Pati-Salam Representation Volumes and the Lenz Proton-Electron Ratio in TCG

Abstract

The 1951 Lenz observation $m_p/m_e \approx 6\pi^5$ is one of the longest-standing unexplained sub-percent identities involving fundamental dimensionless ratios. Within Twistor Configuration Geometry (TCG), in its FPA realization [Zhang FPA], the formal identity $6\pi^5 = 6! \cdot \mathrm{Vol}_{\mathrm{FS}}(\mathbb{CP}^5)$ is suggestive but resists a literal $n=4$ stratum extension of postulate P0: extending the FPA flag from $\mathbb{CP}^3$ to $\mathbb{CP}^4$ produces $r_4! \cdot \mathrm{Vol}(\mathbb{CP}^4) = 30\pi^4$, not $6\pi^5$, and would corrupt the fine-structure functional $\mathcal{I}_{\mathrm{bulk}}[1] = \pi + \pi^2 + 4\pi^3$.

We show that the Lenz expression admits a clean structural reading via the Pati–Salam $SU(4)$ Weyl-lift of the wall-deletion paper [Zhang Wall Deletion]: under the new postulate P7 (Weyl-lift), the FPA chamber count $r_3! = 24 = |W(A_3)|$ at the existing top stratum $n=3$ is identified with the un-broken $A_3$ root-system data, whose Lie algebra is $\mathfrak{su}(4)$, the Pati–Salam unification algebra with lepton number as a fourth color. The antisymmetric two-index representation $\wedge^2\mathbf{4}$ has dimension $\binom{4}{2} = 6$, and its projectivization $\mathbb{P}(\wedge^2\mathbf{4})$ is canonically isomorphic to $\mathbb{CP}^5$ — the Plücker ambient space for $G(2,4)$, the same twistor-line moduli appearing in P5’. Combining these:

We propose this as the postulate $P_H'$: the first hadronic representation-volume invariant in TCG. Like the dimensionless replacement P5’ for the original electroweak P5 [Zhang EW], $P_H'$ is a phenomenological boundary condition with a sharply specified derivation target — in this case, a Pati–Salam-based hadronic state-sum or boundary action whose representation-slot weighted Fubini–Study measure produces $\dim(\wedge^2\mathbf{4})! \cdot \mathrm{Vol}_{\mathrm{FS}}(\mathbb{P}(\wedge^2\mathbf{4}))$ and whose normalization yields the proton-electron ratio with the electron as the lepton-sector reference.

$P_H'$ is not yet a theorem. In v2, a pre-registered second-observable audit closes the “absence of a second hadronic prediction” gap negatively: the strict grammar $X_R = \dim(R) \cdot \pi^{\dim(R)-1}$ is tested against kaon/pion $(m_K/m_\pi)^2 \approx 4\pi$, Schwinger $\alpha/(2\pi)$, and top Yukawa $y_t \approx 1$, and none survives without post-hoc generalization. $P_H'$ is reclassified from candidate generative representation-volume rule to single-anchor phenomenological structural reading of the Lenz ratio. Three derivation gaps remain (electron normalization, baryon-state construction from $\wedge^2\mathbf{4}$, the $S_6$ representation-slot chamber rule), plus one ongoing audit-discipline constraint.

The active TCG/FPA postulate ledger is updated to read P0–P4, P5’, P6, P7, $P_H'$.

DOI

https://doi.org/10.5281/zenodo.20102322