K₂ Universality Across q-State Systems

Published: DOI 10.5281/zenodo.23640792 Companion to: [[connectivity-scaling-law]] (DOI 10.5281/zenodo.23529072)

The result

The spectral efficiency formula and its absolute baseline are universal across q-state systems in the sopfr formulation:

c(d) = d · sin(π/d) / (2(d−1))    [substrate-independent]

K₂ = π²/12 = ζ(2)/2               [universal baseline]

Numerically confirmed for q ∈ {2, 3, 5, 6, 7, 10} at N = 200,000: K_q = K₂ to within 0.05% in all cases. The correction is analytically O(1/N) → 0 as N → ∞.

The q-restricted function

For each positive integer q, the q-restricted sum-of-prime-factors function is:

qsopfr_q(n) = Σ_{p coprime to q, p^a ∥ n} a · p

This is [[sopfr-binding-energy]] restricted to primes that do not divide q. Primes dividing q are excluded because they create degenerate interactions in a q-state system (their period divides q, producing no new information).

Ramanujan coefficient: For prime d = p coprime to q, the coefficient is p/(p−1) — the same as for unrestricted sopfr. For prime d = p dividing q, the coefficient is 0. Excluded primes carry no Ramanujan spectral content.

Why K_q = K₂ for all q

The alternating spectral sum is dominated by large primes. Any fixed q has only finitely many prime divisors. Large primes (p ≫ q) are almost all coprime to q. Excluding the finitely many primes that divide q removes an O(N) contribution from a sum that scales as O(N²/lnN). The correction therefore vanishes:

|S(N) − S_q(N)| = O(N) = o(N²/ln N)

and so K_q → K₂ as N → ∞.

Physical reading: The spectral baseline is set by the distribution of large primes, not by the state count q. Whether the hardware uses binary pbits (q=2), ternary trits (q=3), or any higher-dimensional state space, the same prime spectral structure governs the efficiency at each connectivity degree d.

Substrate independence

The corollary follows directly: the formula c(d) = d·sin(π/d)/(2(d−1)) is substrate-independent. It applies to:

Binary pbits (q=2): the Extropic Z1 at degree d=16 gives 1/c(16) = 9.6 ≈ 10× FLOPs penalty
Ternary trits (q=3): same formula, same efficiency ratios
Any q-state clock model in the sopfr formulation: same formula, same K₂ baseline

This means the [[thermodynamic-computing-membrane]] analysis and the [[auditory-system-prime-resonance]] prediction both hold regardless of the specific state-space dimensionality of the underlying hardware or biological system.

What genuine c(d,q) would require

A formula that genuinely varies with q — where different state counts produce different spectral baselines — requires starting from the q-state Hamiltonian directly:

E_q = −J Σ_{<i,j>} cos(2π(s_i − s_j)/q)

The path from this Hamiltonian to spectral constants leads to Dirichlet L-functions L(2, χ_q) for characters χ of Z/qZ:

q=2: L(2, χ_trivial) = π²/6, K₂ = π²/12
q=3: L(2, χ₃) ≈ 0.7813 (OEIS A086724)
General q: K_q^{Hamiltonian} = L(2, χ_q^{principal}) / (normalization)

This derivation — connecting the q-state Hamiltonian to spectral constants via Dirichlet characters — is the remaining open problem for the c(d,q) programme.

ONM reading

In the ONM framework, the universality of K₂ = π²/12 across q-state systems reflects a deeper property: prime structure is substrate-independent. The primes do not care whether the state space has 2, 3, or q levels. They impose the same spectral skeleton regardless.

The q-state extension changes the SHAPE of possible states (more options per node) but not the DISTRIBUTION of prime-indexed spectral weights. The membrane tax c(d) is the same tax whether paid in binary currency or q-ary currency.

K₂ Universality Across q-State Systems

K₂ Universality Across q-State Systems

The result

The q-restricted function

Why K_q = K₂ for all q

Substrate independence

What genuine c(d,q) would require

ONM reading

See also

Connections