PRH | Observ | 2.2 • Breaking the Multiplicative Speed Limit in Sieve Dynamics

Breaking the Multiplicative Speed Limit in Sieve Dynamics Level-0 Time Smoothing and an RH-Sensitive Diagnostic Aleksandar Periˇsi´c August 2025 Abstract We give a self-contained, elementary formulation of multiplicative causality for sieve dynamics, Π>(L+r)P(Sp) Π≤L= 0, where Sp denotes “multiply by p ,” P is any polynomial in the {Sp} , the level is log n , and r is the total log-cost Pkplog p of a monomial. We then introduce a level-0 (critical-line) time-smoothing filter F0 acting as a Gaussian convolution in log-time (a Fourier multiplier in the Mellin–Fourier variable). Such F0 is time-nonlocal and does not commute with level projectors, so inserting it inside prime steps ( S′ p := F0Sp ) produces sliced leakage across the classical boundary L + r . Finally, we separate two notions of causality (engine vs. observational) and state an RH-sensitive diagnostic: persistent, high-confidence early predictions in a purely observational scheme signal RH failure; boundedness is consistent with RH. 1 Prerequisites (brief) Riemann zeta and RH. ζ ( s ) = Pn≥1n−s ( ℜs > 1), meromorphic continuation, functional equation; nontrivial zeros lie in 0 <ℜs < 1. The Riemann Hypothesis (RH) asserts all nontrivial zeros satisfy ℜs=1 2. Infinite multiplicative Eratosthenes’ sieve (engine intuition). (We use the multiplicative version—not the additive “mark multiples” pass.) Work on the prime–exponent lattice. Start from the unit 1 and process primes in increasing order p = 2 , 3 , 5 , . . . . At the p –step: (i) mark all prime powers pk ( k≥ 1); (ii) close under multiplication with everything already marked (mark every n·pk with n using only primes < p ). After the p –step the marked set is {n : pmin ( n ) ≤p} ; the arrival level of n is log pmin ( n ). Each use of p raises level by log p , so the sieve obeys the no-teleporting law in log-level. Bird’s-eye view (qualitative contrast). Fix N, M ∈N and start from L = 0. In the classical multiplicative sieve that uses only the first N primes with at most M copies of each, the flagged integers are exactly those whose prime factors lie in { 2 , . . . , pN} with exponents ≤M ; in particular nothing appears above Cmax =Y p≤pN pM. By contrast, if we insert a level-0 time-smoothing filter F0 inside each step (so S′ p := F0Sp ), then—after normalizing detector strength—for any fixed sensitivity 0 < τ < 1 one sees a detectable signal from some composites in arbitrarily thin bands just above Cmax , i.e. flagged early, before their least prime has entered the engine. As N and M grow, these premature detections persist and strengthen. This “superluminal” leakage is a purely operator-kinematic effect of time nonlocality; a distinct observational diagnostic links persistent early signals to RH failure. 1 2 Analytic setup and operator dictionary Log-time picture. Let t = log x∈R and Ht := L2 ( R, dt ). The (unitary) Fourier transform on Htis (Ff)(ξ) = ZR f(t)e−2πitξ dt, ξ ∈R. The translation generator is N:= id dt (self-adjoint on the standard domain). Level projectors on Ht .Write Π ≤L for multiplication by 1 (−∞,L] ( t ) on Ht , and Π >L := I−Π≤L. For a Borel B⊂R, set ΠB=1B(t). Embedding of integers. Embed |n⟩ as a localized wave-packet at t = log n ; fix a Schwartz window gwith Rg= 1, and set Jσ|n⟩:= gσ(· − log n), gσ(t) = σ−1g(t/σ), so Jσ : ℓ2 ( N ) →Ht is bounded and J∗ σ Π [a,b]Jσ converges to the discrete level projector as σ↓ 0 (our sliced statements are stable under this regularization). Prime steps and cost (as shifts on Ht ). On Ht , ( Spf )( t ) = f ( t−log p ) is a shift. For a monomial M = QpSkp p define cost ( M ) = Ppkplog p and for P = PjcjMj put cost ( P ) = maxjcost(Mj). Mellin as Fourier on the critical line. With the standard half-density normalization, (MF)(1 2+iξ) = Z∞ 0 F(x)x(1 2+iξ)−1dx =ZR f(t)e−2πitξ dt with f(t) = et/2F(et). Thus Fourier multipliers in ξcorrespond to time-convolutions in t. 3 Level space, prime steps, and causality Let K=ℓ2(N≥1) with orthonormal basis {|n⟩}n≥1and level operator L |n⟩= (log n)|n⟩,Π≤L: projection onto span{|n⟩: log n≤L},Π>L := I−Π≤L. For each prime p, define the prime step Sp|n⟩=|pn⟩(so LSp=Sp(L+ log p)). Amonomial is M=QpSkp pwith cost cost(M) = X p kplog p= logY p pkp, and for P(Sp) = PjcjMj, set cost(P) = maxjcost(Mj). Proposition 3.1 (Prime causality (PC)).For any polynomial P ( Sp )and any L∈R , with r= cost(P), Π>(L+r)P(Sp) Π≤L= 0.(1) Interpretation: starting from numbers ≤eL , after applying any finite recipe of prime multiplications of total cost r, nothing can appear above eL+r. Proof. It suffices to check monomials M , for which LM = M ( L + r ). Hence M Π ≤L has support ≤L+r, and Π>(L+r)kills it. Finite sums preserve the bound. 2 Borel slice of levels (on K ). For any Borel B⊂R , the spectral (Borel) projector is ΠB:= 1B(L). As a special case, for an interval (a, b] we have Π(a,b]= Π≤b−Π≤a. 4 Level-0 time smoothing and two causalities Define the level-0 filter F0by F0:= F−1 mε(ξ)·F, mε(ξ) = e−2π2ε2ξ2.(2) Then F0 acts on Ht as a convolution by a real two-sided Gaussian kernel kε∈L1 ( R ) with kε(u)>0 for every u∈R: (F0f)(t)=(kε∗f)(t) = ZR kε(t−u)f(u)du. Lemma 4.1 (Schur bound and one-sided norm).If F is convolution by k∈L1 ( R )on Ht , then  Π>L FΠ≤L ≤Z∞ 0 |k(u)|du for all L∈R. In particular m+(F) := supL∥Π>LFΠ≤L∥<∞. Proof. For f supported in ( −∞, L ] and t > L , ( Ff )( t ) = RL −∞ k ( t−u ) f ( u ) du with t−u > 0. Schur’s test gives the bound by R∞ 0|k(u)|du. Lemma 4.2 (Perfect OC ⇒ time–local).Let F be a bounded operator on Ht . If for every L∈R, Π>L FΠ≤L= 0 and Π≤LFΠ>L = 0, then Fis multiplication by some ϕ∈L∞(R); i.e., (Ff)(t) = ϕ(t)f(t). Proof. The relations say F preserves both halves of every cut ( −∞, L ] and ( L, ∞ ), so F lies in the commutant of the von Neumann algebra generated by { Π ≤L : L∈R} , which is exactly the algebra of multiplication operators L∞(R). Remark 4.3 (What preserves the bound).Time multipliers Mϕ commute with all level projectors Π ≤L and preserve the hard boundary. Fourier multipliers f ( N )(hence F0 ) commute with the shifts Sp but do not commute with Π ≤L ; they can therefore create one-sided mass across level cuts, quantified by m+(f(N)) as in Lemma 4.1. Engine vs. observational causality. We distinguish: •Engine (multiplicative) causality (PC): Proposition 3.1. •Observational causality (OC): a readout Ris causal if Π>L RΠ≤L= 0 for all L. 5 Level-0 steps and Borel-sliced leakage Define the level-0 prime step by “multiply then smooth”: S′ p:= F0Sp.(3) Proposition 5.1 (Borel-sliced leakage for level-0 steps).Let F0be as in (2). 1. For every prime p and every Borel set B⊂R with nonempty interior, there exists L∈R such that  ΠBS′ pΠ≤L >0. 3 2. More generally, for every Borel set B with nonempty interior there exist a nonzero polynomial Pin the {S′ p}and a level L∈Rsuch that  ΠBP(S′ p) Π≤L >0. Idea. F0 has kernel kε with some u > 0 where kε ( u ) > 0. Starting with |n⟩ ( log n≤L ), Sp shifts the level by log p , then F0 smears by kε , creating amplitude at log ( pn ) + u . Picking L so that log ( pn ) + u∈int ( B ) yields (1). Products add log-costs; finite sums keep nontrivial mass in B. Lemma 5.2 (Quantitative one-slice leakage).Assume kε≥ 0and RRkε = 1. Fix ∆ > 0. For every prime pand every η > 0there exists Lsuch that  Π(L+log p+η, L+log p+η+∆] F0SpΠ≤L ≥Zη+∆ η kε(u)du. Finite-stage (budgeted) multiplicative sieve. Fix a starting cap eLand a finite budget B=X j log pj= log M, where M is the product of the primes used so far (counting multiplicities). For any finite recipe of cost ≤B, Π>(L+B)(sieve after budget B) Π≤L= 0, i.e. at that finite stage nothing can appear above eL+B. Inductive limit (“projecting to infinity”). The infinite sieve is the inductive limit of these finite stages; the speed bound holds at every finite budget B . Any genuine “superluminal” effect must already occur at a finite B . The unfiltered engine has none; with S′ p := F0Sp , Borel-sliced leakage already appears at finite B(Prop. 5.1), hence persists in the limit. 6 Eratosthenes expectations and an RH-sensitive diagnostic Consider the smoothed prime readout Sa,σ(t) = X n≥1 Λ(n)e−(log n−t)2 2σ2eia log n, where Λ is the von Mangoldt function. A standard explicit-formula heuristic gives Sa,σ(t) = main term + X ρb ϕσa+i(1 2−ρ)e(ρ−1 2)t+ tiny tails, with ρ over nontrivial zeros and b ϕσ a rapidly decaying factor from the Gaussian window. Hence: • If RH holds, each zero contributes a bounded oscillation (no e(β−1 2)t growth). Appropriate normalizations make supt|e−t/2Sa,σ(t)|<∞. • If RH fails (some ρ = β + iγ with β > 1 2 ), choosing a to phase-lock that zero yields a term growing like e(β−1 2)t ; thus for any fixed threshold τ > 0, there are arbitrarily large t with |e−t/2Sa,σ(t)|> τ. 4 Proposition 6.1 (RH-sensitive early-flag diagnostic (observational)).Fix σ > 0and define the early-flag score for nby EFa,σ(n) := sup t<log pmin(n)e−t/2Sa,σ(t)e−(log n−t)2 2σ2. Then: • If RH holds, supa∈RsupnEFa,σ ( n ) <∞ (in particular, for any large enough δ > 0, {n: EFa,σ(n)> δ}=∅). •If RH fails, then for every δ > 0there exists awith #{n: EFa,σ(n)> δ}=∞. Sketch. Insert the explicit-formula decomposition and use the bounds above; the Gaussian window localizes tnear log n. Theorem 6.2 (Engine PC, level-0 leakage, and RH signal).Let F0 be the level-0 Gaussian time-smoothing filter (2), and define S′ p:= F0Sp. 1. (Engine causality) For the usual steps Sp , Proposition 3.1 holds for all polynomials P and levels L. 2. (Borel-sliced leakage under level-0 steps) For any Borel set B⊂R with nonempty interior, there exist a polynomial Pin the {S′ p}and a level L∈Rsuch that  ΠBP(S′ p) Π≤L >0. In particular, one can realize arbitrarily thin slices immediately above the classical boundary by taking B= (L+ cost(P) + ε, L + cost(P) + ε+ ∆] after fixing Pand L. 3. (RH-sensitive diagnostic, engine unchanged) If we keep the engine as in (1) and apply time-smoothing (e.g. F0 )only to observational readouts, then Proposition 6.1 applies: unbounded/persistent early flags at fixed thresholds imply RH failure; boundedness is consistent with RH. Remark 6.3 (Why Gaussian and normalization).The Gaussian multiplier yields a strictly positive kε∈L1 and explicit one-sided bounds (Lemmas 4.1, 5.2). For sensitivity comparisons we use the intrinsic normalization e F0:= F0/m+(F0)with m+(F0)from Lemma 4.1. Remark 6.4 (Tone and scope).“Superluminal” here is a metaphor. Formally, it means: there exist Land a Borel slice B⊂(L+r, ∞)of arbitrarily small width such that ΠBAΠ≤L= 0, i.e. support appears beyond the cost bound L + r (equivalently, the inequality Π >(L+r)A Π ≤L = 0 fails). No physics is implied. The only moving parts are: (i) the multiplicative engine Sp (which obeys the bound), and (ii) a time–nonlocal filter F0 inserted inside a step (so S′ p = F0Sp ), which need not commute with level projectors and thus can leak. The RH link enters only via the observational diagnostic (Prop. 6.1); the engine-side statements are purely operator-kinematic. 7 A “multiplicative speed of light” and its violation under level-0 steps Speed-limit in log–level. Define the advance of an operator Aat level Lby AdvL(A) := sup{t−L: Π>t AΠ≤L= 0 } ∈ [0,∞]. For P(Sp) with cost r, c×:= 1 AdvLP(Sp)≤rfor all L∈R, equivalently Π>(L+r)P(Sp)Π≤L= 0 (Prop. 3.1). 5 Superluminal offset under level-0 steps. Let S′ p := F0Sp . Since F0 has a two-sided kernel with kε ( u ) > 0 for u > 0, for any monomial M′ ( S′ p ) of cost r and any η > 0 there exists L with Π>(L+r+η)M′(S′ p) Π≤L= 0 ⇐⇒ AdvLM′(S′ p) r>1. By linearity the same holds for some polynomial P(S′ p). Proposition 7.1 (Finite-stage superluminal flagging at any subunit sensitivity).Fix eL and a finite set P={p1, . . . , ps}with exponent cap M∈N. Let SP,M (S) = Y p∈P I+Sp+· · · +SM p, r =X p∈P Mlog p= log Y p∈P pM, Cmax =eL+r. Then: (PC) Finite-stage causality. Π>(L+r)SP,M (S) Π≤L= 0. (L0) Subunit-sensitivity leakage. Let e F0 := F0/m+ ( F0 )(Lemma 4.1). Put S♯ p := e F0Sp and SP,M (S♯) = Y p∈P I+S♯ p+· · · + (S♯ p)M. Then for every sensitivity τ∈ (0 , 1), every slice width ∆ > 0, and some ε > 0, there exists Lsuch that   Π( log Cmax+ε, log Cmax+ε+∆ ] SP,M (S♯) Π≤L  ≥τ. Comment. Every integer above Cmax requires at least one “forbidden” step (a new prime >max P or an extra exponent > M), so the signal is an early composite flag. Remark 7.2 (Cost yardstick).Classically, the cost per τ -level detection above the frontier is infinite (no hits at all); with level-0 leakage it becomes finite, ≈q/Yτ , where q is the scan cost per location and Yτis the empirical hit-rate in a thin band just beyond the cutoff. 8 Conclusion (plain words) •The (multiplicative) sieve engine obeys a strict speed limit in log–level: c×= 1 i.e. Π>(L+r)P(Sp) Π≤L= 0 ⇐⇒ AdvLP(Sp)≤r. No composite can be produced “earlier” than the accumulated prime–cost. • Which insertions break the bound. Time–local multipliers Mϕ preserve the hard boundary. Fourier multipliers f ( N ) (time-convolutions), such as the Gaussian F0 , need not commute with level projectors and thus can leak; there exist P , L , and arbitrarily thin Borel slices B⊂(L+ cost(P),∞) with ∥ΠBP(S′ p)Π≤L∥>0. • If you keep multiplication untouched and apply F0 only to the observational readout, the speed limit c× = 1 remains intact. In that purely observational regime, persistent high-confidence early signals are a clean diagnostic for RH failure; under RH, such early signals remain uniformly bounded. 6