Predictions from the Fundamental Quadrant: A Spectral Decomposition of Physical Couplings

The canvas model derives the gauge and Yukawa couplings of the Standard Model from definite integrals over the fundamental angular domain [0, π/2]. The integrand is sin^n θ, where the power n is determined by the physical process: n=1 for the U(1) and SU(3) gauge couplings, n=2 for the spherical measure governing mass hierarchies, and n=3 for the SU(2) gauge coupling. What this paper shows: We show that these integrals form a complete spectral sequence. The first three powers (n=0,1,2,3) correspond to known Standard Model parameters. The higher powers (n=4,5,6,...) are currently unassigned. We propose that each unassigned integral corresponds to a physical coupling not yet observed—either a new particle, a new interaction, or a higher-order correction to known processes. The predicted coupling strengths for the first six unassigned levels are: · n=4: I₄ = 3π/16 ≈ 0.589 → α₄ ∼ 0.0177 (new scalar self-coupling)· n=5: I₅ = 8/15 ≈ 0.533 → α₅ ∼ 0.0161 (new five-point interaction)· n=6: I₆ = 5π/32 ≈ 0.491 → α₆ ∼ 0.0148 (gauge absolute scale, partially assigned)· n=7: I₇ = 16/35 ≈ 0.457 → α₇ ∼ 0.0138 (new interaction threshold)· n=8: I₈ = 35π/256 ≈ 0.429 → α₈ ∼ 0.0129 (new interaction threshold)· n=9: I₉ = 128/315 ≈ 0.406 → α₉ ∼ 0.0122 (new interaction threshold) Why this matters: These predictions are falsifiable. If no new particles or interactions are observed with couplings of these magnitudes at the corresponding energy scales, the spectral decomposition hypothesis is ruled out. Conversely, the discovery of new physics matching any of these predicted couplings would provide strong evidence for the canvas model. The periodic table of fundamental integrals is presented as a completeness check on the theory and as a source of genuine, testable predictions for physics beyond the Standard Model. Keywords: spectral decomposition, fundamental integrals, gauge couplings, Yukawa couplings, physics beyond the Standard Model, canvas model, testable predictions, new physics