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TNFR Arithmetic Dynamics — the Pulse Recurrence Theorem (R2)

Status: DERIVED (Kronecker + circulant + cyclotomy) + MEASURED (exact rational, zero discrepancies on the tested domain). Not a proof of any open problem and not a primality test. Modules: src/tnfr/mathematics/arithmetic_pulse.py, src/tnfr/mathematics/krylov.py · Tests: tests/mathematics/test_arithmetic_pulse.py · Depends on: R1 (pointed graphs, §4), C4 (spectral rank), C5 (claim manifest).

1. The pointed object

The static cyclotomy law (TNFR_NUMBER_THEORY.md) states that the k-th power residue Cayley operator L_rw = I − D⁻¹W on ℤ/pℤ (p prime) has exactly s_k(p) = gcd(k, p−1) + 1 distinct eigenvalues. R2 gives that number a dynamic reading through the pointed network (G_{p,k}, 0) seeded at the additive identity

\[e_0 = (1, 0, \dots, 0)^\top .\]

The point is the neutral element 0, so it is chosen without using any factor — the translation symmetry is broken by declaration (R1 §4, ADR-006), not clandestinely.

2. The recurrence (DERIVED)

The pulse moments \(\mu_m = e_0^\top L_{rw}^m e_0\) obey a linear recurrence. Three classical facts chain together:

\[\underbrace{\operatorname{rank}[\mu_{i+j}]}_{\text{Hankel}} \;=\; \underbrace{\dim\operatorname{span}\{e_0, L e_0, L^2 e_0, \dots\}}_{\text{Krylov}} \;=\; \underbrace{\#\{\text{distinct eigenvalues}\}}_{\text{spectral}} \;=\; \underbrace{\gcd(k, p-1) + 1}_{\text{cyclotomy}} .\]
  • Hankel = reachable-and-observable dimension holds for the scalar moment sequence. Equality with the full Krylov dimension additionally requires that the chosen output see every reachable mode. Here the pointed circulant has nonzero weights m_lambda/n on every distinct eigenvalue, so that condition holds. It is false for an arbitrary square matrix: with L=[[0,0],[1,0]] and v=(1,0), the Krylov dimension is two but v^T L^m v=(1,0,0,...) has Hankel rank one.
  • Krylov = #distinct eigenvalues holds because L_rw is a circulant, so e_0 has a non-zero projection on every Fourier mode and the cyclic subspace it generates picks up exactly one dimension per distinct eigenvalue.
  • = gcd(k, p−1) + 1 for primes is the cyclotomy law (Gauss periods).

For a fixed common nu_f>0, this turns the static spectral rank into the order of the temporal pulse

\[h(t) = e_0^\top e^{-\nu_f L t} e_0 = \sum_{j=1}^{s} a_j\, e^{-\nu_f \lambda_j t}, \qquad s = \gcd(k, p-1) + 1 .\]

3. Verification (MEASURED, exact)

Everything is computed over ℚ with :class:fractions.Fraction (no floating-point ambiguity). For every prime p ∈ {5,…,23} and k ∈ {1,2,3,4,6} the three computable ranks agree and equal gcd(k, p−1) + 1 (35/35). The Krylov dimension matches the numerical distinct-eigenvalue count, and the pointed-pulse rank is independent of the chosen point (translation is an automorphism of the Cayley graph).

Composite controls (outside the prime cyclotomy theorem). Hankel = Krylov still holds for these pointed circulant constructions, but the value gcd(k, n−1) + 1 does not — e.g. n = 15, k = 2 gives rank 9 ≠ 3. The identity is prime-specific; an occasional coincidental composite match (n=15, k=3) is why this is a control, not a primality test.

3b. The amplitudes (DERIVED, N12)

The recurrence order fixes how many tones the pulse has; the amplitudes a_j fix how loud each is (pulse_amplitudes.py). Because L_rw is a circulant it is diagonalized by the Fourier basis f_j[x] = ω^{jx}/√p, and the pointed seed has uniform Fourier weight, |⟨e_0, f_j⟩|² = |f_j[0]|² = 1/p for every j. Grouping the p modes by distinct eigenvalue λ of multiplicity m_λ,

\[h(t) = \sum_{j} \tfrac1p\, e^{-t\lambda_j} = \sum_{\lambda} \frac{m_\lambda}{p}\, e^{-t\lambda}, \qquad a_\lambda = \frac{m_\lambda}{p}.\]

Theorem (NT-P02b). The pointed-circulant pulse amplitudes are the normalized spectral multiplicities a_λ = m_λ/n (n = p) — exact rationals summing to 1. Two independent confirmations: the orthogonal spectral projector gives e_0^* P_λ e_0 = m_λ/p (basis-invariant — a random unitary rotation inside a degenerate eigenspace leaves it unchanged, ~1e-16), and Σ_λ (m_λ/p)\,λ^m reconstructs the exact rational moments μ_m = e_0^\top L^m e_0 (~1e-13). Example (p,k)=(11,2): the spectrum is a = 1/11 at λ=0 plus 5/11 on each of the two conjugate eigenvalues 1.10 ∓ 0.332i (real pulse, 1/11 + 5/11 + 5/11 = 1). The number of tones is the R2 rank gcd(k,p−1)+1, so the amplitude spectrum refines the rank without changing it.

4. Two faces (do not mix)

  • Dissipative face: the heat semigroup e^{−ν_f L t} — valid with complex eigenvalues (the directed, p ≡ 3 (mod 4) case) under the overdamped reading. The recurrence order s is face-independent.
  • Conservative face: frequencies ω = √λ — only for real / self-adjoint operators (or with an explicit conservative-substrate derivation). Do not call the root of a complex eigenvalue a "tone" without that derivation.

5. Honest scope

The word "pulse" denotes the response of the declared fixed linear generator to an imposed localized initial condition. It does not derive the occurrence of AL, phase angular velocity, autonomous sustained oscillation or an arithmetic substrate from the nodal identity. The heat response need not oscillate.

At zero capacity, the actual heat response is constant and does not retain the nonzero temporal rates; the moments of the unscaled matrix remain a separate algebraic diagnostic.

This is a recurrence-order identity connecting the pointed pulse to the cyclotomy law — a structural bridge, not an algorithm. Building and analysing the p-node network is poly(p) = poly(2^{L}), i.e. exponential in the input size L = log₂ p bits; there is no factoring or primality speedup and no cryptographic consequence. The result is DERIVED (from Kronecker, the circulant structure and cyclotomy) and MEASURED exactly on the stated domain.