A Minimal-Breaking Principle Selects the Pati–Salam Wall: A Conditional Closure of the Wall-Selection Question in Twistor Configuration Geometry

Abstract

The Wall Deletion note promoted the chamber decomposition of Twistor Configuration Geometry (TCG) to a full type-$A$ Weyl-arrangement structure (postulate $P_7$) and showed that a single wall deletion at stratum $n = 3$ — the parabolic Levi reduction induced by deleting one simple root from the $A_3 \cong \mathfrak{su}(4)$ Dynkin diagram — lands at the Pati–Salam color subalgebra $\mathfrak{su}(3)\oplus\mathfrak{u}(1)_{(B-L)/2}$ provided the deletion is at an end node; the center node instead gives $\mathfrak{su}(2)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$. That note pre-registered, as its principal open question (Q2), what within TCG selects which simple root to delete?

This note supplies a candidate selection principle. We organize the three single-node deletions as the sectors of a wall-selection functional and show: (i) the bare Weyl-arrangement data admit several inequivalent natural invariants that do not agree on a single deletion — the gauge-level analog of the order-parameter ambiguity of the substrate obstruction theorem — so no selection principle is forced by $A_3$ alone; (ii) conditional on a minimal-breaking (maximal-residual-symmetry) self-energy ansatz, the end-node deletion $SU(4)\to SU(3)_C\times U(1)_{(B-L)/2}$ is selected uniquely up to charge conjugation.

The three wall deletions

The end-node generator $\varpi_3 = \tfrac14(1,1,1,-3)$ is proportional to the Pati–Salam $(B-L)/2$ generator; the end nodes $\{\alpha_1,\alpha_3\}$ are exchanged by charge conjugation (the same color breaking). The center node $\alpha_2$ is the inequivalent left–right breaking.

Minimal breaking selects the end node

Bare $A_3$ does not force a selector: the cubic $U(1)$ anomaly $\sum q^3$ on the $\mathbf 4$ favors the center node ($\sum q^3 = 0$), while the broken-generator count favors the end nodes ($6 < 8$). Conditional on a minimal-breaking self-energy ansatz $w(S) = \phi\!\left(\dim(\mathfrak{su}(4)/\mathfrak{l}_S)\right)$ with $\phi' > 0$ — a standard maximal-little-group heuristic in the Michel stratification of spontaneous symmetry breaking, though not forced (different invariant potentials can select different little groups) — the end-node orbit carries the minimal self-energy and is selected.

The natural invariants split into two classes. Wall-self-energy criteria (minimal broken-generator count, maximal residual symmetry, minimal Richardson orbit $[2,1,1]$) all select the end node; canonicity / representation / consistency criteria (diagram-automorphism fixedness, self-duality of $\mathrm{Gr}(2,4)$, balanced branching, vanishing cubic anomaly) favor the center. The anomaly entry is genuinely physical but is not a self-energy of the wall-deletion sector; a dynamical, energy-minimizing functional selects the Pati–Salam end node.

What is and isn’t closed

The result is gauge-arc-internal and conditional, not forced — minimal breaking is a supplied ansatz, and the competing center-node selectors are exhibited explicitly. The selection lands at the Pati–Salam color subalgebra, not the full Standard Model: the missing $\mathfrak{su}(2)_R$ / hypercharge is untouched. The deeper action-level residual $X_{\rm wall\text{-}pol}$ of the Pure-Spinor Condensation Obstruction — which pure-spinor polarization representative the $\mathrm{Spin}(10)$ condensate selects — is orthogonal to Q2 and remains open. The configurable-universe reading is preserved unchanged.

Verdict

Conditional-closure construction note — conditional closure of the wall-selection question Q2 of the Wall Deletion note via a minimal-breaking principle; the Pati–Salam end-node deletion is selected uniquely up to charge conjugation; the closure is conditional, not forced. No new postulate and no new active-ledger residual are introduced; the maturity register is that of the AHS substrate closure. The natural upgrade is to derive the minimal-breaking energetics from the active ledger, which would turn the conditional selection into a forced one.

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

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