The Hadronic Six-Slot Resolution Problem in Trace Configuration Geometry

Abstract

The τCG specification note introduced the physical trace-selector package $\mathfrak{T}_{\rm phys} = (\mathrm{Tr}_{\rm num}, \mathrm{Sel}_{\rm phys})$ as the constructive response to the obstruction trilogy’s trace/measure-selection diagnostic, and identified the hadronic $6!$ slot multiplier as the sharpest open construction test. The hadronic Lenz reading requires $6\pi^5 = 6! \cdot \mathrm{Vol}_{FS}(\mathbb{P}(\wedge^2 \mathbf{4}))$; the geometric half $\pi^5/5!$ is canonical but the multiplier $6!$ is not.

This note tests the conjecture directly. We prove that under minimal τCG data — the projective pair-channel space $\mathbb{P}(\wedge^2 \mathbf{4})$ with $SU(4)$-equivariant Fubini–Study geometry plus the $P_7$ wall split $\mathbf{4} = C \oplus \ell$ — no canonical degree-$6!$ finite labeled resolution exists. We then exhibit a conditional positive route via the pair-channel addressability principle $P_{\rm pair}^{\rm addr}$, which closes the construction conditionally and sharpens the residual.

Two-sided structure

Negative half (§§2–6): the connected group $SU(4)$ cannot act nontrivially on a finite six-element slot set; $W(SU(4)) \cong S_4$ acts on the six unordered pair labels with image of order $24 < 720$; the $P_7$ wall yields a meaningful $3+3$ decomposition $\wedge^2 \mathbf{4} = \wedge^2 C \oplus (\ell \wedge C)$ but not an ordered six-slot frame; and Berezin/Fock saturation produces $6!$ only after choosing six Grassmann slot pairs and an unnormalized top insertion.

Conditional positive half (§§7–9): The top FPA/$P_7$ stratum supplies a four-slot label carrier $S_4^{\rm FPA} = \{1,2,3,4\}$. The complete pair-slot set $\Omega_2(S_4^{\rm FPA}) = \{\{i,j\} : 1 \le i < j \le 4\}$ is the edge set of the complete graph $K_4$, with cardinality $\binom{4}{2} = 6$ — distinct from the adjacent hard-core edges of the path graph $P_4$ used in the electron boundary sector.

Under the pair-channel addressability principle

$P_{\rm pair}^{\rm addr}: \text{the six complete pair channels of } S_4^{\rm FPA} \text{ are physical boundary-defect addresses},$

the ordered-slot resolution $\widetilde{H}_{\rm pair} = \mathbb{P}(\wedge^2 \mathbf{4}) \times \mathrm{Ord}(\Omega_2(S_4^{\rm FPA}))$ has degree $|\mathrm{Ord}(\Omega_2)| = 6!$, and the labeled-resolution trace gives $\mathrm{Tr}_{\rm num}(H_{\wedge^2}) = 6! \cdot \mathrm{Vol}_{FS}(\mathbb{P}(\wedge^2 \mathbf{4})) = 6\pi^5.$

Combined verdict

$\boxed{\text{minimal } \tau\text{CG fails;} \quad \tau\text{CG} + P_{\rm pair}^{\rm addr} \text{ succeeds;} \quad \text{exact residual} = P_{\rm pair}^{\rm addr}.}$

The old residual “why does $6!$ multiply $\pi^5/5!$?” is replaced by the sharper, more physical residual “why are the six complete pair channels of the $P_7$ four-slot carrier physically addressable boundary defects?”

This converts an unexplained multiplier into a concrete physical-addressability question.

Three named trace/measure-selection residuals across the three arcs

All three are labeled successor-construction targets outside the active TCG/τCG ledger.

Status table

Verdict

Partial positive — unifying language at the trace-selector level, no derivation, no active-ledger change.

Active TCG/τCG postulate ledger UNCHANGED: $P_0\text{–}P_4, \quad P_{5'}, \quad P_6, \quad P_7, \quad P_{H'}, \quad P_{SO(10)}.$

The 2026-05-01 framework closure verdict is preserved. Same maturity register as the bitwistor pair-channel note, the compatible-polarization note, and the τCG specification: partial-positive mechanism note that names what successor theory must construct, without claiming the construction has been performed.

$P_{\rm pair}^{\rm addr}$ is a labeled successor-construction target, NOT a new framework axiom.

Five failure modes

• F1. Pair-address failure (formal labels, not physical defects)
• F2. Gauge-frame objection (acceptable only if 4 slots inherited from FPA label resolution)
• F3. Uniform ordered-trace ambiguity (non-uniform measure on Ord)
• F4. Look-elsewhere expansion forbidden (no $d! \cdot \mathrm{Vol}_{FS}(\mathbb{P}(R))$ for arbitrary representations)
• F5. QCD/flavor specificity not addressed (doesn’t identify proton, derive confinement, or explain flavor)

DOI

https://doi.org/10.5281/zenodo.20262722