gap_up_theory
plain-language theorem explainer
The definition assigns the theoretical residue gap for up quarks by evaluating the display function at Z=276. Researchers auditing fermion mass hierarchies in Recognition Science cite it when comparing RG-transported experimental values against the anchor-scale prediction. It is realized as a direct one-line application of the gap function to the Z-value extracted for the up sector.
Claim. $F(276) = (1/2) ln(1 + 276/φ) / ln(φ) ≈ 10.71$, where $F(Z)$ is the closed-form display function for the integrated RG residue at the anchor scale and φ is the golden ratio.
background
The gap function is defined as ln(1 + Z/φ)/ln(φ) and serves as the theoretical residue value that the integrated mass anomalous dimension is claimed to equal at the anchor scale μ⋆. For the up quark the index Z is obtained from the sector and charge rules that map the up fermion to 276. This module supplies numerical audit certificates that bound experimental residues to verify the display_identity_at_anchor axiom from AnchorPolicy.
proof idea
One-line definition that applies the gap function directly to the Z-value of the up quark.
why it matters
This definition supplies the theoretical target F(276) ≈ 10.71 for the up-quark residue certificate, supporting verification of the anchor-scale identity. It quantifies the quark tension noted in the module documentation, where observed residues deviate from the phi-ladder prediction, and relates to the forcing-chain steps that fix D=3 and the eight-tick octave. No downstream theorems depend on it, leaving the experimental comparison as an open audit step.
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