PRH | Inter | 5.13 • Blurred Polynomials between Addition and Multiplication

Blurred Polynomials between Addition and Multiplication Two Lenses, One Switch, a Hopf View, and Soft Operations ⊞,⊠ Aleksandar Perišić October 2025 Abstract Addition is native to the Fourier/translation lens; multiplication is native to the Mellin/dilation lens. After the log change u = log x , the Mellin transform on the midline is a Fourier transform, yielding a Heisenberg tradeoff: the two lenses cannot be simultaneously sharp, forcing an explicit blur whenever we mix +and × . We formalize a “blurred polynomial calculus” that (i) regularizes each monomial xn in the multiplicative channel and (ii) performs additions after a single controlled channel switch. In parallel we adopt a bi–Hopf (Hade/Hide) operator picture that keeps channels separate. Most of this can be phrased as working with soft addition ⊞ and soft multiplication ⊠ , which encode the blur/switch budgets while recovering the classical laws in a sharp limit. We illustrate with quadratic and quartic polynomials and with the product of two polynomials. 1 Two channels and why blur is compulsory Let x > 0and set u = log x . With g ( u ) = eu/2f ( eu )and g∧ = Fg , the Mellin–Fourier bridge implies a Mellin–Heisenberg inequality σ2 xσ2 t≥1 4, with equality for log–Gaussians. A scale–neutral multiplicative blur is Gaussian in log x ; in Mellin it becomes multiplication by a Gaussian, i.e. uniform high–frequency damping [7, 9, 1]. 2 Blurs and a type–correct channel switch Definition 1 (Scale–respecting blurs).The multiplicative blur B× ε acts by a log–Gaussian kernel Kεon dx/x, (B× εf)(x) =Z∞ 0 Kε x uf(u)du u, Kε(r) = 1 √4πε exp−(log r)2 4ε, so that M [ B× εf ]( t ) = e−εt2Mf ( t ). The additive blur B+ τ is convolution with a positive, normalized approximate identity (e.g. Gaussian of variance τ ), with F [ B+ τf ]( ξ ) = e−τξ2Ff ( ξ ) [9, 10]. Definition 2 (Unitary bridge and named switches). (Uf)(u) = eu/2f(eu),(U−1g)(x) = x−1/2g(log x), so that FU =Mand MU−1=F. The explicit channel switches are Switch×→+:= B+ τ◦U◦B× ε,Switch+→× := B× ε◦U−1◦B+ τ, giving controlled passage between the multiplicative and additive channels [13, 12]. 1 Theorem 1 (No free lunch for mixing +and × ).For Gaussian (or Paley–Wiener) blurs with variances ε, τ, the switch defect on compacts obeys  Switch×→+−Id K≤CKΦ(ε, τ),Φ(ε, τ)↓0only if ετ → 0. Thus mixing the channels forces a positive blur budget consistent with Mellin–Heisenberg [13]. 3 A Hopf/Hade/Hide view: keeping channels separate Let G+= (R,+) and G×= (R>0,·). Package them bi–Hopf: H:= Hadd ⊕Hmult, with comultiplications ∆ +, ∆ × and a bridge Φ : Hmult →Hadd intertwining structure maps; scaling on Mellin corresponds to translation on the log side [ 14 , 12 ]. In a Hilbert model, the generators satisfy the ax+b commutator [Hide,Hade] = iHade, pinning the channel mismatch as a first–class operator relation [ 14 ]. See also [ 2 , 3 , 4 , 5 , 6 ] for Hopf structures behind polynomial operators. 4 Soft operations ⊞,⊠ “Soft” addition/multiplication encode the blur/switch semantics while remaining associative/commutative and recovering + ,× in a sharp limit. We fix parameters ( ε, τ )and define: a⊞b:= U−1B+ τUa+Ub, a⊠b:= B× εa·b, so that U(a⊞b)≈ Ua+Ub(additive lens, blurred), a ⊠b≈a·b(multiplicative lens, blurred). When a final readout on the additive side is desired, we use one switch: Read+[h]:=B+ τUh. Heuristically, ⊞ performs addition in the +channel with τ -blur, while ⊠ performs multiplication in the × channel with ε -blur. Both reduce to the classical laws as ε, τ → 0(subject to the Mellin–Heisenberg constraint) [10, 11, 7]. 5 Blurred polynomial calculus (standard notation → soft notation) We regard p ( x ) = PN n=0 anxn as a superposition of multiplicative exponentials in u . Define blurred monomials ϕn(u):=enu κε(u), pε(u) := N X n=0 anϕn(u). Standard route (keep classical notation). Act on fmultiplicatively, then switch once: Evalε,τ (p)[f] := Switch×→+ pε·f. 2 Soft-ops route (expose the multiplication): write p(x)⊠f(x) := a0⊞a1x⊞··· ⊞aNxN⊠f(x), evaluate in the multiplicative channel, and apply a single Read+ at the end. This makes explicit that polynomial multiplication with a signal is ⊠ -native, while coefficient addition is ⊞ -native [13, 14]. 6 Examples 6.1 Quadratic Let p(x)=a0+a1x+a2x2. Multiplicative lens: pε(u)=a0κε(u)+a1euκε(u)+a2e2uκε(u). Standard readout: Evalε,τ (p)[f] = B+ τUB× εpε·f. Soft readout (exposing operations): a0⊞a1x⊞a2x2⊠fRead+ −→ additive output (one switch). On the Mellin side, this multiplies Mf by a sum of Gaussian-shifted bumps, mirroring ⊠↔ dilation and ⊞↔translation [14, 13]. 6.2 Quartic For q(x) = P4 k=0 bkxk, qε(u) = 4 X k=0 bkekuκε(u),Evalε,τ (q)[f]=B+ τUB× εqε·f. Soft-ops version: b0⊞b1x⊞b2x2⊞b3x3⊞b4x4⊠fRead+ −→ output. Smaller ε sharpens scale localization but forces a larger τ if you plan to switch (Heisenberg tradeoff) [7, 9]. 6.3 Multiplying two polynomials (Hopf-first & soft-ops) Let p(x) = Pnanxn,r(x) = Pmcmxm. In the multiplicative lens, xn·xm=xn+m⇐⇒ enu ·emu =e(n+m)u. Hence (p·r)(x) = X k≥0X n+m=k ancmxk,(p·r)ε(u) = X k≥0X n+m=k ancmekuκε(u). Hopf-first workflow (no switch until the end): 1) Form pε, rεin the ×channel; multiply: pεrε. 2) Optionally reblur to the canonical κε. 3) Switch once with Switch×→+for additive readout. 3 Soft-ops wording: M⊞ |{z} coeff sum anxn⊠M⊞ |{z} coeff sum cmxmRead+ −→ output, where the internal multiplication of monomials is handled natively by ⊠ (one × -channel), and only the final presentation pays the switch budget [14, 13, 4]. 7 Nonzero switch error for p ( x )=5 x3 + 2 x2 + 8: three complementary views Fix blur parameters ε, τ > 0. Write the one–switch evaluation operator on the additive side as Evalε,τ (p)[f] := B+ τUB× εpε·f, pε(u)=5e3uκε(u)+2e2uκε(u)+8κε(u), and compare it to the sharp (no–blur, no–switch) additive–side evaluation Eval0,0 ( p )[ f ] = Up·f . Define the switch error (additive readout) for the cubic example by Errε,τ (p;f):=Evalε,τ (p)[f]−Eval0,0(p)[f] = (B+ τ−Id) Upf+B+ τU(B× ε−Id)pf.(1) Quantitative upper bound (band–limited case). Assume the Mellin spectrum of f is contained in |t| ≤ T . Then the spectrum of pf sits in |t| ≤ T + 3 (the largest degree). In Mellin and Fourier, [ B+ τh(ξ)=e−τξ2b h(ξ),M[B× εg](t) = e−εt2M[g](t). Thus, with C= 1 (unitary U, B+ τ) and T⋆:= T+ 3,  Errε,τ (p;f) L2(Ru)≤sup |ξ|≤T⋆1−e−τξ2 U(pf) 2+ sup |t|≤T⋆1−e−εt2 U(pf) 2 ≤1−e−τT2 ⋆+ 1 −e−εT2 ⋆ U(pf) 2.(2) Both attenuation factors are strictly positive for any ε, τ > 0and any T⋆>0. Strict nonvanishing for the cubic. If U ( pf )has any nonzero energy at a frequency ξ0 = 0 (true for all nontrivial f once multiplied by 5 x3 + 2 x2 + 8), then e−τξ2 0 = 1 and the first term in (1) is nonzero. Alternatively, if M ( pf )has any nonzero energy at t0 = 0 (again typical after the x2, x3factors), then e−εt2 0= 1 and the second term is nonzero. Hence: Errε,τ (5x3+ 2x2+ 8; f)≡ 0for all (ε, τ)∈(0,∞)2and nonzero f. (3) (i) Hade/Hide operator view Let Hade generate translations in u (additive lens) and Hide generate multiplicative flow (Mellin lens) with the ax+b commutator [ Hide,Hade ] = iHade [ 14 ]. Gaussian blurs realize heat semigroups, B+ τ=e−τHade2, B× ε=e−εHide2. The one–switch operator acting on pf may be written (up to the unitary shuttle by U) as Sε,τ := e−τHade2e−εHide2. By the Baker–Campbell–Hausdorff expansion, Sε,τ = exp−τHade2−εHide2+1 2ετ [−Hade2,−Hide2]+O(ε+τ)3.(4) 4 Since [ Hide,Hade ] = iHade  = 0, we have [ Hade2,Hide2 ]  = 0, so the mixed ετ -term in (4) is intrinsically nonzero. Therefore S ε,τ  = Id for all ε, τ > 0, and the evaluation of the cubic (which engages Hide through the x2, x3 factors) incurs a strictly nonzero switch error; cf. (3) . This makes precise, at the generator level, that one cannot null the final error unless both blurs vanish. (ii) Soft operations view (⊞,⊠) Define soft addition/multiplication by a⊞b:= U−1 B+ τ(Ua+Ub), a ⊠b:= B× ε(a·b). Then for the cubic, (8 ⊞2x2⊞5x3) | {z } coeff sum in the +lens ⊠fRead+ −→ B+ τUB× ε(8 + 2x2+ 5x3)f, and the soft error equals the decomposition in (1) . The sup–norm attenuations 1 −e−τξ2 and 1 −e−εt2 are strictly positive on any nontrivial frequency content introduced by x2, x3 ; hence the final error is nonzero for all ε, τ > 0. The explicit band–limited bound is exactly (2). (iii) Hopf bookkeeping view Work entirely in the multiplicative Hopf channel to form the product p⊠f : exponents add ( xn·xm = xn+m ), coefficients combine in the additive Hopf component, and we attach the common blur profile κε to each monomial shell. This algebraic stage incurs no cross–channel loss. However, the single switch needed for additive readout multiplies (Fourier side) by e−τξ2 and (Mellin side, already applied) by e−εt2 , so—even in this optimal “Hopf–first, one–switch” workflow—the same strictly positive attenuation appears, giving the nonvanishing error bound (2) and the strict inequality (3) . In short: Hopf bookkeeping postpones but cannot eliminate the switch loss for 5x3+ 2x2+ 8 unless ε=τ= 0 (the forbidden sharp limit) [14, 13]. 8 Why this understanding emerged only now One may ask why this unified view of addition and multiplication—and the need to treat them as connected through a Mellin bridge and an intrinsic blur—did not appear earlier, despite centuries of mathematical progress. The answer lies partly in how mathematics historically treated its most basic operations. (1) The legacy of discrete axiomatization Since antiquity, addition and multiplication were viewed as discrete and exact processes, extended from counting and measurement to reals and beyond by continuity and axioms. The foundational laws—existence of 0and 1, commutativity, associativity, and distributivity—were taken as primitive facts. Yet each of these can be derived once one acknowledges that multiplication is logarithmically connected to addition: log(ab) = log a+ log b, and conversely the Mellin transform formalizes this relationship as an operator bridge between the two. If the field admits an abstract “logarithm” (not necessarily real), then the Mellin–Fourier connection is inevitable, and so is the uncertainty principle. 5 (2) Why this link was historically hidden Mathematicians repeatedly encountered the consequences of this constraint—from analytic number theory to Fourier analysis—but always after the fact, as inequalities or asymptotic limits (Heisenberg, Hardy, Paley–Wiener). Each time, the principle was rediscovered as a technical obstacle rather than recognized as a structural necessity. We assumed that both +and × could be made perfectly sharp; yet the moment one connects them through logarithms, the uncertainty bound ετ ≥cbecomes unavoidable. Historically we worked around it rather than with it. (3) The hidden assumption of commutativity The statement 3 + 3 = 6 and 2 × 3 = 6 appears identical, but conceptually differs: 2 × 3means “three objects two times,” while 3 × 2means “two objects three times.” Declaring these equal requires an implicit logarithmic symmetry: both 2and 3are mapped to a common additive domain, effectively invoking log. In that sense, 2×3≡elog 2+log 3, and the distributive law follows from the very existence of this bridge. The moment multiplication is treated as “repeated addition,” commutativity is imposed by the hidden logarithm and we have already entered the Mellin framework. (4) From discrete repetition to continuous flow For small integers, the distinction is invisible: we can add and multiply as if they were symbolic shortcuts. But in abstract or continuous settings—polynomials, analytic functions, Hopf algebras— this implicit switch of lens (+to × ) incurs a measurable cost. The Heisenberg–Mellin uncertainty shows that we cannot know both the additive and multiplicative localizations perfectly. The “blur” simply acknowledges this cost up front and budgets it quantitatively instead of leaving it hidden inside an inequality discovered later. (5) A modern reconciliation Understanding multiplication as the Mellin shadow of addition, or equivalently through soft operations ⊞,⊠ , restores coherence. The soft formulation does not claim new algebraic laws; it makes explicit that every application of distributivity or commutativity already performs a hidden switch between lenses. The Hopf and Hade/Hide formalisms merely express this separation structurally: two channels coupled by a controlled blur. In this sense, the present framework is not a new arithmetic but a clarification of why the old one, despite centuries of success, has always carried an invisible uncertainty budget that we can now quantify and work with from the start. Acknowledgments and provenance. Soft operations ⊞,⊠ summarize the same blur/switch calculus and recover classical + ,× in sharp limits [ 10 , 11 ]. The Mellin/Fourier bridge and the one-switch policy underpin all examples [ 7 , 13 ]. The Hade/Hide and bi–Hopf framing explain why keeping channels separate is structurally natural [14, 12]. References [1] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. [2] M. E. Sweedler. Hopf Algebras. W. A. Benjamin, 1969. [3] C. Kassel. Quantum Groups. Springer, 1995. 6 [4] M. Aguiar and S. Mahajan. Monoidal Functors, Species and Hopf Algebras. CRM Monograph Series, AMS, 2010. [5] S. Roman. The Umbral Calculus. Academic Press, 1984. [6] K. Ebrahimi-Fard and L. Guo. Rota–Baxter Algebras and Hopf Algebras. In: Handbook of Algebra, Vol. 6, Elsevier, 2009, pp. 173–245. [7] A. Perišić. Soft-Addition and Soft-Multiplication and the Channel–Switch Error. Zenodo, 2025. [8] A. Perišić. Blur at the Finite–Infinite Interface. Zenodo, 2025. [9] A. Perišić. Galois Theory as Harmonic Analysis; Anabelian as Blurred Spectral Reconstruction. Zenodo, 2025. [10] A. Perišić. Blur as a Universal Principle: Number Theory, Probability, Dynamics. Zenodo, 2025. [11] A. Perišić. Riemann Hypothesis and the Multiplicative Choice. Zenodo, 2025. [12] A. Perišić. Multiplicative Closure of the World Zenodo, 2025. [13] A. Perišić. Additive–Multiplicative Bi–Hopf Object on R and R>0. Zenodo, 2025. [14] A. Perišić. A Gentle First Course in Adelic–Quantum Arithmetic. Zenodo, 2025. 7