geometric_residue
plain-language theorem explainer
The geometric residue assigns to each fermion its structure-derived F(Z) value via the gap map. Researchers comparing geometric mass ladders to Standard Model running would reference this when quantifying the recognition strength ratio. The definition is a direct composition of the pre-established gap function with the Z assignment for the fermion.
Claim. For a fermion species $f$, the geometric residue is $F(Z_f) := (1/2) (Z_f / phi + (Z_f / phi)^{-1}) - 1$ in equivalent form, or explicitly $F(Z_f) = ln(1 + Z_f / phi) / ln phi$, where $Z_f$ is the effective index obtained from the sector and quantum numbers of $f$.
background
The gap function, defined upstream as $F(Z) = ln(1 + Z/phi) / ln phi$, gives the closed-form residue at the anchor scale for the mass anomalous dimension integral. ZOf extracts the integer Z for each fermion species according to rules that differ by sector (lepton, up, down). The module documents the large discrepancy between these geometric values (e.g., ~13.95 for the electron) and the small perturbative residues obtained from SM beta-function integration.
proof idea
One-line definition that applies the gap function to the Z value produced by ZOf for the input fermion.
why it matters
This supplies the numerator for the recognition_strength ratio and the left-hand side of the structural_dominance_holds predicate. It completes the geometric side of the comparison introduced in the module doc, where the mass formula is claimed to lock to the geometric residue rather than the perturbative shadow. The construction sits inside the Recognition Science phi-ladder mass formula and the eight-tick octave structure.
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