{
  "table": "T710",
  "checked_entries": 22,
  "total_entries": 22,
  "all_checkpoint_coefficients_equal": true,
  "verification_kind": "Generator.verify(sample=None) with unchanged exact coefficient checkpoint; no minpoly rebuild",
  "rigour_details": "Computed by exact arithmetic in $\\mathbb Q(\\zeta_{\\operatorname{lcm}(4,k)})$ from CITE{formula-cyclotomic}, then clearing denominators and taking the primitive positive-leading-coefficient polynomial. Every result has native exact integer coefficients, is irreducible over $\\mathbb Q$, and annihilates the defining algebraic number.\n\nAll 22 polynomials are independently recovered from the finite Estermann double sum CITE{formula-estermann} in $\\mathbb Q[t]/(\\Phi_L)$. Multiplying over all Galois embeddings and removing repeated factors reproduces every primitive polynomial. This uses neither the generator\u2019s cotangent rational expression nor its minimal-polynomial routine; Sage polynomial arithmetic and factorization remain shared dependencies.\n\nA separate direct cotangent evaluation in 512-bit Arb arithmetic checks the 22 values $\\alpha_k=c_0(1/k)$ for $3\\leq k\\leq24$. The balls are finite, have radii below $7.50\\times10^{-152}$ and retain at least 505 bits of relative accuracy, exceeding 100 significant decimal digits. Every polynomial encloses zero at its ball; signs at enclosing endpoints and a nonvanishing derivative isolate the root locally. Exact Galois sign identities and the modular-inverse Vasyunin conversion were checked for all 178 reduced numerators in this prefix. Known polynomials at $k=2,3,4,6$, wrong signs and scales at $k=4$, perturbed polynomial constants, and invalid enclosure controls test the checker. Code, argument-selection rationale and detailed evidence are attached."
}