bottom_to_strange_eq_phi6
plain-language theorem explainer
The predicted bottom-to-strange quark mass ratio equals φ^6 exactly in the no-PDG forward pipeline. Cite it when checking generation-ladder spacing inside the down sector. The proof applies the structural same-sector ratio lemma (shared charge band, gap cancels) and reduces the integer rung difference 21−15 to 6 by unfolding counting-layer constants.
Claim. Under the Convention A forward mass law, the predicted bottom and strange quark masses satisfy $m_b/m_s=\varphi^6$, where $\varphi$ is the golden ratio. Both share the down-sector yardstick and the same charge-band correction $Z=-1/3$, so only the integer rung gap remains.
background
The module runs a single forward pipeline for all six quark masses with no PDG targets. Each mass is $m_i=A_s,\varphi^{r_i-8+\mathrm{gap}(Z_i)}$, built from a sector yardstick $A_s$ (cube geometry), an integer rung $r_i$ (baseline plus generation torsion), and the charge-band gap.
Within one sector at fixed $Z$, absolute calibration cancels and the mass ratio collapses to a pure power of $\varphi$ equal to the rung difference. The doc-comment records that difference as $21-15=6$ for bottom versus strange.
Counting-layer integers fix the anchors: spatial dimension $D=3$, cube edges $D\cdot 2^{D-1}$, passive field edges (total minus one active edge per tick), and wallpaper groups $W=17$. The down-sector rung map and generation torsion $\tau$ supply the integer rungs that enter the exponent.
proof idea
Apply the structural same-sector ratio lemma on the down-quark sector with rungs $r_{\mathrm{down}}(\mathrm{s})$ and $r_{\mathrm{down}}(\mathrm{b})$ and shared $Z=-1/3$. That lemma yields $m_b/m_s=\varphi^{r_b-r_s}$ after gap cancellation. Unfold the bottom and strange mass abbreviations, rewrite by the lemma, then reduce the exponent by simplifying the rung map, $\tau$, wallpaper count $W$, passive-edge and cube-edge constants, active edges per tick, and $D=3$. Finish with push_cast; norm_num to obtain the integer $6$.
why it matters
A clean internal consistency check of the RS mass formula $m=\mathrm{yardstick}\times\varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$: two down-type quarks related by a pure $\varphi$-power forced only by generation torsion. It lives in the verification layer that audits dimensionless ratios without external mass fits.
Framework landmarks in play: $\varphi$ from T5/T6 (J-uniqueness and the self-similar fixed point), $D=3$ from the forcing chain, and the counting integers ($E_p=11$, $W=17$) that pin the anchors. The exponent $6$ is exactly the strange-to-bottom rung step ($15$ to $21$). No downstream consumers are recorded; the theorem stands as a standalone ratio certificate inside the quark forward pipeline.
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