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Machine-certificate renewal · 13 August 2026 · MASS-PHI-LADDER-20260813

The plate’s half-step recurrence is the shifted Planck–φ scale lattice and is now attached to this existing citizen rather than duplicated.

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Deterministic half-step form

$$ \frac{m(N+\tfrac12)}{m(N)}=\phi^{-1/2},\qquad m(2.5)=M_P $$

implies uniquely

$$ m(N)=M_P\phi^{-(N-2.5)}. $$

Using $M_P=1.22091\times10^{19}\,\mathrm{GeV}$ and $\phi=(1+\sqrt5)/2$, Python deterministically reconstructs:

• $m(90)=6.313426503\,\mathrm{GeV}$;
• $m(91.5)=3.067493624\,\mathrm{GeV}$;
• $m(89)=10.215338667\,\mathrm{GeV}$;
• $m(102.5)=15.414151596\,\mathrm{MeV}$;
• unoccupied levels $70.01697$, $89.06297$, $296.59666$, and $776.50013\,\mathrm{GeV}$.

No particle mass enters the recurrence. The displayed particle values are downstream holdout comparisons.

Synthesis route: Professor’s Cube Exhibit — Shared-Carrier S₃ Symmetry Bridge

Canonical 𝒬 record

$\mathcal Q[\mathsf{Planck\text{-}\phi\ Scale\ Lattice}\mid m_N=m_{\mathrm{Pl}}\phi^{-N};\ \mathbb R_{>0};\ N(m)=-\ln(m/m_{\mathrm{Pl}})/\ln\phi;\ E0;\ \iota\text{ required for }E_{47}]$

Exact lattice

$\phi=\frac{1+\sqrt5}{2},\qquad m_N=m_{\mathrm{Pl}}\phi^{-N}.$

$N(m)=-\frac{\ln(m/m_{\mathrm{Pl}})}{\ln\phi}.$

Conditional E₄₇ route

$N\xrightarrow{\iota}x(N)\in V_2^{\otimes3}\xrightarrow{P_{47}}P_{47}x(N).$