TNFR Additive Dynamics — Controlled Additive-Character Number Theory (R6)

Status: finite cyclic convolution identity DERIVED, with numerical residual checks; the selected phase statistic does not outperform the selected power statistic on the recorded controls. No additional information beyond the complete invertible Fourier transform is possible for a deterministic summary of the same input. A useful TNFR-specific dynamics/measurement bridge remains OPEN / not established (NT-P06). No proof of Goldbach or any open problem; the target property is never used to define the phase. Module: src/tnfr/mathematics/additive_resonance.py · Tests: tests/mathematics/test_additive_resonance.py · Benchmark: benchmarks/additive_resonance_controls.py · Depends on: C5 (claim manifest, circularity audit).

1. The object and the honest question

The implemented cyclic representation function is the self-convolution of the finite prime indicator on Z/NZ,

\[r_2(m) = (1_{\mathbb P} * 1_{\mathbb P})(m) = \#\{(a, b) : a + b \equiv m,\; a, b \text{ prime}\}.\]

Its additive-character (Fourier) transform factorises by the finite convolution theorem: \hat r_2(k) = \hat{1_{\mathbb P}}(k)^2. Modular wraparound counts representations of m modulo N; identifying ordinary integer Goldbach counts requires a separate no-aliasing range or padding. R6 does not try to prove Goldbach. It asks a falsifiable internal question:

Does a TNFR reading of the additive phase χ_k(n) = e^{2πi k n / N} supply a useful summary for a declared discrimination task compared with a particular power-spectrum summary, under explicit control distributions?

The phase is derived only from the additive characters; the target property (r_2, Goldbach) is never used to define it.

2. The exact reduction (DERIVED + MEASURED)

In exact arithmetic the cyclic function equals the inverse transform of the squared spectrum:

\[r_2 = \operatorname{IFFT}\big(\hat{1_{\mathbb P}}^2\big).\]

Measured residual ~1e-14 for N ≤ 1024 (goldbach_fourier_residual). This measured residual is not a proved universal FFT error bound. Both r_2 and the power spectrum are quadratic, not linear, in the input. More generally, the complete DFT is invertible: every deterministic input statistic, linear or nonlinear, is a function of that spectrum. A nonlinear statistic can improve a particular compressed read-out or task cost; it cannot add information beyond the full DFT. Such utility requires a specified comparison and does not derive new nodal dynamics.

3. The controlled negative (MEASURED)

The candidate is the mean squared second difference of an unwrapped FFT phase, implemented by np.unwrap, np.angle and np.diff(...,2),

\[\mathcal{E}_{K_\phi}(1) = \big\langle (\Delta^2 \arg \hat 1(k))^2 \big\rangle_k,\]

on the interior frequency indices. It is not the canonical graph circular-mean K_phi: no graph-neighbor curvature owner is called, and the FFT's zero coefficients do not supply a physically defined phase. The naming in the legacy API does not establish that bridge. It is compared against power concentration max_{k≠0}|\hat 1|^2 / \text{mean}. For each observable a z-score measures how far the primes sit from a control distribution. Shuffle and matched-random preserve the finite prime count. The Cramér branch draws independent Bernoulli values using 1/log(m) and does not preserve that count; it is a different baseline. The historical table is retained with those actual control semantics:

control	classical z	TNFR z	TNFR exceeds
matched	167.8	−0.58	No
Cramér	148.5	−0.21	No
shuffle	190.6	−0.51	No

The classical Fourier observable strongly detects the arithmetic structure of the primes in these samples (\|z\| ≫ 1); it does not compute or prove a singular-series asymptotic. The candidate phase statistic shows no large z-score in the reported samples (\|z\| < 1), and never exceeds the classical observable (excess_over_fourier(...).tnfr_exceeds == False). Prime constellations (twin-prime starts) also stand out — but again under the power statistic. These are comparisons of selected summaries, not proofs of the absence of all possible useful phase statistics.

Invariance controls. The observables respect the required conventions: power_concentration is invariant under translation n → n + c and under character conjugation (n → −n), and cyclic_goldbach is translation-equivariant with the output shifted by 2c for a shift c of the input. These properties do not certify origin invariance of the unwrapped-phase second-difference statistic, which has its own branch and endpoint conventions.

4. Claim ledger

Claim	Basis	Status
r_2 = IFFT(\hat 1^2) (additive reading is Fourier)	convolution theorem	DERIVED + MEASURED (~1e-14)
power concentration distinguishes the supplied samples	selected control distributions	MEASURED (\|z\| ≫ 1)
TNFR phase-curvature exceeds Fourier under controls	—	NEGATIVE (\|z\| < 1, never exceeds)
extra information beyond the complete DFT of the same input	DFT invertibility	EXCLUDED for deterministic summaries
useful canonical nodal contribution for this task	no dynamics/measurement bridge	OPEN (NT-P06)

R6 retains a finite convolution baseline and a comparison of two compressed statistics. The selected phase statistic showed no advantage in the recorded experiment. A canonical phase-curvature interpretation, a useful nodal evolution and any claim about ordinary Goldbach representations require separate work; this note does not open another active research queue.