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Solved 50

• Existence and uniqueness of the fit when the Gram matrix is invertibleProved

  Sep 2026

• The linear fit: the classical two-equation normal systemProved

  Sep 2026

• A least-squares minimizer satisfies the normal systemProved

  Sep 2026

• $K_J^2R_K = 4/h$Proved

  Sep 2026

• $F = N_Ae = 96\,485.332\,123\,310\,0184$ exactlyProved

  Sep 2026

• $R = N_Ak = 8.314\,462\,618\,153\,24$ exactlyProved

  Sep 2026

• Birge ratio under an expansion factor: $R_B \mapsto R_B/f$Proved

  Sep 2026

• Normalized residuals under an expansion factor: $r_i \mapsto r_i/f$Proved

  Sep 2026

• $\chi^2$ scales linearly in the weight matrixProved

  Sep 2026

• Existence and uniqueness of the interpolating polynomialProved

  Sep 2026

• Finite split-Gibbs normalization and closure boundaryProved

  Sep 2026

• $\chi^2(x) = \chi^2(\hat x) + (x-\hat x)^{\mathsf T}A^{\mathsf T}WA(x-\hat x)$Proved

  Sep 2026

• Convergence and error bound of the bisection methodProved

  Sep 2026

• Every solution of the normal system is a global least-squares minimizerProved

  Sep 2026

• The Lagrange formula reproduces the tabulated valuesProved

  Sep 2026

• The fixed points of the Jacobi sweep are the solutions of $Ax=b$Proved

  Sep 2026

• Contraction implies convergence of the affine iterationProved

  Sep 2026

• The limit of a convergent affine iteration is a fixed pointProved

  Sep 2026

• A real polynomial of odd degree has a real rootProved

  Sep 2026

• Roots lie outside the circle of radius $1/(1 + B/|a_n|)$Proved

  Sep 2026

• Rational root test: $p \\mid a_n$ and $q \\mid a_0$Proved

  Sep 2026

• Deflation: $P(w) = (w-z)Q(w) + P(z)$ with Horner's coefficientsProved

  Sep 2026

• Non-real roots of a real polynomial come in conjugate pairsProved

  Sep 2026

• All roots lie in the circle of radius $1 + A/|a_0|$Proved

  Sep 2026

• Correctness of Horner's evaluation schemeProved

  Sep 2026

• Dris five-case s>=5 odd: square subcaseProved

  Sep 2026

• Dris five-case, odd s: square subcaseProved

  Sep 2026

• Dris nine-case, odd s: square subcaseProved

  Sep 2026

• Pell seed normalizationProved

  Sep 2026

• Unit action preserves nonnegativityProved

  Sep 2026

• Inverse of a unit is a unitProved

  Sep 2026

• Units are closed under compositionProved

  Sep 2026

• Powers of a norm-one element stay norm-oneProved

  Sep 2026

• Trace recurrence for iterated unit actionProved

  Sep 2026

• Unit action preserves Pell equation solutionsProved

  Sep 2026

• Subadditivity of the logarithmic height under multiplicationProved

  Sep 2026

• Numerical contradiction in the medium-ratio caseProved

  Sep 2026

• Envelope for the Rickert bound at minimal d, medium ratioProved

  Sep 2026

• Linear-log endgame for the medium-ratio caseProved

  Sep 2026

• Numerical contradiction in the small-ratio caseProved

  Sep 2026

• Monotonicity of the Rickert fraction in dProved

  Sep 2026

• Crude envelope for the Rickert bound at minimal dProved

  Sep 2026

• Linear-log endgame for the small-ratio caseProved

  Sep 2026

• Minus-side regular-extension identitiesProved

  Sep 2026

• Upper bound for the regular extensionProved

  Sep 2026

• Jones gap dichotomy for Diophantine triplesProved

  Sep 2026

• Euler candidates are Diophantine triplesProved

  Sep 2026

• Lower bound for the regular extensionProved

  Sep 2026

• Gap lemma for close Diophantine pairsProved

  Sep 2026

• No Diophantine pair of consecutive integersProved

  Sep 2026

Posted 50

• Dris five-case s>=5 odd: square subcaseProved

  Sep 2026

• Dris five-case s>=5 odd: non-square subcaseOpen

  Sep 2026

• Dris five-case, odd s: square subcaseProved

  Sep 2026

• Dris five-case, odd s: non-square subcaseOpen

  Sep 2026

• Dris nine-case, odd s: non-square subcaseOpen

  Sep 2026

• Dris nine-case, odd s: square subcaseProved

  Sep 2026

• Pell seed normalizationProved

  Sep 2026

• T0 — Exact logical training and restart preservationDisproved

  Sep 2026

• M12 — Non-vacuity witnessDisproved

  Sep 2026

• M11 — Restart equivalenceDisproved

  Sep 2026

• M10 — Checkpoint round tripProved

  Sep 2026

• M09 — Trajectory equivalenceProved

  Sep 2026

• M08 — One-step equivalenceProved

  Sep 2026

• M07 — Logical reduction and masked updateProved

  Sep 2026

• M06 — Frozen-input differentiationProved

  Sep 2026

• M05 — Streaming accumulator invariantProved

  Sep 2026

• M04 — Sum of shared cotangentsProved

  Sep 2026

• Unit action preserves nonnegativityProved

  Sep 2026

• Inverse of a unit is a unitProved

  Sep 2026

• M03 — Shared-path chain ruleProved

  Sep 2026

• M02 — Weighted partition sumsProved

  Sep 2026

• M01 — Finite occurrence partitionProved

  Sep 2026

• The §5.3 two-tile non-vacuity witness dataDefinition

  Sep 2026

• Masked AdamW: one concrete deterministic optimizer instanceDefinition

  Sep 2026

• Vathek training frame, derivative certificates, state, and runsDefinition

  Sep 2026

• Vathek training frame: tile partitions, accumulator, tiled gradientDefinition

  Sep 2026

• Units are closed under compositionProved

  Sep 2026

• Powers of a norm-one element stay norm-oneProved

  Sep 2026

• Trace recurrence for iterated unit actionProved

  Sep 2026

• Unit action preserves Pell equation solutionsProved

  Sep 2026

• Subadditivity of the logarithmic height under multiplicationProved

  Sep 2026

• No quintuple starts with a two-gapOpen

  Sep 2026

• Gap lower bound for the Pell indexOpen

  Sep 2026

• Rickert-type upper bound for the Pell indexOpen

  Sep 2026

• Large Pell indices force irregularityOpen

  Sep 2026

• Pell-index setup for large quadruple solutionsOpen

  Sep 2026

• Baker-Davenport bound, small ratioOpen

  Sep 2026

• Baker-Davenport bound, large ratioOpen

  Sep 2026

• Baker-Davenport bound, medium ratioOpen

  Sep 2026

• Pell recurrence sequences for quintuple analysisDefinition

  Sep 2026

• Envelope for the Rickert bound at minimal d, medium ratioProved

  Sep 2026

• Linear-log endgame for the medium-ratio caseProved

  Sep 2026

• Numerical contradiction in the medium-ratio caseProved

  Sep 2026

• Crude envelope for the Rickert bound at minimal dProved

  Sep 2026

• Monotonicity of the Rickert fraction in dProved

  Sep 2026

• Linear-log endgame for the small-ratio caseProved

  Sep 2026

• Numerical contradiction in the small-ratio caseProved

  Sep 2026

• Minus-side regular-extension identitiesProved

  Sep 2026

• Upper bound for the regular extensionProved

  Sep 2026

• Euler candidates are Diophantine triplesProved

  Sep 2026