TNFR CRT Fractality — the Multiscale Composition Theorem (R3)¶

Status: DERIVED (CRT unit bijection + Kronecker product) + MEASURED (exact rational residual 0; eigenvalue composition to ~4e-15). A structural synthesis result: it uses the known factors a, b to assemble a network from its sub-networks. It is not a factoring or discovery algorithm and makes no complexity, cryptographic, or Millennium claim. Module: src/tnfr/mathematics/crt_multiscale.py · Tests: tests/mathematics/test_crt_multiscale.py · Benchmark: benchmarks/crt_multiscale_composition.py · Depends on: R2 (residue Cayley operator, exact rational linear algebra), C4 (spectral tolerance), C5 (claim manifest, circularity audit).

1. The multiscale object¶

For coprime moduli a, b, the Chinese Remainder Theorem gives the declared arithmetic product carrier

\[\mathbb{Z}/ab\mathbb{Z} \;\cong\; \mathbb{Z}/a\mathbb{Z}\times\mathbb{Z}/b\mathbb{Z},\]

and the unit group splits multiplicatively, \((\mathbb{Z}/ab\mathbb{Z})^\ast \cong (\mathbb{Z}/a\mathbb{Z})^\ast\times(\mathbb{Z}/b\mathbb{Z})^\ast\). The two factors parameterize the parent network. This exact product construction is useful for scale comparisons; it does not instantiate the engine's nested-EPI hierarchy, execute REMESH, or prove its U5 contract.

2. A sufficient product-compatible connection family¶

Write S_m = the connection set of the residue Cayley digraph Cay(ℤ/mℤ, S_m), and L_m = I − (1/|S_m|) W_m for its random-walk Laplacian (R2). The theorem needs the connection set to CRT-factor exactly, and that is guaranteed for the k-th powers of the units:

\[S_m = \{\, u^k \bmod m : \gcd(u, m) = 1 \,\} = \texttt{unit\_power\_residue\_set}(m, k).\]

Under the CRT bijection of unit groups, S_ab maps onto the full product S_a × S_b. For a prime m every non-zero residue is a unit, so this coincides with the R2 set power_residue_set; for composite m it is a proper subset (unit_power_residue_set(15, 2) = {1, 4} vs power_residue_set(15, 2) = {1, 4, 6, 9, 10}). The unrestricted set includes non-units and does not factor under CRT — it is the non-factorizing control (§5). Product factorization is the actual hypothesis, not a characterization of unit sets: any prescribed nonempty T_a and T_b with parent connection set CRT^-1(T_a × T_b) gives the same product identity, including suitable non-unit sets.

3. The Kronecker identity (DERIVED, exact over ℚ)¶

Let σ be the CRT permutation (node r ↦ (r mod a, r mod b), crt_ordering). Because S_ab CRT-factors as S_a × S_b, the adjacency, transition and Laplacian operators are exact Kronecker products (up to σ):

\[A_{ab} = A_a \otimes A_b, \qquad P_{ab} = P_a \otimes P_b, \qquad \boxed{\,L_{ab} = I - (I - L_a)\otimes(I - L_b)\,}.\]

The last identity holds because P_m = I − L_m and P_{ab} = P_a ⊗ P_b = (I−L_a)⊗(I−L_b). It is verified exactly over ℚ using fractions.Fraction: crt_kronecker_residual(a, b, k) == 0 for every tested coprime pair and power (no floating point).

4. Eigenvalue composition and the nonzero-modulus bound¶

Kronecker structure fixes the parent spectrum as the child eigenvalue composition. If λ ∈ spec(L_a) and μ ∈ spec(L_b) then

\[\lambda_{\text{parent}} = \lambda + \mu - \lambda\mu = 1 - (1-\lambda)(1-\mu).\]

The random-walk Laplacian always has the constant mode λ = 0 (the trivial / neutral EPI), so setting μ = 0 gives λ unchanged: every child eigenvalue embeds in the parent spectrum. Hence the parent's non-zero spectrum is the union of the child non-zero spectra and the nonzero cross modes λ + μ − λμ. Define g(L)=min{|lambda|:lambda in spec(L),lambda!=0} when that set is nonempty. The exact modulus bound is

\[g(L_{ab})=\min\!\left(g(L_a),g(L_b), \min_{\substack{\lambda,\mu\ne0\\\lambda+\mu-\lambda\mu\ne0}} |\lambda+\mu-\lambda\mu|\right) \le\min(g(L_a),g(L_b)).\]

An empty cross-mode set contributes no minimum. Excluding zero cross modes matters: additional stationary modes can appear in a product, and the smallest nonzero modulus does not certify connectedness or convergence to one constant. Equality holds if both child spectra lie in [0,1], because lambda+mu-lambda*mu >= max(lambda,mu) there. Symmetric connection alone only places the random-walk Laplacian spectrum in [0,2]; it does not imply that stronger hypothesis.

The implementation's compatibility-named spectral_gap uses eigenvalue moduli above a numerical tolerance. For directed heat flow, decay depends on real parts, so that value cannot generally be interpreted as a coherence timescale. The recorded 4 × 9, k=2 modulus is about 0.518 versus child moduli 1.000; it is a finite spectral comparison. Numerical eigenvalue composition matches to about 4e-15 on the reported cases. Neither this thresholded statistic nor the exact product theorem certifies U5.

5. Non-factorizing control for the unrestricted power-residue family¶

The unrestricted set is the control. residue_set_factors(a, b, k, unit=False) returns False — even for coprime primes (3 × 5), because the non-unit powers {6, 9, 10} of ℤ/15ℤ have no CRT product preimage in S_3 × S_5 = {1} × {1, 4}. Consequently the Kronecker identity fails for the unrestricted operator (full_power_residue_laplacian): the measured residual is 0.30, 0.09, … — non-zero. These controls distinguish the two implemented families; they do not make unit membership necessary for every product graph.

6. Honest scope — synthesis, not factoring¶

This is explicitly a structural (synthesis) branch: the composition uses the known factors a, b to build the parent from its children. The C5 circularity audit therefore records graph_construction_requires_answer = True, yielding verdict CIRCULAR and permits_discovery_claim = False, and the experiment manifest sets uses_known_factors = True (claim NT-P03). The algebraic product theorem (Kronecker identity + eigenvalue law, §3–4) stands on its own and is tested separately from any inverse use: recognising a given L_ab as a product I-(I-L_a) ⊗ (I-L_b) is not an inverse operation implemented here. This code is supplied a,b; the statement is an implementation dependency, not a lower bound on every possible inverse algorithm. The forward direction assembles a declared product transport model; the inverse direction (recovering unknown factors) is not claimed and is out of scope.

7. Claim ledger¶

Claim	Basis	Status
`L_ab = I − (I−L_a)⊗(I−L_b)` (unit set, coprime)	CRT unit bijection + Kronecker	DERIVED + MEASURED (exact `0` over ℚ)
`λ_parent = λ + μ − λμ`	Kronecker eigenvalues	DERIVED + MEASURED (`~4e-15`)
`g(L_ab) ≤ min(g(L_a), g(L_b))`	trivial-mode embedding, exact nonzero moduli	DERIVED; finite thresholded comparisons are separate
unrestricted family can fail to factor	control residual `≠ 0`	MEASURED; not a universal necessity claim
factoring / discovery	—	NEGATIVE (synthesis only; `NT-P03` forbids discovery claim)

R3 supplies an exact finite product identity for a declared family of unit residue networks. The operator and spectrum compose as shown; joint nodal emergence, full-tetrad transport, U5 and generic decay-time claims do not follow. It closes no open problem and provides no factoring capability.