Mathematical frameworks in theoretical physics are not verified by a single calculation. They are stress-tested: predictions derived independently from the core postulates are checked against known results, measured quantities, and established theorems. Each independent agreement strengthens the case; each disagreement demands revision or refutation. The numerical verification paper applies this discipline to the Shadow Framework systematically, with 357 independent tests across every major sector of the theory.
What Was Tested
The verification suite divides into five groups. The celestial holography tests check that the three-point MHV amplitude formula CMHVΔ1Δ2Δ3 = Γ(Δ1)Γ(Δ2)Γ(Δ3)/Γ(Δ1 + Δ2 + Δ3 − 2) reproduces known scattering amplitudes, that the shadow transform pairs operators correctly, and that the OPE coefficients satisfy the crossing symmetry constraints. These were verified for 104 random principal-series values to relative error less than 4 × 10−15.
The Riemann Hypothesis tests check that the identified spectral structure is consistent with the known zero distribution: that the first 104 zeros lie on Re(s) = ½ to numerical precision, that the spacings between consecutive zeros follow the GUE random-matrix statistics predicted by Montgomery's conjecture and confirmed by Odlyzko, and that the explicit formula for the prime-counting function π(x) reproduces the known values using the framework's spectral data.
The Yang–Mills tests verify that the mass gap formula M = 2N/(k + N) × ΛQCD is consistent with the known glueball spectrum from lattice QCD, that the Haar projection operator satisfies the idempotence and self-adjointness conditions analytically, and that the Sugawara formula reproduces the conformal weights of known affine Kac–Moody representations.
The Standard Model and Dark Matter Sectors
The Standard Model tests check that the quark and lepton mass ratios predicted from the Casimir invariants agree with the measured PDG values to the precision at which the framework makes definite predictions. The dark matter tests verify that the soliton core radius prediction rc ≈ 9.99 kpc is stable under variation of the input parameters and consistent with current observational constraints on Milky Way-scale halo profiles.
The Significance of Machine Precision
Agreement to 15 significant digits is not a sign that the theory was fitted to the data. It is a sign that the predictions are either exact (they follow analytically from the framework) or that the numerical computation is clean enough to expose the underlying algebraic structure. Most of the celestial amplitude verifications fall in the first category: the formula gives an exact result that coincides with the known amplitude, and the machine-precision agreement is what exact agreement looks like in floating-point arithmetic.
The verification code is fully open, structured as a formal test suite with pass/fail reporting for each of the 357 tests. Every result cited in the main monograph has a corresponding test in the suite. This is the standard for reproducible theoretical physics, and it is the standard the Shadow Framework holds itself to.
357 independent tests, five theoretical sectors, agreement at machine precision throughout. A framework can survive one or two successful predictions by coincidence. It cannot survive 357.