IndisputableMonolith.RSBridge.GapFunctionForcing
The module develops affine-log families on the reals as candidate forms for the gap function in the RS bridge to particle physics. Researchers deriving Z-to-mass mappings or fermion spectra would reference its forcing lemmas. The argument consists of sequential algebraic lemmas that impose normalization, step, and three-point conditions to collapse the family onto the canonical gap.
claimAn affine-log gap candidate takes the form $F(Z) = a \log(b + c Z)$ on the reals, with parameters $a, b, c$ fixed by zero normalization, unit-step scaling, and three-point closure to recover the canonical $F(Z) = \ln(1 + Z/\phi)/\ln\phi$.
background
The module sits inside the RSBridge domain and imports the time quantum $\tau_0 = 1$ tick from Constants together with the core bridge objects from Anchor. Anchor defines the 12 Standard Model fermions, the charge-indexed integer $Z_i = \tilde{q}^2 + \tilde{q}^4$ (+4 for quarks), the display function $F(Z) = \ln(1 + Z/\phi)/\ln\phi$, and the anchor-scale mass map. The present module examines the larger affine-log family that contains this canonical gap.
Sibling declarations introduce gapAffineLogR and gapAffineLog, then prove a chain of forcing statements (zero_normalization_forces_offset, unit_step_forces_log_scale, minus_one_step_forces_phi_shift, affine_log_parameters_forced, affine_log_collapses_to_gap) that terminate in three_point_forces_canonical_gap and the ThreePointClosure theorem.
proof idea
The module is organized as a sequence of lemmas rather than a single theorem. Each lemma applies an algebraic identity or normalization condition imported from Anchor and Constants: zero normalization fixes the offset, the unit step forces logarithmic scaling, the minus-one step forces the $\phi$ shift, and the three-point condition collapses the remaining parameters. The final ThreePointClosure packages these steps into a single statement that the canonical gap is the unique affine-log function satisfying the listed conditions.
why it matters in Recognition Science
The module supplies the explicit forcing argument that selects the canonical gap function used by massAtAnchor and the Z-map in Anchor. It therefore occupies the step that converts the abstract recognition composition law into a concrete functional form for fermion masses. Although no downstream declarations are yet recorded, the module directly supports any later theorem that invokes the gap function inside the RS forcing chain (T5–T8).
scope and limits
- Does not derive numerical mass spectra for specific fermions.
- Does not extend the affine-log form beyond real Z.
- Does not incorporate running or quantum corrections above the anchor scale.
- Does not claim uniqueness without the three-point closure hypothesis.
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