EXTRACTED TEXT — equation transcription may be incomplete; use the PDF and TeX for mathematics. PAGE 1 Finite-Jet Sign Rigidity on Hyperplane Arrangements and Denjoy–Carleman Realization Brian Theory Reviewed revision — 27 September 2026 Quantitative additions — 13 October 2026 Scope clarifications and revision record — 14 October 2026 Abstract LetA be a finite real affine hyperplane arrangement. We study which{±1}-valued chamber labelings can occur as the sign of a function with prescribed regularity. A first nonzero homogeneous Taylor term inherits the chamber signs weakly, and factor multiplicities force a constant crossing parity. This gives a short arrangement-level proof of the known smooth signature-flatness obstruction and, in finite regularity, explicitC d orthant andC N arrangement rigidity statements. For normalized compactwise Roumieu Denjoy–Carleman classesE {M }, the Denjoy–Carleman criterion yields an exact realization dichotomy. In the quasianalytic case the realizable chamber laws are precisely the signed parity laws; in the non-quasianalytic case every chamber labeling is realized by explicit selector formulas. Thus an arrangement ofm distinct hyperplanes withr chambers admits exactly2m+1 and 2r realizable sign laws in the two regimes. We also record corner-margin consequences and a fixed-dimensional polynomial comparison on a cube with coordinate-wall strips of widthδ removed: signed parity has constant exact degree, non-affine laws have degreeΘ(log(1/δ)), and any nonconstant law requires degreeΘ(1/δ) under a global amplitude bound and fixed margin0<γ < 1. The smooth obstruction, positive-flat existence, and approximation ingredients are classical; the contribution is their explicit formulation and synthesis, with finite-order statements and complete reductions. 2020 Mathematics Subject Classification.26E10, 52C35. Keywords. finite jets; hyperplane arrangements; Denjoy–Carleman classes; quasianalyticity; sign distributions; polynomial sign representation. 1 Introduction A Boolean truth table is finite. Requiring the same truth value throughout an entire orthant of Rn is not. This distinction becomes sharp when the Boolean value is extracted from the sign of a smooth real-valued function. For example, the polynomial F (x,y ) =xy is positive in the first and third quadrants and negative in the other two. With positive interpreted as 1 and negative as0, it realizes equivalence (XNOR), an affine Boolean function overF2. By contrast, the AND sign pattern is positive only in the positive-positive quadrant and negative in the other three. The nonsmooth functionmin(x,y ) has exactly that sign pattern, but the results below show that anysmooth realization of AND on all four quadrants must be flat at the origin. In a quasianalytic class this is impossible for a nonzero function; in a non-quasianalytic class it is possible by using flat one-sided selectors. 1 PAGE 2 The first purpose of this paper is to isolate the local mechanism behind that phenomenon. It is not coordinate division, resolution of singularities, or a special feature of one ultradifferentiable class. It is the first nonzero homogeneous Taylor term. If a smooth function has a nonzero finite jet at the common corner, its leading polynomial already inherits the orthant signs in a weak sense. Polynomial factorization across coordinate hyperplanes then forces an affine parity law. The last factor-multiplicity step is elementary and has clear antecedents; the point of the finite-jet formulation is that it transports that obstruction from polynomials to arbitrary smooth germs and makes the subsequent regularity classification immediate. The second purpose is to carry that mechanism to a finite affine hyperplane arrangement. If a smooth function is nonzero off the arrangement and has no flat point on it, the local leading polynomial at each point fixes a crossing parity for every incident hyperplane. Those parities propagate along each connected affine hyperplane and hence determine the sign on every chamber. The third purpose is to combine these smooth statements with the classical Denjoy–Carleman theorem. We use throughout one compactwise Roumieu convention, |Dαf(x)|≤ CKρ|α| K|α|!M|α|, x ∈K ⋐U, and no derivative-stability assumption. In this normalization the quasianalytic criterion is ∞∑ k=0 Mk (k + 1)Mk+1 =∞. On the divergent side, quasianalyticity rules out the flatness required by a non-affine orthant law. On the convergent side, a classical compactly supported cutoff yields a one-sided selector, from which arbitrary Boolean laws are assembled without cancellation. Prior work and contribution boundary.The C∞ arrangement obstruction belongs to the signature-flatness theory of Broglia and Tognoli [12]; Remark 4.6 gives the precise reduction from the hypotheses used here, including the homological point in their theorem. Their factorization analysis also contains the parity-of-multiplicity mechanism underlying Lemma 3.1, and their examples already use a first nonzero Taylor coefficient to detect incompatible crossing parities. Thus neither the smooth obstruction nor the leading-term idea is claimed as new. The broader historical setting includes distributions of signs and signatures [5, 1, 2, 4, 3], the Varchenko–Gelfand algebra of chamber functions and its later arrangement-theoretic extensions [7, 8, 6], and the classical Denjoy–Carleman theory of quasianalyticity, cutoffs, and positive-flat functions [14, 13, 15]. Stronger division and monomialization frameworks are also available [11, 9, 10], but they are not needed for the elementary proofs below. Exact Boolean sign operations also have a classical constructive theory in theR-function literature. Shapiro’s survey [ 16, Sections 3.3 and 3.6] describes root-based conjunction and disjunction and their finite-differentiability upgrades. These supply direct examples at the boundary of the finite-jet statements; they are credited explicitly in Section 8.3. The note combines explicit finite-order rigidity, a self-contained proof for affine arrangements, and exact realization classifications with constructive formulas and chamber counts. Section 7 adds quantitative consequences: Taylor estimates turn vanishing jets into corner-margin bounds, while classical rational approximation, Sherstov’s majority-conjunction lower bound [17], and Bernstein–Jackson inequalities [18] give a fixed-dimensional degree classification after excluding the walls. These are derived consequences, with no claim of historical firstness or new approximation machinery. They distinguish exact realization near the common corner from sign correctness at a positive distance from the walls. 2 PAGE 3 Organization. Section 2 fixes the sign and Denjoy–Carleman conventions. Section 3 proves the weak-sign polynomial lemma. Section 4 establishes finite-jet rigidity and the smooth arrangement theorem. Section 5 constructs one-sided selectors and proves the whole-orthant dichotomy. Section 6 gives the arrangement classification and exact counts. Section 7 derives corner-margin and polynomial-degree bounds. Section 8 records counterexamples and comparisons withR- functions and normal-crossing results. 2 Definitions, conventions, and classical input 2.1 Orthants and Boolean sign laws Forα = (α1,...,α n)∈{ 0, 1}n, put εi(α) = 2αi− 1∈{− 1, +1} and define the open orthant Oα ={x∈ Rn :εi(α)xi> 0 for everyi}. A functionF defined on an open setU meeting all these orthantsrealizes a Boolean law Θ : {0, 1}n→{ 0, 1} on the orthants inU if F is nonzero on their intersections withU and sgnF (x) = 2Θ(α)− 1, x ∈O α∩U. Local realization at the common corner means realization on a centered ball contained inU; global realization meansU = Rn. We callΘ affine if Θ(α) =b⊕e1α1⊕···⊕ enαn, b,e i∈{ 0, 1}, where⊕ denotes addition inF2. There are2n+1 affine Boolean functions. Whenn = 1, these are all four unary Boolean functions. A smooth function isflat atp if DαF (p) = 0 for every multi-indexα. Definition 2.1 (Taylor-quasianalytic class). For the two generic corollaries below, a smooth function class is calledTaylor-quasianalytic if, at every point of its domain, a function whose full Taylor jet vanishes has zero germ there. For a vector space of germs, this zero-jet implication is equivalent to injectivity of the Taylor map. Only the zero-jet implication is used in the two generic corollaries below. No division, composition, or sheaf axiom is intended by this shorthand. The divergent Denjoy–Carleman classes used later have this property by the Denjoy–Carleman theorem. 2.2 Affine arrangements Let A ={H1,...,H m}, H j ={x∈ Rn :ℓj(x) = 0}, be a finite family of distinct proper affine hyperplanes, where each nonconstant affine defining form ℓj is fixed once and for all. Achamber is a connected component of Rn\ m⋃ j=1 Hj. 3 PAGE 4 We write Ch(A) for the chamber set. On a chamberC, every sgnℓj is constant. A chamber labeling h : Ch(A)→{± 1} is asigned parity lawif h(C) =c m∏ j=1 sgn(ℓj|C)ej, c ∈{± 1}, e j∈{ 0, 1}. (1) The word “signed” is deliberate: whenc =−1, this is not literally a group character, and the realized chamber sign vectors need not form a group. 2.3 One Denjoy–Carleman convention Throughout Sections 5–6, anormalized weightmeans a sequenceM = (Mk)k≥0 such that M0 = 1, M k > 0, with M nondecreasing and log-convex. For an open setU⊆ Rn, define the compactwise Roumieu classE{M}(U) by requiring that for every compactK ⋐U there existCK,ρK > 0 such that |Dαf(x)|≤ CKρ|α| K|α|!M|α| (x∈K). (2) This convention keeps the factorial outsideM. To compare with the common total-weight convention, set Wk =k!Mk. ThenW is a weight sequence in the sense used in [14]: it is positive and log-convex,W0 = 1≤W1, and W 1/k k →∞ . Its quotient sequence satisfies Wk+1 Wk = (k + 1)Mk+1 Mk , so the Denjoy–Carleman series is precisely ∞∑ k=0 Mk (k + 1)Mk+1 . (3) The Denjoy–Carleman theorem says thatE{M} is quasianalytic exactly when(3) diverges; see [14, Theorem 3.6]. In the convergent case, nontrivial compactly supported functions exist; the cutoff form we use is supplied, for example, by [14, Corollary 3.12]. We will need only elementary closure properties in addition to this classical input. Lemma 2.2(Elementary closure). For a normalized weightM, the classE{M} is closed under finite sums, finite products, translations, and precomposition with affine linear functionals. For functions on R it is also closed under the primitiveG(t) = ∫t 0g(s)ds. Proof. Onlyproductsandprimitivesneedcomment. Log-convexitywith M0 = 1impliesMjMk−j≤ Mk. Combining this with Leibniz’ rule and the bounds(2) gives the same form of estimate with a larger radius constant. For a one-variableg∈E {M}(R), defineG(t) = ∫t 0g(s)ds. On a compact interval,G itself is bounded. Fork≥ 1, G(k) =g(k−1), and monotonicity ofM gives |G(k)|≤ Cρk−1(k− 1)!Mk−1≤C′(ρ′)kk!Mk 4 PAGE 5 for suitable enlarged constants. For an affine functionalℓ(x) =a·x +b and a one-variable function g, the chain rule gives Dα(g◦ℓ)(x) =aαg(|α|)(ℓ(x)). The image of a compact set is compact, and the factors froma are absorbed into the radius constant. Translations are the one-variable special case of slope one. 3 A weak-sign polynomial lemma The finite-jet argument rests on an elementary fact: even aweak chamber sign prescription for a nonzero polynomial has parity form. Lemma 3.1(Polynomial weak-sign lemma). LetA ={H1,...,H m} be a finite family of distinct proper affine hyperplanes inRn, with fixed defining formsℓj. LetP be a nonzero real polynomial. Suppose numbershC∈{± 1} are assigned to the chambersC∈ Ch(A) and satisfy hCP (x)≥ 0 ( x∈C). (4) Then there are uniquec∈{± 1} and e1,...,e m∈{ 0, 1} such that hC =c m∏ j=1 sgn(ℓj|C)ej (C∈ Ch(A)). (5) The polynomial is allowed to vanish at points inside a chamber. Proof. For eachj, letaj be the largest exponent for whichℓaj j divides P, and write P =ℓaj j Qj, ℓ j ∤Qj. The restriction ofQj to Hj is therefore a nonzero polynomial onHj and cannot vanish on a relatively open subset ofHj. Consider two chambers sharing a facet contained inHj. In the relative interior of that facet choose a pointp outside every other hyperplane and withQj(p)̸= 0. In a sufficiently small ball about p, the sign ofQj is fixed, whileℓj changes sign whenHj is crossed. Hence the sign ofP changes by the factor(−1)aj. WhereverP̸= 0, the weak inequality(4) forces its sign to equal the prescribed chamber sign. Thus every adjacency acrossHj has the same ratio hC′ hC = (−1)aj. Put ej≡aj (mod 2). Now divide the chamber label by∏ j sgn(ℓj)ej. The quotient is unchanged across every chamber adjacency. The chamber adjacency graph is connected: choose interior points of two chambers and perturb a segment, or use a two-segment polygonal path, so that it meets the hyperplanes transversely and never meets two at the same time. The quotient is therefore a single constant c∈{± 1}, proving (5). For uniqueness, fixj and choose a point ofHj outside all other hyperplanes. The two local chambers on its two sides differ only in the sign ofℓj, so their label ratio determinesej. One chamber then determinesc. Remark 3.2. The content of Lemma 3.1 is elementary polynomial factorization plus chamber adjacency. We use it as a transparent engine for the smooth results; we do not attach an independent novelty claim to it. 5 PAGE 6 4 Finite-jet rigidity 4.1 Orthants Theorem 4.1(Finite-jet rigidity). LetF beC∞ on a neighborhood of the origin inRn. Assume that in some ballB centered at the origin, F (x)̸= 0 whenever every coordinate ofx is nonzero. Because eachB∩Oα is connected, continuity makes the sign ofF constant there; write the resulting local orthant sign law ash :{±1}n→{± 1}. If at least one derivative ofF at the origin is nonzero, then there are uniquec∈{± 1} and e∈{ 0, 1}n such that h(s) =c n∏ i=1 sei i . (6) Equivalently, every smooth realization of a non-affine Boolean orthant law is flat at the origin. Proof. Let d be the least order of a nonzero derivative ofF at the origin;d = 0 is allowed. The first nonzero homogeneous Taylor term is Pd(x) = ∑ |α|=d DαF (0) α! xα, P d̸= 0. For anyv∈ Rn with no zero coordinate, Taylor’s formula along the raytv, t↓ 0, gives F (tv) =tdPd(v) +o(td). By the nonvanishing hypothesis and connectedness just noted,F (tv) has signh(sgnv) for every sufficiently smallt> 0, h(sgnv)Pd(v)≥ 0. Thus the nonzero polynomialPd satisfies a weak sign prescription on the coordinate-hyperplane arrangement. Lemma 3.1 gives (6) and its uniqueness. Under the identificationsi = 2αi− 1, a sign law of the form(6) is equivalent to an affine Boolean functionb⊕e1α1⊕···⊕ enαn. The contrapositive gives the flatness statement. Remark 4.2 (Finite regularity). The proof uses only the first nonzero Taylor order. If the first nonzero jet has orderd, then aCd Taylor expansion is enough. The theorem does not assert that nonflatness is necessary for an affine law: flat functions can also realize affine laws. Theorem 4.3(Finite-order orthant rigidity). Letd≥ 0 and letF beCd near the origin, nonzero whenever all coordinates are nonzero in some centered ball. If every derivative of order< d vanishes at the origin and at least one derivative of orderd is nonzero, then the local orthant sign law is affine parity. Proof. The order-d Taylor expansion supplies the same nonzero leading homogeneous polynomial used in Theorem 4.1; Lemma 3.1 then applies verbatim. Corollary 4.4(Quasianalytic orthant rigidity). LetF belong to a Taylor-quasianalytic class on a neighborhood of the origin and suppose it realizes a whole local orthant law there. Then that Boolean law is affine. 6 PAGE 7 Proof. If every derivative ofF vanished at the origin, the defining zero-jet property would make the germ ofF at the origin zero, contradicting nonvanishing on every local open orthant. Apply Theorem 4.1. 4.2 Affine hyperplane arrangements Theorem 4.5(No-flat-point arrangement rigidity). LetA ={H1,...,H m} be a finite family of distinct proper affine hyperplanes inRn. LetF∈C∞(Rn) be nonzero onRn\⋃ jHj. Assume that F is not flat at any point of⋃ jHj. Then there are uniquec∈{± 1} and e1,...,e m∈{ 0, 1} such that sgnF (x) =c m∏ j=1 sgnℓj(x)ej, x / ∈ ⋃ j Hj. (7) Proof. Fix p∈⋃ jHj. Choose a ballBp small enough that every hyperplane meetingBp contains p. After translatingp to the origin, the incident hyperplanes form a central arrangement. LetPp be the first nonzero homogeneous Taylor polynomial ofF at p. For every directionv outside the incident hyperplanes, F (p +tv) =td(p)Pp(v) +o(td(p)), t ↓ 0. The functionF has a fixed sign on each local chamber because it is continuous and nonzero there. HencePp inherits those signs weakly. Lemma 3.1 gives, for every hyperplaneHj throughp, a local crossing parityej(p)∈{ 0, 1}. We claim thatej(p) is locally constant asp varies inHj. Fixp∈Hj and its ballBp. Forp′∈Hj sufficiently close top, choose the local ballBp′ contained inBp and avoiding all hyperplanes not containingp′. The relatively open setHj∩Bp′ contains a pointq outside every other hyperplane: the excluded intersections form a finite union of proper affine subspaces ofHj. In dimension one, Hj is a point and distinct hyperplanes do not meet it. A sufficiently small transverse pair nearq, on opposite sides ofHj, lies inside both balls and meets no other hyperplane. The ratio of the two actual signs ofF is both (−1)ej(p) and (−1)ej(p′). Thus the bits agree. SinceHj is connected, ej(p) is a single global bitej. Now consider the global chamber adjacency graph. Across a generic facet contained inHj, the sign ofF changes by(−1)ej. The same is true of the product∏ j sgn(ℓj)ej. Their quotient is therefore unchanged across every chamber adjacency. As in Lemma 3.1, the chamber adjacency graph is connected, so the quotient is a global constantc∈{± 1}. This proves(7). Uniqueness follows by crossing a generic point of eachHj. Remark 4.6 (Earlier signature-flatness theorem). Theorem 4.5 is an elementary affine-arrangement case of the signature-flatness theory of Broglia and Tognoli [12, Theorem 4.2]. To compare hypotheses, set G =F m∏ j=1 ℓ2 j. Then G has zero set exactly⋃ jHj and the same chamber signs asF. If F has a first nonzero Taylor term at a point of the arrangement, the first nonzero term ofG is its product with a nonzero polynomial, so nonflatness is preserved. Forn≥ 2 their homological condition is automatic onRn because Hn−1(Rn; Z2) = 0; the hyperplane components admit global affine equations and satisfy the required connectivity. The casen = 1 is immediate from the one-dimensional crossing picture. 7 PAGE 8 Their admissible signatures reduce, for hyperplanes, to the signed parity laws in(1). This remark records the historical implication explicitly; the proof above is retained because it isolates the finite-jet mechanism needed for Theorem 4.7. Theorem 4.7 (Uniform finite-order arrangement rigidity). Let N≥ 0, letA ={H1,...,H m} be a finite family of distinct proper affine hyperplanes inRn, and letF∈CN(Rn) be nonzero on Rn\⋃ jHj. Assume that for everyp∈⋃ jHj the N-jet ofF at p is nonzero; equivalently, for each such p there is a multi-indexα with|α|≤ N and DαF (p)̸= 0. Then the chamber sign law ofF is a unique signed parity law of the form(7). Proof. At eachp on the arrangement choose the leastd(p)≤N for which some derivative of order d(p) is nonzero. TheCd(p) Taylor expansion gives a nonzero leading homogeneous polynomial and therefore the same local crossing parities as in the proof of Theorem 4.5. The propagation of those parities along a hyperplane, and then across the connected chamber adjacency graph, uses only continuity and nonvanishing off the arrangement. Thus the remainder of that proof applies unchanged. Remark 4.8 (A finite-order obstruction). Equivalently, if aCN function is nonzero off the ar- rangement and realizes a chamber labeling that is not signed parity, then at some point of the arrangement its entireN-jet must vanish. The smooth theorem is a stronger all-orders analogue: it yields a single flat point. The finite-order statements alone do not imply that conclusion by letting N tend to infinity, because the point at which theN-jet vanishes may depend onN. Corollary 4.9(Quasianalytic arrangement rigidity). LetF be a nonzero globally defined member of a Taylor-quasianalytic class onRn, and supposeF is nonzero off a finite affine arrangement. Then its chamber sign law has the form(7). Proof. At a flat point, the defining zero-jet property would make the germ ofF there zero. Every neighborhood of a point on a finite hyperplane arrangement contains points off the arrangement, contradicting the assumed nonvanishing there. ThusF has no flat point on the arrangement, and Theorem 4.5 applies. 5 Denjoy–Carleman realization on orthants The finite-jet theorem supplies the rigidity half of the classification. The converse direction in a non-quasianalytic class is constructive. Lemma 5.1(One-sided selector). LetM be a normalized weight. If ∞∑ k=0 Mk (k + 1)Mk+1 <∞, (8) then there existsψ∈E{M}(R) such that ψ(t) = 0 ( t≤ 0), ψ (t)> 0 ( t> 0). Proof. With Wk = k!Mk, condition (8) is the non-quasianalytic Denjoy–Carleman condition∑Wk/Wk+1 <∞. By the classical cutoff theorem, choose a nonzero real-valued compactly 8 PAGE 9 supported functionf in the corresponding Roumieu class; see [14, Theorem 3.6 and Corollary 3.12]. Translatef so that the left endpoint of its support is0. Then f(t) = 0 ( t≤ 0), and every interval(0,t ), t> 0, contains a point at whichf is nonzero: otherwisef would vanish on a right-neighborhood of0, contradicting that0 is the left endpoint ofsuppf. By Lemma 2.2,f 2 and its primitive belong to the same class. Define ψ(t) = ∫t 0 f(s)2ds. Fort≤ 0 the integral is zero. Fort> 0, the interval(0,t ) contains a point wheref̸= 0; continuity then gives an interval on whichf 2> 0, and thereforeψ(t)> 0. Classical constructive antecedent. A stronger existence statement is classical. Rainer’s Remark 3.13, citing work of Langenbruch, records positive-flat ultradifferentiable functions for prescribed closed sets in the non-quasianalytic setting [14, Remark 3.13]; see also [13]. Taking the closed set to be the arrangement and changing the sign of such a function independently on each chamber already gives arbitrary chamber labelings, since flatness makes the chamberwise pieces glue with zero jet on every wall. The selector construction above is retained because it gives an explicit formula in the defining affine forms and makes the later counting argument transparent. Definition 5.2. For a normalized weightM, letRM(n) be the set of Boolean functionsΘ : {0, 1}n→{ 0, 1} realized by some globally definedF∈E {M}(Rn) that is nonzero whenever all coordinates are nonzero. Theorem 5.3(Whole-orthant classification). LetM be a normalized weight. Then RM(n) =    {affine Boolean functions}, ∞∑ k=0 Mk (k + 1)Mk+1 =∞, {all Boolean functions on{0, 1}n}, ∞∑ k=0 Mk (k + 1)Mk+1 <∞. Proof. Assume first that the series diverges. By the Denjoy–Carleman theorem,E{M} is quasiana- lytic. IfF realizes a whole-orthant law, thenF is not the zero function in any neighborhood of the origin, so its Taylor jet at the origin cannot vanish identically. Theorem 4.1 implies that the law is affine. Conversely, every affine law is realized by the polynomial F (x) =c n∏ i=1 xei i for suitablec∈{± 1} and ei∈{ 0, 1}, and every polynomial lies in the class. Now assume the series converges and choose the selectorψ from Lemma 5.1. Forα∈{ 0, 1}n, define Ψα(x) = n∏ i=1 ψ ( εi(α)xi ) . For an arbitrary Boolean functionΘ, set FΘ(x) = ∑ α∈{0,1}n ( 2Θ(α)− 1 ) Ψα(x). (9) 9 PAGE 10 By Lemma 2.2,FΘ∈E{M}(Rn). Ifx∈O α, then Ψα(x)> 0, while for everyβ̸=α at least one factor of Ψβ(x) is zero. Hence sgnFΘ(x) = 2Θ(α)− 1. Thus every Boolean function is realized. Corollary 5.4 (Gevrey dichotomy). For the Gevrey classGs, s≥ 1, with derivative growth Cρk(k!)s, the whole-orthant laws are exactly the affine Boolean functions whens = 1, and all Boolean functions whens> 1. Proof. In the normalization (2), takeMk = (k!)s−1. Then Mk (k + 1)Mk+1 = (k + 1)−s. The corresponding series diverges exactly ats = 1 for s≥ 1. Apply Theorem 5.3. Remark 5.5. Forn = 1, the two sides of Theorem 5.3 contain the same four Boolean functions because every unary Boolean function is affine. The strict expressive difference begins at arity n≥ 2. 6 Affine arrangements: classification and exact counts The orthant construction extends directly once orthants are replaced by the realized sign cells of a finite affine arrangement. Theorem 6.1(Arrangement classification). LetM be a normalized weight. LetA be an arrange- ment ofm distinct proper affine hyperplanes inRn, and writer =| Ch(A)|. If the series(3) diverges, the chamber sign laws realized by globally definedF∈E{M}(Rn) that are nonzero off the arrangement are exactly the signed parity laws(1); there are exactly2m+1 such laws. If the series(3) converges, every one of the2r chamber labelings is realizable by such a function. Proof. On the quasianalytic side, Corollary 4.9 gives the signed-parity form. Every choice of (c,e 1,...,e m) is realized by the polynomial c m∏ j=1 ℓj(x)ej. These 2m+1 restrictions are distinct. Indeed, for eachHj choose a generic point ofHj outside all other hyperplanes. The adjacent chamber pair at that point differs only in the sign ofℓj, so its label ratio recoversej; one chamber then recoversc. On the non-quasianalytic side, letψ be the selector from Lemma 5.1. For each chamberC, let sj(C) = sgn(ℓj|C) and define ΨC(x) = m∏ j=1 ψ ( sj(C)ℓj(x) ) . 10 PAGE 11 Table 1: The classification in the common notation of this paper. All exact realizability counts refer to the full normalized Roumieu classesE{M}. Setting Mechanism Sign laws Smooth germ with a nonzero jet at the common orthant corner Leading homogeneous Taylor polynomial and Lemma 3.1 Affine parity only CN function on an arrangement with nonzero N-jet at every wall point Local leading terms plus parity propagation Signed parity only QuasianalyticE{M} on orthants Taylor injectivity excludes the flatness re- quired by Theorem 4.1 Exactly 2n+1 affine Boolean laws Non-quasianalytic E{M} on orthants One-sided flat selector All 22n Boolean laws QuasianalyticE{M} on m distinct hyperplanes No flat points plus Theorem 4.5 Exactly 2m+1 signed-parity laws Non-quasianalytic E{M} on an arrangement withr chambers Chamber selectors All 2r chamber labelings Distinct chambers have distinct sign vectors: a nonempty fixed sign cell is an intersection of open half-spaces and is therefore convex and connected. Hence ifx lies in a chamberD, thenΨD(x)> 0 and ΨC(x) = 0 for everyC̸=D. Given any labelingh : Ch(A)→{± 1}, set Fh(x) = ∑ C∈Ch(A) h(C)ΨC(x). Elementary closure givesFh∈E {M}(Rn), and on chamberD the sum reduces toh(D)ΨD(x). ThusFh has the prescribed sign and is nonzero off the arrangement. Example 6.2(Three concurrent lines). Take three distinct lines through the origin inR2, for example x = 0, y = 0, x −y = 0. There arer = 6 chambers. A full quasianalytic normalized Roumieu classE{M} therefore realizes 23+1 = 16 signed-parity chamber laws, whereas a full non-quasianalytic such class realizes all 26 = 64 labelings. Remark 6.3 (When the counts coincide). Each added distinct hyperplane splits at least one existing chamber, sor≥ m + 1. The numerical gap is strict exactly whenr > m+ 1. For m parallel hyperplanes one hasr =m + 1, so the two counts coincide. Thus the quasianalytic/non- quasianalytic distinction is structural, but its cardinality shadow depends on the arrangement. 11 PAGE 12 7 Margins and exclusion of the coordinate walls The qualitative classification imposes correctness arbitrarily close to every wall. Two quantitative consequences clarify this requirement. Throughout this sectionh :{−1, +1}n→{− 1, +1}, with positive meaning true, andn is fixed. A non-affine law means one not of the formc∏ i∈Ssi. 7.1 Corner margins under derivative bounds Proposition 7.1(A normalized corner estimate). Letk≥ 0, 0<α ≤ 1, and letF∈Ck,α(BR(0)) realize a non-affine orthant law. Suppose thatDkF has Hölder seminorm at mostH, using the Euclidean multilinear operator norm (absolute value fork = 0). Then its entirek-jet at zero vanishes and |F (x)|≤ H Γ(α + 1) Γ(k +α + 1)∥x∥k+α (x∈BR(0)). (10) In particular, if√nδ 0 satisfy ∥DjF (x)∥op≤ABjj!Mj (j≥ 0), Taylor remainder at every order gives |F (x)|≤ AhM(B∥x∥), h M(t) = inf j≥0 Mjtj. (11) Forj = 0 use the amplitude bound; forj≥ 1 the factorial cancels in the remainder. This is the usual associated-function estimate in the factorial-out convention; compare [14, Section 7]. Coordinate partial-derivative bounds of the form(2), with fixed constants on the same ball (or on a fixed compact subball), imply the corresponding operator bounds after replacingB byB√n. In the Gevrey caseMj = (j!)s−1,s> 1, optimization inj giveshM(t)≤C exp(−ct−1/(s−1)) for small t> 0. A matching optimal-cutoff assertion for general weights requires additional assumptions and is not used here. These estimates concern spatial derivatives and do not assert failure of an optimization or learning procedure. 12 PAGE 13 7.2 Polynomial signs at positive distance from the walls Put Qn = [−1, 1]n, D δ,n ={x∈Qn :|xi|≥ δ for alli}, 0<δ < 1. Let dh(δ) be the least total degree of a real polynomialP satisfying h(sgnx)P (x)> 0 ( x∈Dδ,n). There is no amplitude or coefficient bound in this definition. For fixed0<γ < 1, letdbd h,γ(δ) be the least total degree subject instead to ∥P∥Qn,∞≤ 1, h (sgnx)P (x)≥γ (x∈Dδ,n). (12) The amplitude bound covers the entire cube, including its excluded strips. Theorem 7.3(Fixed-dimensional degree comparison). For fixedn≥ 1 the following statements hold. (i) Ifh(s) =c∏ i∈Ssi, thendh(δ) =|S| for every 0<δ < 1. (ii) Ifh is non-affine, thendh(δ) = Θh(log(1/δ)) as δ↓ 0. (iii) Ifh is nonconstant and0<γ < 1 is fixed, then dbd h,γ(δ) = Θh,γ(1/δ) ( δ↓ 0). In fact,dbd h,γ(δ)≥γ √ 1−δ2/δ. Constants may depend on the fixed dimension, law, and margin. No uniform growing-arity, coefficient-size, or evaluation-cost assertion is intended. Proof. Parity. The polynomial c∏ i∈Sxi proves the upper bound. On Boolean vertices, fix coordinates outsideS. Every polynomial of degree less than|S| has zero correlation with parity, by expanding into monomials and averaging over the remaining independent signs. Strict sign agreement would give positive correlation. This proves (i), including constants withS = ∅. A logarithmic upper bound for every law.Let m =⌈log2(1/δ)⌉ and choose an integerL≥ 1. Define p(t) = m∏ j=0 (t + 2−j)2L, a (t) =p(t)−p(−t), q (t) =p(t) +p(−t), r (t) = a(t) q(t). The two nonnegative polynomialsp(t) and p(−t) have disjoint real zero sets, soq >0 and|r|≤ 1 on R. For 2−m≤t≤ 1, some dyadic nodeu = 2−j satisfies u≤t≤ 2u. Therefore 0≤ p(−t) p(t) = m∏ j=0 ⏐⏐⏐⏐⏐ 2−j−t 2−j +t ⏐⏐⏐⏐⏐ 2L ≤ 3−2L, |r(t)− 1|≤ 2 3−2L. Oddness gives the negative-interval estimate. This is a dyadic rational construction of Newman type; see also [17, Theorem 2.4]. Foru∈ [−1, 1]n, define the multilinear expectation extension Hh(u) = ∑ s∈{−1,+1}n h(s) n∏ i=1 1 +siui 2 . 13 PAGE 14 It is bounded by one. If|ui−si|≤ ε for every coordinate, regardHh(u) as the expectation ofh under independent signs with meansui. Each sign differs fromsi with probability at mostε/2, so the union bound gives h(s)Hh(u)≥ 1−nε. (13) Choose L with 2n3−2L < 1. Substitution of r(xi) and multiplication by∏ iq(xi) > 0 give a polynomial of total degree at most2Ln(m+1), strictly correct onDδ,n. Hencedh(δ) =On(log(1/δ)). The non-affine lower bound.Every non-affine Boolean law has a non-affine two-dimensional face: otherwise every second Boolean difference of itsF2-valued truth function would vanish identically, forcing its multilinear algebraic normal form to have degree at most one. Fix the other coordinates at the corresponding±1 values. Each of the eight non-affine binary laws is AND after input sign flips and possibly output negation, all of which preserve degree and the guarded square. Choose an odd integerN≤ 1/δ with N≍ 1/δ. In a polynomial for binary AND substitute x = 1 N N∑ i=1 zi, y = 1 N N∑ i=1 wi, z i,wi∈{− 1, +1}. Each magnitude is at least 1/N ≥ δ. Thus the substituted polynomial sign-represents the conjunction of two independentN-variable majorities on a finite Boolean cube, without increasing degree. Sherstov’s lower bound [17, Theorem 1.10] isΩ(logN). His convention uses−1 for true; input and output sign changes align the conventions. This proves (ii). Bounded margin: lower bound.A nonconstant law changes along some Boolean edge. Fix the other coordinates at that edge’s±1 values. The resulting univariate slicep is bounded by one on [−1, 1] and takes opposite values of magnitude at leastγ at±δ. The mean value theorem gives ξ∈ (−δ,δ ) with|p′(ξ)|≥ γ/δ. Bernstein’s inequality [18, Theorem 24.1] gives, withd = degP, γ δ≤|p′(ξ)|≤ d√ 1−ξ2≤ d√ 1−δ2. Bounded margin: upper bound.The clipped rampvδ(t) = max(−1, min(1,t/δ )) is 1/δ-Lipschitz. Jackson’s inequality yields a polynomialu of degree at mostk with ∥u−vδ∥[−1,1],∞≤ C kδ. For completeness, the trigonometric form [18, Theorem 21.1] applies to the even periodic function vδ(cosθ). Symmetrizing its approximant preserves the error and leaves only cosine terms;cos(jθ) = Tj(cosθ) converts it to an algebraic polynomial of the same degree. Choose a fixed errorη >0 and k≥C/(ηδ). The polynomialr =u/(1 +η) is bounded by one on the whole interval and differs from sgnt by at mostε = 2η/(1 +η) when|t|≥ δ. Chooseη small enough thatnε≤ 1−γ. Then Hh(r(x1),...,r (xn)) satisfies (12), by(13), and has total degree at mostnk =On,γ(1/δ). This proves (iii). Remark 7.4 (Interpretation and endpoints). Positive denominator clearing preserves signs but need not preserve a fixed margin after normalizing the polynomial onQn. This explains why (ii) and (iii) concern different resources. Constants have degree zero in both models; whenγ = 1, no nonconstant law satisfies(12), since the polynomial would equal a constant on an open box and hence everywhere. The theorem is a synthesis of established approximation inputs. It does not follow from qualitative finite-jet rigidity alone, nor does it apply that rigidity to a polynomial required to be correct only onDδ,n. 14 PAGE 15 8 Scope, counterexamples, and relation to nearby theory 8.1 Real zero support is weaker than a normal-crossing divisor If one already has a factorization F =u ∏ i xai i with a nonvanishing unitu, then the sign law is immediately the sign ofu times a parity product of coordinate signs. Normal-crossing and monomialization theories therefore provide important neighboring context; see [11, 10]. They should not, however, be substituted for the weaker hypothesis used in Theorems 4.5 and 6.1. Indeed, F (x,y ) =xy(x2 +y2) (14) is real analytic and has real zero set exactly equal to the union of the two coordinate axes. Its sign off those axes issgn(xy). But after extracting the generic first-order factorsx and y, the residual factor x2 +y2 still vanishes at the origin and is not a unit. Thus set-theoretic real zero support on an arrangement does not imply a unit-times-monomial factorization in the original coordinates. This is why the arrangement theorem above is stated and proved directly at the sign level. Resolution and monomialization results involve stronger divisor hypotheses and/or transformations of source and target; they remain relevant background, not a replacement for the present argument. 8.2 A common corner is essential for local orthant rigidity Theconnectednessofadomainalonedoesnotreplacethecommon-cornerhypothesisinTheorem4.1. On the punctured planeU = R2\{ 0}, define F (x,y ) =x +y + √ x2 +y2. (15) This function is real analytic onU. It is positive in the positive-positive quadrant and in both mixed-sign quadrants, while it is negative in the negative-negative quadrant. In a mixed-sign quadrant, √ x2 +y2>|x +y| because xy <0, whereas in the negative-negative quadrant √ x2 +y2<|x +y| =−x−y. Thus (15) realizes OR on the four quadrants of the punctured plane. There is no contradiction with Theorem 4.1: the origin, at which the finite-jet obstruction would live, has been removed. 8.3 Classical R-functions and finite differentiability The preceding root formula is the classicalR-disjunction. Changing the plus sign to a minus sign gives anR-conjunction; see [16, Section 3.3, equations (3.4)–(3.5)]. In particular, for any integer k≥ 0, putr = (x2 +y2)1/2 and Fk(x,y ) =rk(x +y−r), F k(0, 0) = 0. 15 PAGE 16 This realizes AND on the open quadrants. Its smooth angular factor and positive homogeneity of degreek + 1 giveDjFk =O(rk+1−j) for j≤k; these derivatives extend by zero at the origin. The kth derivative is Lipschitz: for pairs at distance comparable to their radius use itsO(r) size bound, and for closer pairs integrate its bounded derivative along a segment away from zero. Thus Fk∈ Ck,1 and itsk-jet vanishes. It is notCk+1: that regularity and homogeneity would make it its homogeneous Taylor polynomial of degreek + 1, contrary to the non-affine sign law. On (t,t ), t> 0, its value is2k/2(2− √ 2)tk+1. Hence the exponentk + 1 in Proposition 7.1 forα = 1 cannot be increased in general. The construction is classical; the estimate states its relation to the finite-order obstruction. 8.4 Analytic models and restricted domains A finite composition of real-analytic maps is real analytic on its domain. If a scalar output realizes an orthant law on a full neighborhood of the common corner, Corollary 4.4 therefore forces affine parity. The hypothesis concerns independently variable coordinates throughout that neighborhood; it does not cover merely fitting finitely many Boolean representatives or a restricted set of reachable inputs. Even an analytic formula can realize AND on everyDδ,2 when it depends onδ. For example, Fδ(x,y ) =x +y− √ x2 +y2 +δ2 is globally real analytic. In the positive-positive guarded quadrant it is increasing in each coordinate, so its minimum is(2− √ 3)δ >0. If x≤− δ, thenFδ(x,y ) < x+y−|y|≤− δ, and the other negative-input case is symmetric. At the common cornerFδ(0, 0) =−δ, so it fails the positive- positive law sufficiently near zero. This example imposes no global amplitude normalization and does not contradict Theorem 7.3. A nonsmooth alternative ismin(x,y ) =x− max(0,x−y), which realizes AND exactly on every open quadrant. 8.5 Smoothness alone does not imply rigidity The selector construction in(9) givesC∞ realizations of every Boolean law in the non-quasianalytic case. For a non-affine law, Theorem 4.1 says that such a realization must be flat at the origin; the construction is consistent with exactly that requirement. The mathematical boundary is therefore not “smooth versus nonsmooth,” but rather the presence or absence of a nonzero finite jet at the relevant common boundary. 8.6 Signatures and chamber functions The factor-parity mechanism in Lemma 3.1 is classical antecedent material. Broglia–Tognoli’s Proposition 2.4 records how parity of maximal factor multiplicities controls the induced signa- ture [12, Proposition 2.4], while the subsequent signature and sign-distribution literature studies broader realization and approximation questions [5, 1, 2, 4, 3]. A distinct combinatorial tradition studies all functions that are constant on arrangement chambers. Varchenko and Gelfand introduced the algebra generated by hyperplane Heaviside functions [7]; see also [8, 6]. The present analytic question is narrower: regularity constraints select a small family of{±1}-valued chamber functions in the quasianalytic regime, while the non-quasianalytic regime permits them all. 16 PAGE 17 8.7 Established machinery versus the sign classification Stronger division, quasianalytic local-ring, resolution, and monomialization frameworks are avail- able [15, 11, 9, 10]. The proofs here deliberately avoid that machinery: the rigidity input is only the first nonzero Taylor polynomial, and the realization input is only the classical Denjoy–Carleman criterion together with cutoffs or positive-flat functions. This is the sense in which the note is self-contained without claiming foundational priority. 9 Conclusion A first nonzero Taylor term provides a compact bridge from local regularity to global chamber signs. In finite order it yields explicitCd orthant andCN arrangement obstructions; in the smooth setting the same mechanism gives an elementary proof of the established signature-flatness consequence for affine arrangements. Combined with the Denjoy–Carleman criterion, the argument gives an exact realization classification: form distinct hyperplanes withr chambers, the full normalized Roumieu class realizes exactly2m+1 signed parity laws in the quasianalytic case and all2r chamber labelings in the non-quasianalytic case. With fixed derivative bounds, the same vanishing jets control how rapidly an exact non-affine score approaches zero at the corner. Excluding wall strips changes the polynomial problem: at fixed dimension non-affine sign laws require logarithmic degree in the inverse width, whereas a fixed margin under a whole-cube amplitude bound requires reciprocal-width degree for every nonconstant law. The complete reductions make these distinctions explicit. 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