The Representation-Slot Measure Obstruction in the Hadronic P_H' Invariant of Twistor Configuration Geometry

Abstract

The Lenz observation $m_p/m_e \approx 6\pi^5$ has a compact hadronic reading in Twistor Configuration Geometry: $6\pi^5 = 6! \cdot \mathrm{Vol}_{FS}\bigl(\mathbb{P}(\wedge^2 \mathbf{4})\bigr), \quad \mathbb{P}(\wedge^2 \mathbf{4}) \cong \mathbb{CP}^5, \quad \mathrm{Vol}_{FS}(\mathbb{CP}^5) = \frac{\pi^5}{5!}.$ The projective space is the antisymmetric two-bitwistor pair-channel space of the Pati–Salam $SU(4)$ sector, as developed in the bitwistor pair-channel note. This note asks whether the remaining factor $6!$ is forced by the same canonical geometry or is an additional labeled-slot input.

The answer is theorem-level obstructed. Under the TCG/FPA Fubini–Study normalization convention $[\omega_{FS}] = \pi H$ inherited from the Pati–Salam representation-volumes note, the geometric half $\pi^5/5!$ is canonical (within that convention). The factorial half is not.

Theorem 1 (Weyl-symmetry / $S_4$-versus-$S_6$ obstruction, §3). The actual weights of $\wedge^2 \mathbf{4}$ live in the $A_3$ weight lattice; their pair-channel coordinate labels, inherited from a choice of basis of $\mathbf{4}$, are the unordered pairs $\{12, 13, 14, 23, 24, 34\}$. The Weyl group action canonically induced on these six labels is the image of $S_4$ acting on two-element subsets of $\{1,2,3,4\}$, a faithful but proper embedding $S_4 \hookrightarrow \mathrm{Sym}\{12,13,14,23,24,34\} \cong S_6$ of image order $4! = 24$, not $6! = 720$. Passing from the induced pair action to arbitrary slot permutations forgets the incidence structure inherited from four fundamental labels — that forgetting is an extra slot-frame choice, not Weyl symmetry.

Theorem 2 (Projective-geometry obstruction, §4). The canonical $SU(4)$-equivariant projective geometry of $\mathbb{P}(\wedge^2 \mathbf{4}) \cong \mathbb{CP}^5$ derives the factor $\pi^5/5!$ via the standard top-degree Fubini–Study integral. No natural $SU(4)$-equivariant projective invariant constructed from the standard Fubini–Study form and Chern–Weil data ($c(T\mathbb{CP}^5) = (1+H)^6$, $c_j = \binom{6}{j} H^j$) selects the ordered six-slot factor $6!$ without an additional normalization or slot-frame choice.

Theorem 4 (Berezin / action-level trace obstruction, §5). For a six-dimensional complex pair-channel field with kinetic operator $K$, a finite Gaussian gives $Z_B \propto (\det K)^{-1}$, $Z_F \propto \det K$ — for $K = I$, just $1$, not $6!$. Berezin integration with $\eta = \sum_{i=1}^6 \bar\theta_i \theta_i$ produces $\int d^6\bar\theta\, d^6\theta\, \eta^6 = 6!, \quad \int d^6\bar\theta\, d^6\theta\, \frac{\eta^6}{6!} = 1, \quad \int d^6\bar\theta\, d^6\theta\, e^\eta = 1.$ Keeping the factorial requires withholding the canonical normalization. That choice is external to the action.

Proposition 5 (Auxiliary obstruction, §6). Geometric quantization of $\mathbb{CP}^5$ with the hyperplane bundle $\mathcal{O}(k)$ gives $\dim H^0(\mathbb{CP}^5, \mathcal{O}(k)) = \binom{k+5}{5}$. The sequence skips $720$ — between $k = 6$ ($\binom{11}{5} = 462$) and $k = 7$ ($\binom{12}{5} = 792$) — so no integer $k$ yields $6!$. The $P_7$ wall gives the Pati–Salam split $\mathbf{4} = C \oplus \ell$, hence $\wedge^2 \mathbf{4} \cong \wedge^2 C \oplus (\ell \wedge C)$, a meaningful $3+3$ decomposition of color–color and lepton–color pair channels; the naturally derived residual symmetry is a single diagonal $S_3$ fixing the lepton label, and even granting independent block permutations or block exchange the order is bounded by $|S_3 \wr S_2| = 72$, far below $720$.

Corollary 6 ($G3$ obstruction, relative form, §7). Under canonical $SU(4)$-equivariant TCG/FPA structures — projective geometry, Weyl symmetry, Chern–Weil, geometric quantization, Gaussian/Berezin action-level trace, and $P_7$ wall structure — the $6!$ multiplier is not derivable. It remains an FPA-style labeled-slot residual: subgap $G3$ of the original $P_{H'}$ subgap list. This is a relative obstruction: it rules out derivation from the current canonical $SU(4)$-equivariant projective, Weyl, Gaussian/Berezin, quantization, and $P_7$-wall data, unless an additional slot frame, boundary trace, or measure normalization is introduced. Any successful future derivation must identify that structure and record it as new input rather than hide it inside $P_{H'}$.

Verdict: theorem-level OBSTRUCTION. $P_{H'}$ remains active as a single-anchor phenomenological structural reading of the Lenz observation; only the factorial slot multiplier is residual. The active TCG/FPA postulate ledger is unchanged: $P_0$–$P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, $P_{SO(10)}$.

Completes three-arc symmetric maturity. The hadronic-side arc (Pati–Salam representation-volumes + bitwistor pair-channel notes → present obstruction note) now has the same maturity register as the electron-side arc (bulk–boundary localization + connected-residues notes → boundary-superselection obstruction note with $P_{\rm BFV}^{\rm sec}$ residual) and the gauge-side arc (Spin(10) downstream-breaking + pure-spinor polarization + compatible-polarization notes → pure-spinor condensation obstruction note with $X_{\rm wall-pol}$ residual). All three arcs now supply theorem-level action-level obstruction notes that name precise residuals outside the active framework ledger.

Cross-arc pattern. The three remaining residuals classify as trace/measure-selection problems, not representation-theoretic problems: the framework’s canonical equivariant geometry produces orbits and projective volumes but does not produce labeled-slot frames, orientation choices, or unit-trace normalizations. Electron $P_{\rm BFV}^{\rm sec}$: sector trace + uniform measure + unit augmentation unforced. Gauge $X_{\rm wall-pol}$: orientation/representative selection within the pure-spinor orbit unforced. Hadronic $G3$: $6!$ labeled-slot count unforced from canonical $SU(4)$ symmetry.

Five failure modes (§8) record where the obstruction does NOT close: multi-trace selection mechanism (corner BV–BFV or factorization-algebra trace, parallel to the electron-side corner gap) (F1); non-equivariant slot frame, equal to the labeled-slot input under review (F2); hadronic-side boundary or auxiliary structure parallel to the gauge-side $X_{\rm wall-pol}$ (F3); honest acceptance of $G3$ as imposed input (F4); look-elsewhere expansion to other representations such as $\mathbb{P}(\mathbf{10})$ or $\mathbb{P}(\mathbf{16})$ explicitly forbidden by anti-evasion discipline (F5). A scope-of-the-theorem paragraph documents the relative-obstruction form.

DOI

https://doi.org/10.5281/zenodo.20149827