coreExponent
plain-language theorem explainer
Packages the integer rung and gap correction into the single real exponent of the sector-yardstick quark mass law: r − 8 + gap. Cited by anyone equating core and residue/quarter mass coordinates on the φ-ladder. The body is the literal arithmetic expression, not a derived identity.
Claim. For an integer rung $r$ and a real gap correction $g$, the core exponent is the real number $r - 8 + g$.
background
In Recognition Science the fermion mass law sits on a multiplicative φ-ladder. The primer form is yardstick times φ raised to (rung − 8 + gap). The −8 centers the octave (eight-tick) offset; the gap term is the species-dependent residue correction.
This module (Quark Coordinate Unification, Pass 2) treats two coordinate conventions for the same positive mass law. Core form writes m = A_sector · φ^(r − 8 + gap) with a sector yardstick A_sector. Residue/quarter form writes m = m_ref · φ^R relative to a fixed reference mass. The core exponent is exactly the power appearing in the core form.
Upstream, Gap45.Derivation.gap is the integer product of closure and Fibonacci factors (main theorem: equals 45). RSBridge.Anchor.gap is the closed display F(Z) = ln(1+Z/φ)/ln(φ) claimed equal to the integrated RG residue at the anchor. Masses.Anchor.yardstick is the sector scale A_s = 2^{B_pow} · E_coh · φ^{r0}.
proof idea
Definitional abbreviation only: cast the integer rung r to ℝ, subtract 8, add the real gap parameter. No lemmas, tactics, or rewriting. Downstream defs unfold it and work with the resulting real power of φ.
why it matters
This is the shared exponent that makes the two quark mass coordinate systems the same law once a reference mass is fixed. It is unfolded by coreMass (sector yardstick form), residueFromCore (R = log_φ(A_sector/m_ref) + coreExponent), and yardstickFromResidue (inverse map). The equivalence theorems core_eq_residue_of_positive and residue_eq_core reduce to φ-power algebra after unfolding this exponent.
In the framework it is the concrete carrier of the mass-formula landmark yardstick · φ^(rung − 8 + gap(Z)), tying the eight-tick octave offset (−8) to the gap residue on the φ-ladder. It does not itself force gap = 45 or fix numerical quark masses; it is the structural hinge for the coordinate-unification claim of the module.
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