Exact laboratory and certified evaluation
The reusable module coupled_lab.py uses Python's standard library and imports the existing exact coefficient, phase and tap operations from family_lab.py. The figure builder additionally uses NumPy and Matplotlib. Neither script contacts an external service. The generic integer and signed catalog rows are formal power series, with no claim of analytic convergence.
Normalization and recurrence
Write
The exact recurrence is
with \(A_0=B_0=1\). Coefficients with even derivative index vanish. Swapping the components sends the parameter tuple \((q,r,a,b)\) to \((b,a,r,q)\). Identical rows \((a,b)=(q,r)\), or exchanged rows \((a,b)=(r,q)\), yield the scalar reduction \(U=V=T_{q,r}\).
Both derivative arrays are polynomials in four scale variables with nonnegative integer coefficients. Their total degree is at most \(n(n+1)\). Indeed, a recurrence term with \(i+j=n-1\) has degree at most \(2n+i(i+1)+j(j+1)\le n(n+1)\). If all four denominators divide \(D\), then \(D^{n(n+1)}A_n\) and \(D^{n(n+1)}B_n\) are integers. The clearing factor is a convenient uniform choice and need not be minimal.
An elementary exact evaluation certificate
This argument uses \(0<q,r,a,b<1\) and does not require the order theorem. Let \(Q=\max(q,r,a,b)\). All ordinary odd Taylor coefficients are positive and at most one: induction in the ordinary recurrence bounds the coefficient of \(z^{2n+1}\) by \(n/(2n+1)\) for \(n\ge1\). Thus both series converge on the unit disk and, for \(0\le x<1\),
Analytic continuation by the integral equations expands each known disk by the factor \(1/Q\), so both solutions are entire. Positive coefficients give \(\max_{|z|\le x}|F(z)|=F(x)\) for either component. The integral equations therefore imply
Start at the first member \(Q^kx<1\) of this finite contraction ladder, use \(x/(1-x^2)\) there, and repeatedly apply \(B(x)=x(1+B(Qx)^2)\) backward. This defines the rational disk_majorant bound for rational parameters and radius. The implementation has explicit recursion and integer size guards, which raise errors rather than returning a weakened result.
If \(F_N\) retains odd indices \(0,\ldots,N\), then for \(S>R\ge0\),
For every omitted exponent \(m\ge2N+3\), use \(R^m\le (R/S)^{2N+3}S^m\), sum positive terms, and bound their sum by \(F(S)\le B(S)\). No extra geometric denominator is needed. For rational \(x\ge0\), certified_evaluation returns the exact interval \([F_N(x),F_N(x)+B(S)(x/S)^{2N+3}]\) for each component.
Worked calls
from fractions import Fraction as Q
import coupled_lab as lab
p = (Q(1,5), Q(2,5), Q(3,5), Q(7,10))
A, B = lab.coupled_odd_derivatives(*p, last=12)
certificate = lab.certified_evaluation(*p, Q(1), last=30, outer_radius=2)
root = lab.pf_order(*p) # floating approximation, not interval arithmetic
derivatives = lab.full_egf_coefficients(B)
phases = lab.phase_lanes(derivatives, 3)
matrix = lab.polyphase_tap_matrix([1, -2, 1], 3)
length = len(derivatives) + 2
output_phases = lab.apply_polyphase_matrix(phases, matrix, length)
output = lab.merge_lanes(output_phases, length)
At \(x=1\), with \(N=30\) and \(S=2\), the common tail bound is below \(1.600\times10^{-13}\). The partial values are approximately \(U(1)=1.027044341827337\) and \(V(1)=1.1467074957737624\). These displayed decimals are rounded illustrations. The returned fractions are the actual certificate.
Run the independent checks and regenerate figures from the package’s research/ directory:
python3 checks/coupled_v11.py
python3 build_coupled_figures.py
The first command passes 1,559 exact assertions, 10 numerical comparisons, and 13 separate input/resource guards. Independent constructions include dense ordinary Taylor Picard iteration, sine/cosine division for tangent, the exact sinh formula, component symmetry, and direct convolution versus phase carry. The test counts describe bounded implementation checks, not universal proofs.