coordinate dependence explicit. Three truth-table coordinate probes remain necessary in the worst case, including for adaptive identification. We place these calculations in a zero-aware certification contract, distinguish fixed test selection from per-instance and adaptive costs, and prove directly that dispatch is unchanged by faithful correspondence-matrix and fixed-frame logical-matrix representations. The general sign quotient and minimum-test-set formulation are established machinery; the contribution is a self-contained worked analysis of how coordinates, permitted observations, and zero handling change exact dispatch. 1 Introduction A continuous model can induce different Boolean operators as its parameters vary. For example, evaluate the real scorex +y at (a,b ), (a,−b), (−a,b ), (−a,−b) with a,b> 0, and declare positive output true and negative output false. Ifa>b , the resulting truth table is the first projection; if b>a , it is the second. Ata =b, two scores vanish and the strict binary interpretation is undefined. Thus even this elementary model requires both a parameter partition and a zero policy. The broad passage from continuous parameters to discrete logical data is not new. DSGRN constructs combinatorial parameter descriptions for switching regulatory networks and attaches dynamical information to them [12, 13, 15]. Logical bifurcation diagrams relate parameter variation in piecewise differential models to logical valuations and attractors [1]. Joint realizability and parameter adjacency of monotone Boolean functions are treated in [11], and recent work embeds monotone Boolean models into multilevel and DSGRN models [14]. These structured dynamical theories have substantially more application-specific content than a generic score-sign partition. Likewise, strict polynomial sign conditions and their connected components are standard real algebraic geometry [4, 5, 16]; collecting sign semantics belongs to abstract interpretation [9]; and selecting a minimum distinguishing test family is a classical optimization problem [17, 6]. Predicate minimization in verification is related but can optimize preservation of a property rather than exact recovery of every operator label [8]. We use these connections to specify the problem, not 1 PAGE 2 to claim a new abstraction theory, a new hardness theorem, or a replacement for existing solvers. The minimum number of polynomial inequalities describing a set is also an established question [3]. Here the optimization instead requires joint decoding of four labels on a promised domain, with a fixed degree bound and fixed original coordinates; the general polynomial-description literature is therefore relevant context, not a priority claim for the concrete calculation. Question and main result. The question is narrower:which observations must be available, and what must be proved about them, before a parameter value can be assigned its exact Boolean operator? For F (x,y ) =x +y +xy, (x,y ) = (ε1a,ε 2b), a,b> 0, Sections 2–3 give a complete answer for three explicit contracts. Three truth-table coordinates distinguish the four realized operators. No finite library of affine tests in(a,b ) suffices. Among polynomial-sign libraries, three tests are optimal at degrees two and three, whereas two quartic tests are optimal once degree four is allowed. The last statement concernstwo polynomial observations, not two queries to the original truth-table-coordinate oracle. Scope. This is a mathematical methods note built around an exact example and its certification requirements. The degree tradeoff is proved by an elementary boundary-divisibility argument. The synthesis of these facts is not offered as theorem-level priority clearance for every constituent observation. In particular, the curved boundaries become straight lines under reciprocal coordinates, and fewer sign tests do not by themselves imply less arithmetic or faster execution. The accompanying implementation is an exact rational reference dispatcher and regression suite, not a general certified guard-synthesis system or an empirical performance study. Correspondence matrices (CMs) and logical matrices (LMs) provide optional output represen- tations. We follow the companion manuscript’s terminology [19], but prove every representation fact needed here. Readers interested only in dispatch may regard a CM simply as a reshaped truth table. Classical trajectory and Walsh calibrations are collected in the appendix rather than used as novelty claims. 2 Score families and the nonlinear four-operator model 2.1 Strict semantics and operator fibers Let Λ be a nonempty open subset ofRp, letn≥ 1, and suppose gα : Λ−→ R, α ∈{ 0, 1}n, are continuous. They may arise by evaluating a continuous model at continuously varying repre- sentatives. For signed magnitudes, the representative forα is ε(α)⊙r, whereεi(α) = 2αi− 1 and ri> 0. Define the partial thresholdq(t) = 1 for t> 0 and q(t) = 0 for t< 0; q(0) is undefined. Set ∆ = ⋃ α∈{0,1}n g−1 α (0), Λ◦ = Λ\ ∆, Θλ(α) =q(gα(λ)) ( λ∈ Λ◦). Thus ∆ is the union of all score-zero sets, not their intersection. It is closed relative toΛ. Neither nonemptiness nor density ofΛ◦ is assumed. The numeric correspondence tensor is[Θ][α] = Θ(α), in true-first order on each axis; forn = 2 its entries are ordered(11, 10, 01, 00). 2 PAGE 3 Proposition 2.1 (Elementary exact quotient). The nonempty fibers ΩΘ ={λ∈ Λ◦ : Θλ = Θ} = ⋂ Θ(α)=1 {gα> 0}∩ ⋂ Θ(α)=0 {gα< 0} form the unique coarsest partition ofΛ◦ into sets carrying constant operators. Each fiber is open in Λ and clopen inΛ◦. Its connected components are open and are precisely connected components of Λ◦. Proof. The displayed finite intersection is open by continuity. Its relative complement is the union of the other fibers and is open as well. Every set carrying a constant operator lies in the corresponding fiber, proving the partition claim. An open Euclidean set is locally path-connected, so its connected components are open. A connected set cannot meet two distinct clopen fibers, proving the component assertion. We call those componentsoperator chambers. There are finitely many operator fibers but, for merely continuous scores, possibly infinitely many chambers. For instance,g1(t) = sint andg0(t) = 1 on R give two fibers and infinitely many interval chambers. IfΛ is open semialgebraic and the scores are polynomial, the fibers are strict sign-condition realizations and have finitely many semialgebraic connected components [4, 5, 16]. This is an application of established sign-condition theory. 2.2 Exact classification and a convex normal form All arithmetic in the following model is ordinary real arithmetic, not arithmetic over a Boolean field. ForF (x,y ) =x +y +xy on signed magnitudes, the four scores are g11 =a +b +ab, h 1 :=g10 =a−b−ab, h2 :=g01 =−a +b−ab, h 3 :=g00 =−a−b +ab. (1) Here g11> 0 everywhere on Λ = R2 >0, while h1 +h2 =−2ab< 0, h 1 +h3 =−2b< 0, h 2 +h3 =−2a< 0. (2) At most one of thehi can therefore be positive. If one is positive, both others are strictly negative; simultaneous zeros of two are impossible. Theorem 2.2 (The four chambers). The zero-free model realizes exactlyO⋆ ={AND,X,Y, XNOR}. Its fibers and rational interior witnesses are as follows. Region Truth vector (11, 10, 01, 00) Operator Witness (a,b ) h1,h 2,h 3< 0 (1 , 0, 0, 0) AND (1/2, 1/2) h1> 0 (1 , 1, 0, 0) X (2, 1/2) h2> 0 (1 , 0, 1, 0) Y (1/2, 2) h3> 0 (1 , 0, 0, 1) XNOR (3, 3) Every fiber is connected. The three discriminant curves are pairwise disjoint in the positive quadrant and are C1 : b = a 1 +a, C 2 : a = b 1 +b, C 3 : b = a a− 1 (a> 1). (3) Each separates AND from exactly one ofX,Y, XNOR, respectively, so regular chamber adjacency is a three-leaf star. 3 PAGE 4 Proof. The sign restriction follows from(2), and direct evaluation at the witnesses proves that all four patterns occur. To prove connectedness without a gluing assumption, apply the smooth involution (a,b )↦−→(u,v ) = (1/a, 1/b). (4) Since ab> 0, dividing the scores byab preserves their signs and gives 1 ab(g11,h 1,h 2,h 3) = (u +v + 1, v−u− 1, u−v− 1, 1−u−v). (5) In the positive(u,v ) quadrant the four regions become X v >u + 1 Y u>v + 1 XNOR u +v <1 AND |u−v|< 1, u+v >1. Each is a nonempty intersection of open half-planes, hence is convex and path-connected. Their inverse images under (4) are therefore path-connected. Solving the three score equations gives (3). Equation (2) excludes intersections of two curves in the domain. In reciprocal coordinates each curve is a line segment or ray with nonzero normal, with AND on one side and the corresponding leaf on the other. This also proves the claimed regular adjacency and that exactly one truth-table coordinate changes at each crossing. For example, (a,b ) = (2, 1/2) gives the scores(7/2, 1/2,−5/2,−3/2) and hence projectionX. The output CM is [X] = [ 1 1 0 0 ] . This is a certified identification, not a conclusion extrapolated from a numerical plot: the witness identifies a region whose uniform signs were proved above. Remark 2.3 (Positive coefficient rescaling). The same conclusions hold forF (x,y ) =Ax +By +Cxy with fixedA,B,C > 0. Setx′ = (C/B)x and y′ = (C/A)y; then F (x,y ) = AB C (x′ +y′ +x′y′). The induced positive linear rescaling of magnitude coordinates preserves total polynomial degree, guard-library cardinality, and the operator labels. This extends the example to a positive-coefficient family by an elementary equivalence, not by a new classification argument. The caseC = 0 is different: the addition model has only the two projection chambers. 3 The polynomial guard-degree tradeoff 3.1 An explicit cost model A binary polynomial-sign testin the original coordinates is pH(a,b ) = 1{H(a,b)>0}, H ∈ R[a,b ]. 4 PAGE 5 One test means one such polynomial, not an arbitrary Boolean formula built from several hidden tests. On the promised domainΛ◦, a finite libraryH1,...,H k is exact when its binary signature determines the operator: there is a decoder d :{0, 1}k−→O⋆ such that d(pH1(a,b ),...,p Hk(a,b )) = Θ(a,b) for every(a,b )∈ Λ◦. Values ofd on unrealized signatures do not matter. Tests need not individually be constant on an operator fiber. Constant polynomials can be discarded without improving a minimum. Definemd to be the minimumk for tests of total degree at most the integerd≥ 1, and set md =∞ when no finite library is exact. This cost counts fixed library size, not arithmetic operations, decision-tree depth, coefficient bit complexity, or the cost of proving that an input is zero-free. Strict-sign dispatch on an unpromised domain requires the additional treatment in Section 4. 3.2 The boundary obstruction Lemma 3.1 (Boundary factors). The three quadraticsh1,h 2,h 3 in (1) are pairwise nonassociate irreducibles in R[a,b ]. If finitely many nonzero polynomial zero sets coverCi, at least one of their polynomials is divisible byhi. After constant tests are discarded, every exact finite polynomial-sign library has such a factor for each ofi = 1, 2, 3. Proof. Writehi =Ai(a)b +Bi(a). The coefficient pairs are (A1,B 1) = (−a− 1,a ), (A2,B 2) = (1−a,−a), (A3,B 3) = (a− 1,−a). In each pair the entries are coprime inR[a]. Thushi is primitive and is linear, hence irreducible, over the fraction fieldR(a). Gauss’s lemma gives irreducibility inR[a,b ]. Comparing coefficients shows that no two are scalar multiples. Parametrize the corresponding positive arc byb =−Bi(a)/Ai(a) on an open interval avoiding the pole. Substituting into any polynomialH(a,b ) gives a rational function ofa. Unless it vanishes identically, it has only finitely many zeros on that interval. A finite collection of such finite sets cannot cover the interval. Therefore someH vanishes identically after substitution. Division in R(a)[b], followed by primitivity, shows thathi divides H in R[a,b ]. Finally, letz∈Ci. If every library polynomial were nonzero atz, all test values would be constant on a sufficiently small neighborhood ofz. That neighborhood contains zero-free points from the two different operator chambers adjoiningCi, contradicting exactness. Hence the library zero sets coverCi, and the preceding argument supplies the required factor. This reasoning uses points on the boundary only to compare nearby promised inputs; it does not assume that dispatch is defined on ∆. Theorem 3.2 (Exact degree–cardinality tradeoff). For the four-operator model in the original magnitude coordinates, md =    ∞, d = 1, 3, d = 2 or 3, 2, d ≥ 4. (6) Furthermore, in any exact library consisting of two polynomial-sign tests,each defining polynomial has degree at least four. 5 PAGE 6 Proof. A polynomial of degree at most one cannot contain any of the quadratic factors required by Lemma 3.1. Consequently no finite affine-sign library suffices. The same argument rules out any finite Boolean combination of affine inequalities as a description of the complete operator atlas: all such combinations over a finite library still depend only on its signature. A polynomial of degree at most three cannot contain two distincthi, since their product has degree four and they are relatively prime. Covering all three boundaries therefore requires at least three tests. The three tests1{hi>0} achieve this bound by Theorem 2.2. For degree four, take the two tests r1 = 1{h1h3<0}, r 2 = 1{h2h3<0}. (7) They have the requiredH >0 form withH =−h1h3 and H =−h2h3. Their signatures are Operator r1 r2 AND 0 0 X 1 0 Y 0 1 XNOR 1 1 Thus two tests suffice. One binary test cannot distinguish four nonempty operator fibers, so two is optimal. For the stronger assertion, an exact two-test library has only four possible signatures. Since four operator labels occur and signatures cannot be shared by different labels, each operator must have exactly one signature and all four signatures are used. Relative to the AND signature, each of the two bits changes on exactly two of the other three signatures. The polynomial defining that bit must consequently vanish along the two corresponding AND/leaf boundaries. Applying the substitution argument in Lemma 3.1 to that polynomial gives two distinct quadratic factors, so its degree is at least four. The argument applies to both tests. The lower bound covers arbitrary real coefficients and arbitrary decoders; it does not merely check a small catalogue of candidate polynomials. The accompanying code checks the displayed factorizations and all four output signatures, but the universal minimum is the consequence of the proof, not of a numerical search. 3.3 What the separation does not imply The affine obstruction is not invariant under arbitrary changes of coordinates. Equations(4)–(5) give an explicit smooth change under which the three original score signs are affine tests. Alternatively, introducing the featurez =ab makes them linear in(a,b,z ) on the feature surfacez =ab. Either transformation changes the permitted input features and therefore the observation contract. Nor does two-test quartic dispatch establish a runtime improvement. Computingh1h3 and h2h3 from the original scores uses those scores and two additional multiplications. The theorem compares sign-information channels under a degree constraint; feature construction, arithmetic sharing, numerical certification, and bit complexity must be charged separately in an implementation study. 6 PAGE 7 4 Zero-aware certification and partial abstraction 4.1 Nonempty guards and explicit coverage A strict sign certificate on a guardG⊆ Λ proves eithergα> 0 throughout G or gα< 0 throughout G. It may be an analytic proof, an exact real-algebraic certificate, or another sound verification object. A sample point or a plot is not a certificate of a uniform sign. Proposition 4.1 (Nonempty-guard soundness). LetG be nonempty. If one strict sign of every gα is proved throughoutG, then G⊆ Λ◦ and G carries a unique exact Boolean operator and correspondence tensor. A collection of such guards is exact on a declared domainD0⊆ Λ◦ when it coversD0. Guards with the same operator may be merged by explicit union; overlapping nonempty guards with certified labels cannot disagree on their intersection. Proof. Strict signs exclude zeros and determine the truth value at every input. Choosing any point of G proves that this truth table is realized, and the uniform certificates make it the same at every point. Coverage and the assertions about unions and intersections follow pointwise. Empty guards must be discarded or marked infeasible. All strict inequalities are vacuously true there, so the operator is not uniquely determined by empty-guard semantics. This is why nonemptiness is part of the statement rather than an implicit implementation preference. Polynomial sign decomposition and quantifier elimination provide exact routes in principle [4, 16]. Interval methods and solver-based methods may leave cells uncertified. Such an outcome reports a limit of the analysis; it is not evidence that both signs actually occur. The present note supplies no general solver for arbitrary continuous scores. 4.2 Certified dictionaries and reduced observation A complete, already proved operator dictionary can justify inference of unobserved signs. Proposition 4.2 (Reduced certificate with an explicit promise). LetD0⊆ Λ◦ be a certified zero-free domain and let the finite setO contain every Θλ for λ∈D0. LetG⊆D0 be nonempty. Suppose strict signs on a selected setS⊆{ 0, 1}n are proved constant throughoutG, and exactly one member Θ∈O has the corresponding restriction toS. Then every point ofG has operatorΘ, including the truth values at unobserved inputs. Proof. At any point ofG an induced total operator exists by the zero-free hypothesis, lies inO by dictionary completeness, and agrees with the certified restriction. Uniqueness of the dictionary match identifies it withΘ. Neither dictionary completeness nor the zero-free promise may be inferred from a sample. Without the latter, even an empty separating test set for a singleton dictionary can be unsafe: the scores1 and t2 induce the constant-true operator for everyt̸= 0, but not att = 0. Boundary rejection in the nonlinear example. At (a,b ) = (1, 1/2), (g11,h 1,h 2,h 3) = (2, 0,−1,−1). All three binary predicateshi > 0 are false, exactly as in the AND chamber; so they cannot by themselves certify membership in Λ◦. A correct unpromised dispatcher uses exact trichotomy, 7 PAGE 8 rejecting a zero rather than silently treating it as negative. The quartic method likewise checks whether either product in(7) is zero. Sinceg11 > 0, both products are nonzero exactly when all hi are nonzero. Thus twoternary product-sign observations suffice for classification together with boundary rejection, but this is a different outcome alphabet from the binary contract in Theorem 3.2. 4.3 Collecting semantics for predicate cells Synthetic representatives supply one score per Boolean input. Predicate abstraction of a concrete state space need not do so. LetD be a state space,P :D→{ 0, 1}n a total predicate map, and F :D→ R a total score. Define Sα ={sgn0F (x) :x∈D, P(x) =α}⊆{− 1, 0, +1}, where sgn0 retains zero. The empty set means an unreachable cell. The singleton sets{−1} and {+1} mean exact false and true outputs. A set containing0 exhibits a tie, and a set containing both strict signs exhibits genuine two-sided variation. The last two properties can hold simultaneously. Proposition 4.3 (Exactness on predicate cells). On a reachable cell,q◦F is everywhere defined and constant exactly whenSα is one of{−1},{+1}. Under the declared full-cube convention, a canonical total Boolean operator is induced exactly when every cell is reachable and satisfies this singleton criterion. Otherwise one retains a reachability mask and exact/partial cell information, unless a separate completion policy is declared. Proof. The two singleton cases make the threshold defined and constant. Conversely, a defined constant strict threshold permits neither a zero nor both strict signs. Values on unreachable inputs are not fixed by the concrete model; adding them is a completion choice. This is a collecting-semantics contract, not a new abstract domain [9]. In particular, checking only existence of a positive score and existence of a negative score cannot distinguish an empty cell from a nonempty all-zero cell: both tests are false in both cases. Reachability, zeros, genuine variation, and a solver’sUNKNOWN response must remain separate. 5 Test selection, coordinate probes, and faithful representations 5.1 Finite predicate selection is a minimum-test-set problem Suppose a certified finite partition{A1,...,A m} of a zero-free domain has nonempty atoms labelled by operators Θi. Letp1,...,p r be admissible binary predicates, each constant on each atom, and let wj > 0 be selection costs. Put Dik ={j :pj(Ai)̸=pj(Ak)} (Θi̸= Θk). A minimum-cost exactfixed predicate basisis the binary optimization problem min r∑ j=1 wjzj, ∑ j∈Dik zj≥ 1 (Θ i̸= Θk), z j∈{ 0, 1}. (8) 8 PAGE 9 Indeed, selected predicates yield exact labels if and only if they separate every differently labelled pair, which is exactly the set of displayed constraints. If any requiredDik is empty, no exact basis exists in that library. If predicates vary within an atom, the partition must first be refined and certified; testing only its representatives does not establish this premise. This is the classical pair-separation/hitting-set formulation [17, 6]. It does not optimize expected path length or adaptive worst-case depth. Its weights account for selected predicates only; costs of computing shared features are not automatically additive in these weights. Predicate abstraction with minimum predicates [8] is an important related precedent, but exact label recovery here is a different specification from preserving a chosen verification property. 5.2 Fixed coordinates versus adaptive and target-specific information For a nonempty finite familyO of Boolean functions, define σCM(O) = min{|S| :S⊆{ 0, 1}n, Θ↦→Θ|S is injective onO}. It is zero for a singleton. This is a fixed truth-table-coordinate test-set number, expressed in CM notation, not a new generic combinatorial invariant. Its exact formulation is(8) with one test per coordinate and unit cost; hence ⌈log2|O|⌉≤ σCM(O)≤ 2n. The lower bound counts binary signatures and the upper bound uses the complete table. It must not be confused with a target-specific specification or teaching set, whose chosen coordinates can depend on the particular function being specified, or with adaptive querying of an unknown function. Threshold-function specification numbers and their chamber-facet interpretation are developed in [2]; the global, fixed separating set above has different quantifiers. Proposition 5.1 (The coordinate-oracle costs of the four-operator family). ForO⋆, the unique minimum fixed coordinate set is{10, 01, 00}, soσCM(O⋆) = 3. Deterministic adaptive identification from truth-table-coordinate queries also has worst-case cost three. AND requires three coordinate values for a target-specific certificate, while each leaf operator has a one-coordinate certificate. Proof. Coordinate 11 is always1, and the other coordinates are 10 01 00 AND 0 0 0 X 1 0 0 Y 0 1 0 XNOR 0 0 1 . Omitting a nonconstant coordinate leaves AND indistinguishable from the leaf differing only there. This proves the fixed minimum and its uniqueness. On the AND input, an adaptive algorithm receives zero at every nonconstant coordinate. Until all three have been queried, the leaf at an unqueried coordinate is still consistent with the transcript; the worst-case lower bound follows. Querying all three proves the upper bound. A positive value at the unique leaf coordinate identifies that leaf, whereas AND needs all three zeros. The adaptive lower bound is for an unknown operator accessible only through the listed coordinate oracle and a known dictionary. An algorithm given numerical(a,b ) and allowed arbitrary derived 9 PAGE 10 predicates has a different contract, as the quartic construction illustrates. Within the coordinate oracle, an early-stop order has depths1, 2, 3, 3 for the four operators and expected cost9/4 under the uniform distribution. Thus “three probes are necessary” means a fixed-basis or worst-case claim, not a claim about every parameter point. 5.3 The fixed-frame LM endpoint is self-contained LetX = (X1,...,X n) be independent formal Boolean generators, and identify formulas up to Boolean equivalence. For a formulaA, setA⟨1⟩ =A and A⟨0⟩ =¬A. Extend a Boolean functionΘ by its disjoint minterms: ˜Θ(A1,...,A n) = ⋁ β∈{0,1}n ( Θ(β)∧ n⋀ i=1 A⟨βi⟩ i ) . The minterms are disjoint, so an iterated XOR, denoted⇕, would give the same function. Define the formula-valued lift LX (Θ)[α] = ˜Θ(X⟨α1⟩ 1 ,...,X ⟨αn⟩ n ). (9) The formal reference frame is not a collection of constrained physical predicates: all Boolean valuations, including the all-true valuation, are available. Proposition 5.2 (Faithfulness and representation-invariant dispatch). All-true valuation of (9) returns [Θ]. ConsequentlyLX is injective. For every Boolean connectiveH :{0, 1}k→{ 0, 1}, the lift preserves pointwise superposition: LX ( H(Θ1,..., Θk) ) =H ( LX (Θ1),...,L X (Θk) ) . Therefore Θλ = Θµ ⇐⇒ [Θλ] = [Θµ] ⇐⇒ LX (Θλ) =LX (Θµ). Any fixed parameter-side observation contract and cost has the same exact-dispatch optimum for these three outputs, provided decoding or output-construction costs are not added differently. Proof. At Xi = 1, the literalX⟨αi⟩ i has valueαi; the minterm extension therefore evaluates to Θ(α). Equal lifted tensors give equal truth tables under this valuation, proving injectivity. At every entry and every valuation of the formal frame, both sides of the superposition identity evaluate the same connectiveH on the same function values. Finally, injective representations induce the same equality relation on parameter space. The same observations consequently identify the same labels; only representational output costs could alter a total implementation cost. For the running point(2, 1/2), the complete output chain is (2, 1/2)↦−→X↦−→ [ 1 1 0 0 ] ↦−→ [ X1 X1 ¬X1 ¬X1 ] . For comparison, the AND and XNOR lifts are [ X1∧X2 X1∧¬X2 ¬X1∧X2 ¬X1∧¬X2 ] , [ X1⇔X2 X1⇕X2 X1⇕X2 X1⇔X2 ] . Continuous scores, numeric truth tensors, and formula-valued lifts are different types. A partial truth tensor from an unreachable or nonexact predicate cell does not determine a full lift without a completion policy. The elementary proof above removes any dependence on an unpublished companion theorem; the companion is a terminology reference only. 10 PAGE 11 6 Reproducibility, limitations, and conclusion The companion archive contains the source of this note, a portable exact rational dispatcher, a deterministic verification script, machine-readable results, reproduction commands, and an audit crosswalk. Both coordinate and quartic dispatch modes return a separate boundary status; floating- point inputs are rejected by the reference implementation to avoid silently replacing exact rational semantics with approximate comparisons. The tests compare direct signed-input evaluation, the expanded scores, the reciprocal normal form, and both dispatch modes. They also check boundary points, the original recorded rational grid, dictionary separation, actual minterm evaluation, and the classical Walsh witnesses. These checks are bounded computations, not formal proof certificates for the topology or universal polynomial lower bound. The latter rest on the proofs in Theorems 2.2 and 3.2. The package documents exactly what was run; it does not claim solver benchmarks, a speedup over direct score evaluation, or a completed general guard-synthesis implementation. The example separates three issues that can otherwise be conflated. Geometry determines where each operator is realized. An observation contract determines which tests may be used and how they are counted. Certification determines when those observations justify a total Boolean output. In the original coordinates, the exact degree–cardinality tradeoff(6) and the three-coordinate lower bound make the contract dependence explicit. Reciprocal linearization prevents interpreting the result as a coordinate-free nonlinear obstruction. Faithful CM/LM representations preserve dispatch distinctions but do not strengthen the information available from the continuous model. The general quotient, collecting semantics, finite test selection, and Walsh arrangement remain classical or elementary components. A stronger algorithms or applications claim would require a motivated model family, an implemented certificate-producing backend, and an evaluation that charges feature construction and certification as well as sign queries. Those are not conclusions of this note. A Classical trajectory and Walsh calibrations A.1 Crossings, event counts, and endpoint-relative intervals Proposition A.1 (Single-entry crossing). Letγ : (−η,η )→ Λ be continuous. Suppose one score has a fixed strict sign on(−δ, 0) and the opposite fixed strict sign on(0,δ ), while all other scores are nonzero on(−δ,δ ). Then the two side operators differ at exactly that entry. If the composed scores are differentiable, a simple zero of one score and nonzero values of all the others at0 suffice after shrinkingδ. Proof. Every other composed score keeps its sign by continuity. The distinguished entry alone changes. A nonzero derivative at a zero gives opposite strict signs on sufficiently short punctured sides. A mere assertion that a continuous function “changes sign at zero” would not exclude oscillation. The fixed-side hypotheses are therefore part of the statement. One-input adjacency is already a feature of structured parameter graphs and threshold arrangements [11, 2]. Proposition A.2 (Trajectory and polynomial event bound). For a continuous pathγ : [0,T ]→ Λ, the setS =γ−1(∆) is closed and the induced operator is constant on each connected component of [0,T ]\S. If all points ofS are isolated relative to[0,T ], thenS is finite. The complementary 11 PAGE 12 components arerelatively openintervals, possibly containing0 or T. Separately, if the scores on an interval are nonzero polynomials of degree at mostd, there are at most2nd discriminant values and at most 2nd + 1 complementary interval components. Proof. Relative closedness of∆ and continuity give closedness ofS. Connected complementary components map into one fiber by Proposition 2.1. An infinite closed subset of a compact interval has an accumulation point in the set, contrary to isolation. For the polynomial assertion, each nonzero polynomial has at mostd roots and the union bound gives the result. Even-multiplicity roots need not change the operator, so this counts possible zero events, not necessarily actual switches. These are elementary score-level observations, not a dynamics theorem or a substitute for logical bifurcation theory [1]. A.2 A centered Walsh arrangement Let N = 2n and x∈{− 1, +1}n. For nonemptyS⊆ [n], putχS(x) =∏ i∈Sxi and letϕ(x) be the vector of theseN− 1 characters. Standard Walsh orthogonality [18] gives ϕ(x)·ϕ(y) = { N− 1, x =y, −1, x ̸=y, ∑ x ϕ(x) = 0. The coefficient familyFc(x) =c·ϕ(x) identifies RN−1 with the zero-sum subspace of score vectors in RN: the displayed Gram matrix isNI−J, which has rankN− 1. Proposition A.3 (Classical Walsh calibration). For everyn≥ 1, precisely the2N− 2 nonconstant strict Boolean sign patterns occur, each in one open convex cone. Forn≥ 2, regular facet adjacency is Hamming-distance-one adjacency on those truth tables, theN-cube with the two constant vertices deleted. Forn = 1 the two indexed zero hyperplanes coincide and ordinary geometric adjacency is not single-entry adjacency. Proof. For a positive setP of size 0