coordinate_systems_equivalent
plain-language theorem explainer
Once a positive sector yardstick and reference mass are fixed, the core rung-gap mass law and the residue (quarter) mass law describe the same positive masses: some residue coordinate R always equates them. Quark-mass and ladder-unification arguments cite this to treat the two conventions as interchangeable. The proof is an existence witness via the explicit log_φ transform, discharged by the positive-parameter equality lemma.
Claim. For all real sector amplitudes $A$ and reference masses $m_{\mathrm{ref}}$ that are strictly positive, and for every integer rung $r$ and real gap $g$, there exists a real residue coordinate $R$ such that $A\,\varphi^{r-8+g}=m_{\mathrm{ref}}\,\varphi^{R}$.
background
This module (Quark Coordinate Unification, Pass 2) records that two quark-mass coordinate conventions are the same multiplicative φ-ladder law once a reference mass is chosen. The core form is the integer-rung architecture $m=A_{\mathrm{sector}},\varphi^{r-8+g}$. The residue (quarter) form is the reference-mass shape $m=m_{\mathrm{ref}},\varphi^{R}$.
The explicit bridge is $R=\log_{\varphi}(A_{\mathrm{sector}}/m_{\mathrm{ref}})+(r-8+g)$. Upstream, coreMass and residueMass are those two shapes; residueFromCore is the log_φ map just written; and core_eq_residue_of_positive states that the core value equals the residue value at that image whenever both amplitudes are positive. The gap parameter may come from the Gap45 product (closure times Fibonacci, equal to 45) or from the RSBridge anchor display $F(Z)=\ln(1+Z/\varphi)/\ln\varphi$; this theorem treats it as an arbitrary real.
proof idea
Term-mode existence proof. Introduce the four parameters and the two positivity hypotheses. Witness the existential with the explicit transform residueFromCore A_sector m_ref r gap. The remaining equality goal is exactly core_eq_residue_of_positive applied to the same positivity hypotheses, which unfolds the three defs and finishes by the log_φ power identity for a positive ratio.
why it matters
In the Recognition mass formula, particle masses sit on a φ-ladder of the shape yardstick times $\varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$. Two lab conventions write that law either with a sector yardstick and integer rung, or with a fixed reference mass and a real residue exponent. This theorem is the structural closure of Pass 2: the quarter coordinate is not a second physical law, only a reparameterization once $m_{\mathrm{ref}}>0$ is fixed.
No downstream consumers are wired yet in the graph, so the result stands as a verification lemma rather than a step inside a larger proved chain. It underwrites any later claim that quark masses computed in core coordinates may be rewritten in residue coordinates (or conversely) without changing the predicted positive mass. It does not itself force the numerical gap 45, the anchor display $F(Z)$, or the α band; those live upstream in Gap45 and RSBridge.
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