TNFR Algebraic Number Fields — Finite Fields & Gaussian Signatures (R5)¶

Status: prime-field regression DERIVED (R2 recovered) + extension behaviour MEASURED (exact trace-fiber counts; tolerance-based spectral counts); Gaussian decomposition detection CONJECTURAL (NT-P05), a descriptive study using the classical type as a ground-truth label. No factoring, complexity, cryptographic, or Millennium claim; no external field package required. Modules: src/tnfr/mathematics/finite_fields.py, src/tnfr/mathematics/algebraic_residue_networks.py · Tests: tests/mathematics/test_finite_fields.py, tests/mathematics/test_algebraic_residue_networks.py · Benchmark: benchmarks/algebraic_field_signatures.py · Depends on: R2 (pulse rank), C5 (claim manifest, circularity audit).

1. Finite fields via the trace character¶

R2 established that on the k-th power residue Cayley network over the prime field F_p the pulse rank equals the cyclotomy count gcd(k, p−1) + 1. R5 lifts the construction to a general finite field F_q (q = p^f) using the additive characters built from the field trace

\[\psi_a(x) = \exp\!\Big(\tfrac{2\pi i}{p}\, \operatorname{Tr}_{F_q/F_p}(a x)\Big),\qquad \operatorname{Tr}(y) = y + y^p + \cdots + y^{p^{f-1}} \in F_p.\]

The eigenvalues of the normalized transition matrix of Cay(F_q, S), with S the non-zero k-th powers, are the normalised Gauss periods

\[\eta_a = \frac1{|S|}\sum_{s\in S}\psi_a(s),\]

constant on cosets of the k-th power subgroup H: multiplication by h in H permutes S, hence eta_(ah)=eta_a. The characters themselves are not equal pointwise. The adjacency eigenvalues are |S|*eta_a, and the random-walk Laplacian eigenvalues are 1-eta_a. There are gcd(k, q−1) such cosets plus \eta_0 = 1, so the distinct count is at most gcd(k, q−1) + 1.

2. Prime regression and extension collisions (DERIVED + MEASURED)¶

Prime fields (f = 1, DERIVED). The trace is the identity, \psi_a(x) = \zeta_p^{a x}, and the construction is exactly R2. Every tested case satisfies

\[\#\{\eta_a\} = \gcd(k, p-1) + 1 \qquad (`prime_field_matches_cyclotomy`).\]

Extensions (f ≥ 2, MEASURED). Here the trace is p^{f−1}-to-one, so distinct cosets of H can share a period value and the bound is not tight. Measured distinct counts (computed from numerical character sums and cross-checked against the explicit graph spectrum, with the implementation's tolerance):

field	`q`	`k`	`gcd(k,q−1)+1`	distinct
`F_4`	4	3	4	2	collision
`F_9`	9	4	5	2	collision
`F_25`	25	3	4	3	collision
`F_49`	49	4	5	3	collision
`F_27`	27	4	3	3	match

So the prime-field independence argument (distinct cosets ⇒ distinct periods) does not transfer to extensions: distinct = gcd(k, q−1) + 1 becomes an upper bound; equal coset character sums can make it strict. Elementwise trace collisions alone are not a sufficient criterion for period degeneracy. The trace-character count and the explicit additive Cayley spectrum agree in every case (period_and_explicit_spectrum_agree), an internal consistency control.

2b. Trace collisions as loss of observability (N10)¶

The extension collisions are not a nuisance — they are the durable R5 result, read through observability (trace_collisions.py). For the k-th powers H = { a^k : a ∈ F_q^* } the trace Tr : F_q → F_p collapses H onto the p residues; the fiber counts N_a = #{ h ∈ H : Tr(h) = a } are the collision structure and #{ a : N_a > 0 } is the number of trace values that stay distinguishable by that static trace. This is not itself the dynamical observation y_m = c^* P^m b: no such input/output pair or Hankel-rank equality is built by trace_collisions.py. R2's pointed circulant sees every distinct spectral mode; a static count of trace values is a different quantity.

Two algebraic facts determine the histogram for the supplied finite field:

Character formula (DERIVED, exact). Fourier inversion on F_p gives N_a = (1/p) Σ_{u∈F_p} ψ(−ua) Σ_{h∈H} ψ(u·Tr(h)) with ψ(t) = e^{2πi t/p}. It reproduces the direct counts to machine precision (residual ≈ 1e-16); N_a = |H|/p + (1/p) Σ_{u≠0} ψ(−ua) S(u), so the character sums S(u) are the exact deviation from equidistribution.
Galois invariance (MEASURED). Tr(h^p) = Tr(h) for every h, so the collision histogram is a representation-free field invariant — not an artifact of how F_q is coordinatized.

Measured: on F_p the trace is the identity, every h keeps its own residue (observed = |H|, no collisions — R2 recovered); on F_{p^f} the trace is p^{f-1}-to-one, e.g. F_8 collapses all 7 non-zero cubes onto {0, 1} (observed = 2, max fiber = 4) and F_25 sends the 12 squares onto 4 of the 5 residues (observed = 4 < p). The type detector is not built here: a k-selective ramified/inert/split separator stays NT-P05c CONJECTURAL; the R5 durable result is this exact, invariant trace-collision structure.

3. Gaussian integers: split / inert / ramified (MEASURED, CONJECTURAL)¶

For a rational prime p the quotient ℤ[i]/(p) has three classical shapes fixed by p mod 4:

p = 2 — ramified, (2) = −i(1+i)^2, a local ring with nilpotent 1+i;
p ≡ 1 (mod 4) — split, ℤ[i]/(p) ≅ F_p × F_p;
p ≡ 3 (mod 4) — inert, ℤ[i]/(p) ≅ F_{p^2}.

Building the additive Cayley network on (ℤ[i]/(p), +) (p^2 nodes) with connection set the non-zero unit k-th powers, the distinct-eigenvalue count at k = 2 separates the three types on the tested primes:

\[\text{ramified} \to 2, \qquad \text{inert} \to 3, \qquad \text{split} \to 6.\]

The inert count agrees with the F_{p^2} finite-field result of §2 (k=2 → 3), as it must — the inert quotient is F_{p^2}. The split count 6 reflects the product structure F_p × F_p (a CRT product network, cf. R3).

Honest scope. The classical decomposition type is used only as a ground-truth label for scoring, never as an input to the observable — a descriptive study in the C5 sense (CircularityAudit(factors_used_only_for_scoring=True)), so it may not be presented as a discovery algorithm. The separation is k-sensitive: at k = 3, 4 the counts no longer separate the three types (signature_separates_types(..., 3) == False). It is tested only on small p and has no proof. The claim "the pulse detects the decomposition type" (NT-P05) is therefore CONJECTURAL, with the k = 2 signature as supporting measured evidence.

4. Claim ledger¶

Claim	Basis	Status
`F_p`: `#{η_a} = gcd(k, p−1) + 1`	trace = identity (R2)	DERIVED + MEASURED
`F_q`: `#{η_a} ≤ gcd(k, q−1) + 1`, strict in some examples	coset character sums	DERIVED upper bound; numerical collision examples
trace-character count = explicit spectrum	character theory	MEASURED (agree)
`N_a` via character formula = exact fiber count	Fourier inversion on `F_p` (N10)	DERIVED + MEASURED (`~1e-16`)
trace collision histogram is representation-free	`Tr(h^p) = Tr(h)` (N10)	MEASURED (Galois-invariant)
`#{a : N_a>0}` = number of attained static trace values	direct fiber counts (N10)	EXACT COUNT; not a proved dynamical realization order
`k=2` count separates ramified/inert/split	Cayley spectrum	MEASURED (small `p`)
pulse detects decomposition type	—	CONJECTURAL (`NT-P05`; k-sensitive, descriptive)

R5 carries the declared arithmetic construction from prime fields to finite fields and Gaussian quotients. It recovers R2 exactly on F_p, quantifies the trace collisions that break the formula on extensions, and exhibits a k = 2 spectral signature that separates the three Gaussian decomposition types. The decomposition detector is a descriptive, k-sensitive, small-p observation — not a proof and not an algorithm. The field, polynomial representation and residue connection set are supplied inputs. These identities do not derive a physical carrier, autonomous NFR formation, or a full joint nodal evolution.