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

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.