The Minimal-Breaking Wall Selector Is Not Entailed by the Postulate Ledger: A Gauge-Arc Obstruction in Twistor Configuration Geometry

Abstract

The wall-selection note closed the pre-registered wall-selection question Q2 of the Wall Deletion note conditionally on a minimal-breaking self-energy ansatz: among the three single-node parabolic deletions of $A_3 \cong \mathfrak{su}(4)$ at stratum $n = 3$, a self-energy $w(S) = \phi\!\left(\dim(\mathfrak{su}(4)/\mathfrak{l}_S)\right)$ monotone increasing in the broken-generator count $2k(4-k) = 6, 8, 6$ selects the end-node (Pati–Salam color) deletion $SU(4) \to SU(3)_C \times U(1)_{(B-L)/2}$. That note flagged the ansatz as supplied, not derived (its failure mode F1), and pre-registered the derivation of minimal breaking from the active postulate ledger as the natural next target. We settle that target negatively.

We prove that the active Twistor Configuration Geometry (TCG) ledger $P_0$–$P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, $P_{SO(10)}$ is kinematic: each postulate supplies a Fubini–Study volume functional, a combinatorial Weyl-arrangement datum, an amplitude-ratio coupling tower, a structural maximal-subalgebra inclusion, or a dimensionless gauge-kinetic normalization, and none supplies a self-energy — a real-valued cost — on the wall-deletion sectors. Consequently the minimal-breaking ansatz is not entailed by the ledger: there exist ledger-computable selector completions selecting either node, so no selection is forced, and restricting to genuine self-energies is itself additional dynamical input the ledger does not define.

The ledger is kinematic

The six active postulates supply, respectively, volume functionals ($P_0$–$P_4$, and the hadronic $P_{H'}$ Fubini–Study reading $6\pi^5 = 6!\,\mathrm{Vol}_{\rm FS}(\mathbb{CP}^5)$), a combinatorial Weyl-arrangement datum ($P_7$), an amplitude-ratio coupling tower ($P_6$), a structural maximal-subalgebra inclusion ($P_{SO(10)}$), and a dimensionless gauge-kinetic normalization ($P_{5'}$). None is a vacuum energy, a Higgs potential, or an action on the gauge-breaking sectors. A wall self-energy is a map $w : \{\alpha_1, \alpha_2, \alpha_3\} \to \mathbb{R}$ assigning a real cost to each deletion. The deletion sectors lie in the domain of $P_7$ as Weyl-arrangement data, but not in the domain of any ledger-supplied cost.

The count is canonical as a number, not as an energy

The broken-generator count $N(S) = \dim(\mathfrak{su}(4)/\mathfrak{l}_S) \in \{6, 8, 6\}$ is a canonical invariant of the deletion. The assertion $w(S) = \phi(N(S))$ with $\phi' > 0$ is strictly stronger: it requires a monotone real map $\phi$ whose existence and monotonicity are not consequences of $N$ being canonical. $P_7$ asserts $6 < 8$; it does not assert $w(6) < w(8)$. Promoting a canonical number to a canonical energy is the supplied step.

The obstruction

The minimal-breaking ansatz is not entailed by the ledger. There are ledger-computable selector completions with opposite selections, both built only from $P_7$ data: $w_{\rm br}(S) = N(S)$ — a genuine self-energy proxy — selects the end node; $w_{\rm an}^+(S) = \left|\sum_a q_a^3\right| = \tfrac38, 0, \tfrac38$ — an anomaly magnitude, not a self-energy — selects the center node. So the ledger forces no selection.

A critic may restrict attention to genuine self-energies and thereby exclude $w_{\rm an}^+$. That restriction is legitimate, and within it minimal breaking may well be natural. But it is not defined by the present ledger: no postulate supplies a Lagrangian, a scalar representation, a mass matrix, a heat-kernel or Plancherel trace, or any positivity functional whose domain is the deletion-sector set. The restriction is itself additional dynamical input — which is the decisive reason the result is unchanged-conditional rather than sharpened-conditional.

No sharpening

Competing canonical invariants of the same data select the center node — diagram-automorphism fixedness ($\alpha_2$ is fixed), the vanishing cubic $U(1)$ anomaly ($\sum q^3 = 0$ on the $\mathbf 4$), and the balanced branching $\mathbf 4 \to \mathbf 2 \oplus \mathbf 2$. So no canonical equivalence class of ledger invariants selects the end node uniformly: there is no canonical selector simpliciter, and a sharpened-conditional reading would itself require an extra axiom defining what counts as a wall self-energy.

Verdict

Gauge-arc obstruction theorem — the minimal-breaking wall selector is not entailed by the active ledger; the wall-selection closure of Q2 remains conditional, and the minimal-breaking principle is irreducibly supplied relative to the present ledger. The obstruction specifies exactly the dynamical input a forced closure would require: an action or energy functional on the gauge-breaking sectors. This is the obstruction half of the gauge arc, methodologically analogous — though narrower — to the substrate-level obstruction. The action-level pure-spinor polarization residual $X_{\rm wall\text{-}pol}$ of the Pure-Spinor Condensation Obstruction is orthogonal and remains open. It sharpens the wall-selection note’s failure mode F1 from a disclaimer into a theorem.

No new postulate and no new active-ledger residual are introduced. The active ledger does not move: $P_0\text{--}P_4, \quad P_{5'}, \quad P_6, \quad P_7, \quad P_{H'}, \quad P_{SO(10)}.$

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