PRH | Inter | 5.8 • Blur as a Unifying Spectral Lens for Factorization

Blur as a Unifying Spectral Lens for Factorization Classical Limits, Quantum Advantage, and Threshold Certificates Aleksandar Perišić December 2025 Abstract We formulate a single “blur” principle that models information extraction from arithmetic objects across both classical and quantum settings. The guiding assumption is descriptor completeness: given a blur family that covers all invariant angles of the object (group actions, symmetries, and spectral coordinates), any feature that survives blur is a legitimate, modelstable invariant and any feature that vanishes carries no usable information. Operationally, blur acts as a controlled causality-breaking operator: it deliberately mixes data nonlocally along the chosen coordinate (time, logarithmic scale, etc.), while faithful deblurring is only allowed within stability and uncertainty (commutator) bounds that forbid the creation of new invariants. This lens unifies (i) the additive–multiplicative “Hade/Hide” representation and the associated prime-band projector in multiplicative log-frequency, (ii) the structural “primecomb” readout that turns factorization into spectral spikes, (iii) lower bounds on classical factoring attempts pursued purely via blur/deblur, and (iv) Shor’s algorithm viewed as time-blur spectroscopy, where a unitary deblur (the QFT) concentrates a periodic comb written into a coherent time register. Two conclusions follow. First, under standard classical resources, a blur-only route to factoring a generic semiprime requires exponential resolution (and therefore steps), so factoring remains as safe as believed: any improvement that would beat the main asymptotic term would require an operator that violates the model’s causal/uncertainty bounds. In our vocabulary, that would be true causality breaking, and blur precisely characterizes what happens as one pushes toward that limit without ever creating new invariant signal. Second, the same lens explains Shor’s speedup: quantum coherence creates an exponential number of time-channels that are deblurred unitarily in polynomial time—achieving very narrow effective blur without violating causality, because interference and the QFT operate within a different (unitary) resource envelope. Finally, we formalize Blur Threshold Certificates—operational statements of the form “if no invariant exceeds a threshold under allowed blur, then the attack surface is bounded”—and show how they provide security guarantees that remain valid even if additional (but modelconsistent) features are uncovered later. Contents 1 Introduction 2 2 Blur mechanics and the Hade/Hide frame 3 2.1 Symmetry, generators, and blur families ....................... 3 2.2 Prime-band projector and multiplicative frequency ................. 3 3 Factorization as spectral readout 3 3.1 A structural readout .................................. 3 3.2 What blur does and does not do ........................... 4 4 Lower bounds for classical blur-only factoring 4 1 5 Shor as time-blur spectroscopy 5 5.1 The dictionary ..................................... 5 5.2 Pipeline ......................................... 5 5.3 Why the speedup fits the blur story ......................... 5 5.4 Complexity accounting (standard) .......................... 6 6 Blur Threshold Certificates (BTCs) 6 7 Security implications and limits 7 7.1 Classical ......................................... 7 7.2 Quantum ........................................ 7 7.3 “If the unknown is below a threshold, you are safe” ................. 7 8 A compact worked example 8 9 Conclusions 8 A Kernels and apertures 8 B Continued fractions decoding condition 8 C Resource bookkeeping (quick reference) 8 D Energetics for an “Oracle-Grade” Universe 8 D.1 (A) Realistic 32–100 logical qubits within our universe ............... 9 D.2 (B) Fundamental floors: Landauer, Margolus–Levitin, Bekenstein ......... 9 D.3 (C) Hypothetical “non-blur” oracle universe ..................... 10 1 Introduction “Blur” is a controlled smoothing/deblurring protocol imposed on an object and its descriptors. It is not a trick to create information: a faithful blur semigroup attenuates or merges features but cannot invent invariant content. Conversely, a stable deblur acts as a spectral sharpening that concentrates already-present features while respecting uncertainty/commutator constraints from the underlying symmetries. This paper develops a coherent story in which: 1. Factorization is framed as a spectral readout on the multiplicative log-frequency axis (§2.2), where prime powers manifest as discrete spectral lines (a “prime comb”). 2. Aclassical blur-only attack that tries to separate the two dominant lines at log p and log q for N = pq faces an intrinsic resolution barrier: to split those lines one needs aperture (or bandwidth) exponential in the bit-length of N(§4). 3. Shor’s algorithm is the same blur protocol but run on a quantum time-channel: a uniform time blur is imprinted with a hidden period and unitarily deblurred by the QFT to yield narrow Dirichlet/Fejér lobes at multiples of M/r ; continued fractions finishes the readout (§5). 4. ABlur Threshold Certificate (BTC) states that if a chosen family of blur-compatible observables never exceeds a security threshold, then no adversary obeying the same resource envelope can extract enough invariant signal to break the system (§6). 2 Throughout, “steps” mean probe operations consistent with the chosen blur interface (classical: smoothed low-degree statistics; quantum: unitaries/measurements bounded by available qubits and gate depth). The narrative is deliberately descriptor-agnostic: once your descriptor is complete (covers all angles), blur gives a complete imagery of the system up to your current knowledge. Anything not resolved under admissible blur carries no usable information energy within that interface. Standing premise. All our claims are orthogonal: they are about a blur-only route to factoring and about quantum period-finding viewed through blur. 2 Blur mechanics and the Hade/Hide frame 2.1 Symmetry, generators, and blur families Let X be a descriptor space for the object (e.g., a function of an additive variable u that encodes multiplicative frequency). A blur family {Bτ}τ≥0 is a completely positive contraction semigroup on Xwith generator G≥0, Bτ=e−τG, B0= Id,d dτ Bττ=0 =−G. Two commuting actions encode the geometry: •Hade (additive/wave): translation Taf(u) = f(u−a). •Hide (multiplicative/dilation): (Dλf)(u) = f(u−log λ). These generate the Lie algebra of the affine group (ax +b) with the commutator [log D, Ta] = a Ta,(1) an uncertainty principle: simultaneously sharpening in u and its Fourier dual ω is limited. A well-conditioned deblur Sτ is a (regularized) pseudo-inverse of Bτ obeying stability bounds compatible with (1). 2.2 Prime-band projector and multiplicative frequency On the multiplicative side, primes and their powers occupy discrete frequencies on the ω-axis: Ω = {klog p:pprime, k ∈N}. Formally, a prime-band projector PΩ keeps only the spectral mass on Ω. Conceptually, PΩ is the arithmetic counterpart of frequency selection in signal processing: it does not create atoms; it discards everything off the prime comb. 3 Factorization as spectral readout 3.1 A structural readout Define, for σ > 0, the Dirichlet observable Dn(σ) = X d|n Λ(d)d−σ=X p|n vp(n) X j=1 (log p)p−jσ,(2) 3 where Λis von Mangoldt and vp ( n )the exponent of p in n . Smearing with an even window f of width τon the u= log d-axis gives Gn,σ,τ (u) = X d|n Λ(d)d−σf(u−log d).(3) The Fourier transform in u places spectral mass at ω = klog p with weights reflecting vp ( n ). In other words, factorization is a spectral readout: PΩb Gn,σ,τ is a blurred prime comb whose line positions are {log p}and line multipliers capture exponents. Key caveat (structural vs. algorithmic). Constructing Dn ( σ )or Gn,σ,τ for an unknown n requires summing over d|n , i.e. knowing the prime-power divisors. Thus (2) – (3) are perfect readouts once the appropriate channel is available; they do not by themselves give a cheap way to manufacture the channel from the bitstring of n. 3.2 What blur does and does not do Blur with parameter τ only merges or attenuates lines; it cannot move a line off ω = klog p or create a new one. Deblurring (regularized inverse) can sharpen the lobe around a true line but cannot concentrate mass where there is none. This is the operational meaning of “blur offers a complete imagery up to current knowledge”: if the descriptor is complete, what you see after admissible blur/deblur is all the invariant information the model lets you access. 4 Lower bounds for classical blur-only factoring Consider N = pq with p≤q large and p≈q (RSA style). Let ∆denote the separation of the two fundamental lines in multiplicative frequency: ∆def =|log p−log q|= log1 + |p−q| q≍|p−q| q. For random semiprimes with q≈√N and a typical prime gap near √N on the order of log N , the heuristic scaling is ∆≍log N √N.(4) Any timeor sample-limited measurement induces a main-lobe width (aperture) Λ −1 : for a window of length M , the Dirichlet/Fejér lobe width is Θ(1 /M ); for a Gaussian of variance τ2 it is Θ(1/τ)in frequency. Resolving two lines needs Λ≳1/∆. Proposition 1 (Two-line resolution requirement).Let A be any blur/deblur pipeline whose effective frequency aperture is Λ. To separate the lines at log p and log q in the sense of producing two disjoint peaks after Awith bounded false-split probability, one must have Λ≥c/∆, for a universal constant c > 0depending only on the chosen lobe-separation criterion (e.g. Rayleigh or any comparable metric). Sketch. This is a standard Fourier resolution statement. For Fejér/Dirichlet lobes in a lengthM window the first zero lies at 1 /M ; two sinusoids closer than c/M cannot be separated without large bias/variance blow-up. The Gaussian case is analogous with the reciprocal variance playing the role of aperture. Combining Proposition 1 with (4) yields: 4 Corollary 2 (Heuristic classical step lower bound).Any classical blur-only route that aims to separate log p and log q for a generic N = pq requires an effective aperture Λ ≳√N/ log N = 2 n/2/poly ( n ), where n = log2N is the bit-length. If the cost in steps grows at least linearly with Λ(as it does for time-length, sample count, or equivalent precision), then the step complexity is exponential in n. Interpretation. In the classical blur interface the admissible probe families are smoothed, low-degree, block-averaged tests with explicit projection losses and stability guards. Those budgets grow at most polylogarithmically with N ; hence they fall far short of the exponential aperture required by Corollary 2. Pushing τ→ 0(or M→ ∞ ) from within the model induces instability rather than free resolution, exactly as the commutator constraint (1) predicts. 5 Shor as time-blur spectroscopy Shor’s algorithm fits the same blur protocol, but on a different channel. 5.1 The dictionary Blur element Shor element Time grid of length M= 2tt-qubit time register 1 √MPM−1 x=0 |x⟩ Uniform time blur Flat window ⇒Dirac at 0in frequency Modulation by a hidden period Compute f(x) = axmod Ninto a second register Periodic comb in time Superposition over a coset of rZ Unitary deblur QFTM: concentrates mass near multiples of M/r Blur kernel Dirichlet/Fejér lobe of width ≍1/M Unsharp readout →exact invariant Measure y, recover rfrom y/M by continued fractions 5.2 Pipeline 1. Prepare a uniform time blur. Put the time register in the uniform superposition. This is maximal blur in frequency. 2. Imprint the period. Coherently compute f ( x ) = axmod N . Discarding or measuring the function register leaves the time register in an equal superposition of an arithmetic progression with unknown period r. 3. Deblur unitarily. Apply QFT M . Peaks appear at integer points closest to k·M/r , each broadened by a Fejér lobe of width ≍1/M. 4. Decode. Measuring yields y ; with M≳N2 one has |y/M −k/r| ≤ 1 / (2 r2 )with constant probability; continued fractions recovers k/r and thus rwith high probability. 5.3 Why the speedup fits the blur story Classically, narrowing the lobe to 1 /M requires M separate samples and an FFT of length M . Quantumly, coherence prepares all M time-channels in one state, writes the periodic comb in one modular-exponentiation pass, and performs the FFT unitarily in poly ( n )gates. The effective blur width shrinks exponentially with the number of qubits ( M = 2 t with t = Θ( n )), but the step count remains polynomial. This is precisely “quantum blurring of time”: the same optics, different resource scaling. 5 5.4 Complexity accounting (standard) Let n= log2N. • Modular exponentiation via repeated squaring: poly ( n )gates per control bit; O ( n )controlled powers ⇒O(n3)gates with textbook arithmetic (improvable with better arithmetic). •QFT on t= Θ(n)qubits: O(n2)two-qubit gates (or O(nlog n)approximately). •Classical postprocessing (continued fractions): O(n2)arithmetic. Thus Shor runs in poly(n)steps while achieving lobe width 2−t. 6 Blur Threshold Certificates (BTCs) We now formalize the “security by threshold” statement you want to use in practice. Definition 3 (Blur interface).A blur interface Iis a tuple I= (O,{Bτ}τ≥0,S,R), where O is a class of observables (tests), Bτ a blur semigroup, S a family of deblurs with stability bound ∥Sτ∥ ≤ Γ( τ ), and R the admissible resource envelope (number of probes, block size, degree, aperture, gate depth, qubits, . . .). Definition 4 (BTC predicate).Given a hidden parameter θ (e.g. a factorization, a period), a BTC is a statement of the form ∀A∈Adv(I,R) : PrScore(A;Bτ,Sτ)≥T≤ε, for all τ in an allowed range, where Adv ( · )ranges over adversaries built from O,S within R , Score is a fixed invariant-sensitive functional, and Tis the security threshold. Theorem 5 (Soundness under descriptor completeness).Assume (i) descriptor completeness: any invariant of θ accessible in the model must pass through O and the prime-band projector (or its quantum analogue); (ii) blur faithfulness: Bτ preserves invariants and cannot create spurious ones; (iii) deblur stability bounded by Γ( τ ); and (iv) adversaries obey R . If a BTC holds at threshold T with failure ε for the allowed τ , then no adversary within the same interface can raise the score above T with probability > ε , even if they discover additional features consistent with the model. Hence any security claim tied to Tis sound within I. Proof idea. Any additional feature must be an invariant routed through the same O and subject to the same blur/deblur map. Faithfulness and stability limit amplification; resource constraints limit aperture/degree. Therefore the BTC upper tail is uniform over all model-consistent strategies. Instantiation for classical factoring. Let O be smoothed low-degree statistics over multiplicative frequency, Bτ a Gaussian blur, Sτ a Tikhonov-stabilized deblur, and R the standard polylog budgets. Define Score as the maximum resolvable line-separation. By Corollary 2, for generic semiprimes the resolvable separation is ˜ O (1 / Λ) with Λ = polylog ( N ), well above ∆. Thus with overwhelming probability the two lines at log p, log q cannot be split, certifying that no blur-only adversary within Rcan factor. Instantiation for quantum period-finding. Let O be unitary circuits on t = Θ( n )qubits, Bτ the finite-time window on the time register (aperture M = 2 t ), and deblur the QFT. The BTC does not forbid recovery of r ; rather, it permits it, since the interface includes coherence and QFT. The certificate moves from “cannot separate” to “will separate with high probability.” 6 7 Security implications and limits 7.1 Classical Under the blur-only classical interface—i.e., probes built from smoothed, low-degree statistics together with stable deblurring constrained by commutator/uncertainty bounds—separating the two fundamental lines at log p and log q for a generic semiprime N = pq requires exponential aperture and thus exponential steps. Consequently, factoring remains as safe as currently believed within this interface. Crucially, the blur calculus covers all of our current operational aspects for interrogating information (time, scale, locality, spectral coordinates) while preserving causality and stability; any attack that respects these premises reduces to a blur/deblur instance and inherits the same resolution limits. A putative “non-blur” breakthrough would therefore have to exit these basic premises—e.g., by violating at least one of causality / temporal order, localization / finite aperture, or stability / uncertainty— which would amount to an operator that creates invariants ex nihilo. Absent such a theory (and its physical realization), Blur Threshold Certificates apply: no adversary confined to the same resource envelope can amplify invariant signal past the certified threshold, so the security claim stands. Stronger perspective (possibility vs. reach). A genuinely non-blur theory would, in effect, communicate with an oracle for the hidden invariant: ask questions and obtain near-direct answers without paying aperture/degree costs or respecting commutator/uncertainty constraints. Even if one boldly posits such a theory, surpassing blur thresholds would require stepping outside the causal/stability envelope of our current physics — via super-causal coordination, information packing beyond finite aperture/locality, or defeating stability/uncertainty tradeoffs. Back-of-theenvelope resource accounting (under any reasonable information–energy bounds) then pushes the spacetime/energy requirements beyond any realizable budget — heuristically, beyond the energy content of the observable universe — because the protocol must, in one way or another, exit the universe’s blur reality to create invariant signal where none exists. Thus, while non-blur is not ruled out in principle, it is not within operational reach: it would entail either (i) effectively creating another universe within our own, or (ii) maintaining a bidirectional channel to an external one, both of which lie beyond the admissible blur envelope set by our information–energy limits. In this sense, within our present premises and physics, we are safe. 7.2 Quantum Shor’s algorithm sits within the same blur story but with different resources: coherence provides exponentially many time-channels; the QFT is a unitary deblur. The complexity is polynomial in n , so a fault-tolerant quantum computer threatens factoring—not because blur violates classical ceilings, but because the interface itself changes. This does not contradict the BTC for the classical interface; it simply invokes a different I. 7.3 “If the unknown is below a threshold, you are safe” BTCs implement your practical maxim. If you can prove (under your chosen interface) that every admissible observable stays below a threshold T (for all allowed blur parameters), then your algorithm cannot be broken by any adversary bounded by the same interface, even if they obtain more data that carries no additional invariant energy. This is precisely how a blur proof upgrades “strong in practice” to “certified strong” under explicit resource and stability assumptions. 7 8 A compact worked example Classical prime-line separation. Let f be Fejér of length M , with main-lobe width ≈ 1 /M . Suppose M = polylog ( N ). Then the smallest resolvable separation is ˜ Θ (1 /M ) = 1 /polylog ( N ) ≫ ∆from (4) . Hence the two lines merge; deblurring cannot unmelt them stably. Quantum period readout. Choose t = Θ( n )so M = 2 t≳N2 ; then the Fejér lobe width 1 /M ≪ 1 /N and a single QFT measurement gives y with |y/M −k/r| ≤ 1 / (2 r2 )with constant probability; continued fractions recovers r with high probability. The same optics; different scaling. 9 Conclusions Blur is a unifying lens: same protocol, different resource envelopes. On the classical side, blur certifies limits—and with BTCs, these limits become operational security guarantees. On the quantum side, the time-channel plus QFT transforms the same blur geometry into an efficient period-finding engine. This reconciles “factorization is safe under blur” with “Shor works” without contradiction. A Kernels and apertures Fejér/Dirichlet. For window length M, the Fejér kernel is FM(ω) = 1 M sin2(πMω) sin2(πω), with main-lobe width Θ(1/M)and side lobes O(1/ω2). Gaussian. For variance τ2 in time, the Fourier transform has variance 1 /τ2 ; the effective aperture is Λ≍1/τ. B Continued fractions decoding condition If M≥ 2 N2 and y satisfies |y/M −k/r| ≤ 1 / (2 r2 ), the convergent of y/M closest to k/r equals k/r. This underlies the post-QFT recovery of r. C Resource bookkeeping (quick reference) • Classical blur interface: probes =smoothed, low-degree, block-averaged tests; aperture Λ = polylog ( N )under standard budgets; deblur stability bounded by a fixed regularization schedule. • Quantum time interface: t = Θ( n )qubits; modular exponentiation poly ( n )gates; QFT O(n2)gates; lobe width 2−t. D Energetics for an “Oracle-Grade” Universe This section quantifies, at three levels, the energy scale required to obtain substantial computational power, contrasting (A) realistic 32–100 logical qubits within our universe, (B) fundamental lower bounds that no device can beat, and (C) a hypothetical “non-blur” oracle universe that would evade our causality/uncertainty premises. 8 Constants used. Boltzmann k = 1 . 380649 × 10 −23 J/K , ln 2 ≈ 0 . 693, Planck h = 6 . 62607015 × 10−34 J·s, reduced ℏ=h/2π, speed of light c= 2.99792458 ×108m/s. D.1 (A) Realistic 32–100 logical qubits within our universe For fault tolerance (e.g., surface code), 32–100 logical qubits typically map to Nphys ∈[ 2d2,104](phys./logical) ⇒(32 logical: 4×104to 3.2×105, 100 logical: 1.25 ×105to 106. taking a representative code distance d≈ 25. At millikelvin operation, practical energy is dominated by infrastructure, not Landauer limits: • Cryogenics: A large dilution refrigerator draws ∼ 10–30 kW wall power (to supply µ W at ∼ 10 mK ), scaling up with wiring and heat load. Multi-fridge systems push to ≳ 100 kW continuous. • Classical control: Room-temperature RF/microwave electronics and FPGA/ASIC racks typically add ∼10–100 kW for 105–106channels (even with heavy multiplexing). Takeaway: For 32–100 logical qubits at useful code distances, today’s order-of-magnitude wallplug power is 10 1 –10 2kW (steady), i.e. ∼ 10 9 –10 10 Jper day. This is large at the lab scale but astronomically tiny relative to cosmological energies. D.2 (B) Fundamental floors: Landauer, Margolus–Levitin, Bekenstein These bounds apply to any universe obeying our premises. Landauer (irreversibility cost). Erasing one classical bit at temperature T dissipates at least Eerase min =kT ln 2. Numerics: Eerase min ≈(2.9×10−21 J (T= 300 K), 9.6×10−26 J (T= 10 mK). Even 10 17 erasures at 10 mK would cost only ∼ 10 −8 J. Hence thermodynamic floors are not the bottleneck; engineering overhead dominates. Margolus–Levitin (time–energy for orthogonalization). The maximum rate of orthogonal state transitions satisfies ops/s≤2E πℏ⇒E≥πℏ 2 ops s. Illustration (energy needed to carry out a target rate in 1 s): For 2128 ops/s: E≳5.6×104J, For 2256 ops/s: E≳1.9×1043 J, the latter ∼ 1 . 6 × 10 9 years of the Sun’s total power output. Thus, demanding oracle-like rates quickly forces cosmic energies. 9