Prime Harmonic Transform (PHT)

Status: active Domain: both (interface) Source: → wiki/research/prime-harmonic-transform/

What We Know

Definition: A weighted matched-filter bank decomposing signals into harmonics indexed by positive integers, with weights from prime factorisation
Basis functions: φₙ(t) = w(n) · e^{-2πi(f₀/n)t}, where w(n) = ∏_{p|n} 1/p (Euler product weight)
Forward transform: P(n) = w(n) · (1/K) Σ x[k] · e^{-2πi(f₀/n)·k/fₛ}
Inverse: Simple synthesis or pseudoinverse with Tikhonov regularisation (basis is non-orthogonal)
Reconstruction quality: correlation >0.99 for both pure prime and Tusk-optimal signals

PRS = (Σ |P(p)|² for prime p) / (Σ |P(n)|² for all n)
5000× discrimination: PRS = 0.997 for pure prime signals vs 0.0002 for composites
PRS ≈ 1/ln(N) for white noise (by Prime Number Theorem)
Extended metrics: prs_tusk (energy at {1,2,3,5,6,7}), prs_scaffold (primorial scaffolds), prs_squarefree (|μ(n)|=1)

PRS of Tusk Series = 0.967 — overwhelmingly a “prime signal”
Amplitude decay follows |P(n)| ∝ 1/n (consistent with known red spectrum)
Dominant peaks: |P(2)| = 249.1, |P(3)| = 97.3, |P(5)| = 24.7 — primes dominate every amplitude rank

Möbius inversion: P(n) = Σ_{d|n} P̃(d) inverts to P̃(n) = Σ_{d|n} μ(n/d)·P(d) — divisor lattice decomposition that FFT completely misses
Dirichlet series: PHT with w(n)=1/nˢ IS a Dirichlet series evaluation; naturally factorises over primes via Euler product
Ramanujan sums: PHT basis at integer times reduces to roots of unity — continuous-time extension of Ramanujan expansion

Configuration	PRS	prs_tusk	Experimental result
Primes {1,2,3,5,7,11}	0.300	0.994	Baseline (+28% vs composites)
Tusk {1,2,3,5,6,7}	0.290	1.000	+24% vs pure primes
Zeta zeros	0.000	1.000	85-90% of prime performance
Composites {4,6,8,9,10,12}	0.000	0.042	Baseline (worst)

Δt · Δp ≥ C · ln(N)/4π — prime-indexed frequencies are sparser (density ~1/ln n), requiring more temporal extent to resolve
Single prime: Δt·Δp = 0.008; Three primes: 0.135; Tusk set: 0.179

Can PHT PRS be validated against v4 live measurements in real-time?
Does the prime uncertainty principle have physical consequences for resonance network design?
Can Ramanujan sums replace Fourier in specific physical applications?
Will Möbius deconvolution noise amplification limit practical use at large N?
Is there a fast PHT algorithm (current is O(N·K) vs FFT’s O(K log K))?

[[tusk-series]] — supports (strong): Tusk Series PRS = 0.967; PHT proves it’s overwhelmingly prime-structured
[[four-factor-theory]] — extends (strong): PRS is necessary discriminator (prime vs composite) but insufficient for Tusk-optimal; composite score combining PRS + scaffold + coprimality needed
[[v4-board-design]] — supports (moderate): PHT provides measurement tool for v4 experiments
[[prime-resonance-computing]] — extends (strong): PHT is the measurement/analysis layer for prime resonance systems
[[coprimality]] — bridges (moderate): Möbius deconvolution reveals coprimality structure in signals
[[zeta-zeros-physical]] — bridges (strong): PHT scores zeros at PRS ≈ 0 (inherently irrational, don’t land on integer indices) — independent confirmation of “lossy encoding”
[[euler-product-rational]] — depends-on (strong): PHT weight function IS the Euler product; Dirichlet series connection is direct
[[six-dimensional-scaffold]] — supports (moderate): prs_scaffold metric detects 6k structure

If combined with v4 measurements, PHT provides real-time prime resonance scoring — a quantitative bridge from math to physics
Möbius deconvolution could reveal hidden scaffold structure in arbitrary physical signals
Ramanujan sum connection opens path to arithmetic Fourier analysis of physical systems

wiki/research/prime-harmonic-transform/pht.py — complete implementation
wiki/research/prime-harmonic-transform/validate_pht.py — 7-test validation suite
PRS discrimination: 0.997 vs 0.0002 (5000× ratio)
Tusk Series PRS: 0.967
Reconstruction correlation: >0.99