IndisputableMonolith.RSBridge.GapFunctionForcing
This module defines the affine-logarithmic candidate family for the gap function on the reals and derives the forcing conditions that select the canonical phi-based form. Researchers deriving fermion masses from the recognition ladder would cite the gapAffineLog construction when mapping Z indices to display scales. The module consists of auxiliary definitions followed by algebraic lemmas that close the three-point parameter system.
claimThe gap function family on the reals is given by the affine-log form $F(Z) = a + b^{-1} (1 + c Z)$ with the canonical instance $F(Z) = ln(1 + Z phi^{-1}) / ln phi$ forced by three-point closure.
background
The module operates inside the RSBridge layer that connects Recognition Science to Standard Model fermions. Upstream, the Anchor module supplies the gap display function F(Z) = ln(1 + Z/phi)/ln(phi) together with the ZOf charge index for each fermion species, while Constants fixes the base time quantum tau_0 = 1 tick. The present module explores affine-log candidates on the reals as candidate realizations of this display function.
proof idea
This is a definition module, no proofs. It introduces gapAffineLogR and gapAffineLog as the real-valued affine-log candidates, then supplies the supporting algebraic identities (phi_eq_one_add_inv_phi, log_one_add_inv_phi_eq_log_phi, zero_normalization_forces_offset, unit_step_forces_log_scale) that prepare the three-point closure argument.
why it matters in Recognition Science
The module supplies the explicit affine-log realization of the gap function used in massAtAnchor calculations. It feeds the fermion species and Z-map machinery in the Anchor module by showing that three-point conditions force the canonical phi form, thereby closing the parameter space for the phi-ladder mass formula.
scope and limits
- Does not extend the gap function to complex or non-real Z values.
- Does not incorporate quantum loop corrections to the affine form.
- Does not compute explicit numerical masses for individual fermions.
- Does not address interactions with the eight-tick octave or spatial dimension forcing.
depends on (2)
declarations in this module (13)
-
def
gapAffineLogR -
def
gapAffineLog -
lemma
phi_eq_one_add_inv_phi -
lemma
one_add_inv_phi_eq_phi -
lemma
log_one_add_inv_phi_eq_log_phi -
lemma
zero_normalization_forces_offset -
lemma
unit_step_forces_log_scale -
theorem
minus_one_step_forces_phi_shift -
theorem
affine_log_parameters_forced -
theorem
affine_log_collapses_to_gap -
theorem
three_point_forces_canonical_gap -
structure
ThreePointClosure -
theorem
three_point_closure