Specification, Not Derivation: τCG and the Trace-Selector Package

The three obstruction theorems of the past two weeks — the boundary-superselection obstruction (Paper #27, electron arc), the pure-spinor condensation obstruction (Paper #32, gauge arc), and the representation-slot measure obstruction (Paper #33, hadronic arc) — closed the three-arc symmetric maturity with a striking common diagnosis. All three named residuals — $P_{\rm BFV}^{\rm sec}$, $X_{\rm wall-pol}$, $G3$ — are trace/measure-selection problems, not representation-theoretic problems. The framework’s canonical equivariant geometry produces orbits and projective volumes (representation theory) but does not produce labeled-slot frames, orientation choices, or unit-trace normalizations (measure selection).

Today’s paper takes that cross-arc diagnostic and turns it into a constructive specification. We introduce Trace Configuration Geometry (τCG), with the convention that the Greek letter $\tau$ (for “trace”) replaces the “T” of “Twistor” in TCG — preserving the configuration-geometry skeleton but shifting the organizing principle from twistor-arena invariants to trace-selector outputs over physically resolved sectors.

The central object: trace-selector package

The single missing object is the physical trace-selector package

$\mathfrak{T}_{\rm phys} = (\mathrm{Tr}_{\rm num}, \mathrm{Sel}_{\rm phys}).$

The split avoids a type mismatch flagged in two independent fresh-session GPT reviews: a single number-valued trace cannot also output a Lie subgroup. The cleanest fix is to separate the two roles:

$\mathrm{Tr}_{\rm num}: \mathcal{C}_{\rm num} \longrightarrow \mathbb{R}_{\geq 0}$

handles numerical sector weights (bulk chambers, boundary matchings, the electron prefactor, the hadronic Lenz invariant), while

$\mathrm{Sel}_{\rm phys}: \mathcal{C}_{\rm pol} \longrightarrow \mathrm{Sub}(G_{\rm PS})$

handles polarization-sector stabilizer outputs over the Pati–Salam group $G_{\rm PS} = SU(4)_C \times SU(2)_L \times SU(2)_R$.

This is not yet a category-theoretic functor — that would require source/target categories, morphism actions, and compatibility axioms not yet defined. We therefore call $(\mathfrak{X}, \mathcal{A}_{\rm bulk}, \mathcal{A}_\partial^{\log}, \mathcal{P}_{D_5}, \mathfrak{T}_{\rm phys}, \mathcal{D}_{\rm spin})$ a τCG specification datum (or pre-datum), with the terminology to be upgraded only when a successor construction provides the categorical structure.

The first candidate: labeled-resolution trace

The simplest candidate for $\mathrm{Tr}_{\rm num}$ is a trace over a physically addressable labeled resolution. For a geometric sector $X$ with measure $\mu_X$ and a finite labeled resolution $\pi_X: \widetilde{X}_{\rm phys} \to X$ of degree $d_X$:

$\mathrm{Tr}_{\rm num}(X) = \int_{\widetilde{X}_{\rm phys}} \pi_X^* \mu_X = d_X \cdot \mathrm{Vol}(X) \quad \text{(finite uniform case)}.$

The note restricts to finite uniform resolutions; non-finite and non-uniform cases require Radon–Nikodym pushforward, left out of scope.

Five test results

Test 1 — Bulk chamber factorials. The configuration space of $r$ labeled points on an oriented interval has $r!$ ordering chambers, so $\pi_0(\mathrm{Conf}_r^{\rm lab}(I)) \cong S_r$ and

$\mathrm{Tr}_{\rm num}(B_r) = |S_r| = r!.$

For the FPA ranks $r_n = 2n-2$, this reproduces the chamber weights $\pi + \pi^2 + 4\pi^3$ in the $1/\alpha$ formula. PASSES.

Test 2 — Hard-core boundary Fibonacci. The hard-core algebra has basis monomials in bijection with matchings of the path graph $P_r$, so $\dim \mathcal{A}_\partial^{\rm hc}(r) = F_{r+1}$. With the uniform square-free basis trace (unit counting functional — a linear unit-counting trace on the basis, not a multiplicative augmentation, the distinction is load-bearing):

$\mathrm{Tr}_{\rm num}(\partial_r^{\rm hc}) = F_{r+1}.$

PASSES conditional on (i) hard-core selection and (ii) uniform basis trace. The uniform basis trace is a trace/measure choice in addition to the hard-core selection itself.

Test 3 — Electron polar-normal connected trace. For $r = 4$, $\mathrm{Match}(P_4) = \{\emptyset, 12, 23, 34, 12|34\}$ with average matching size $\langle|M|\rangle = (0+1+1+1+2)/5 = 1$. The sectorwise connected trace gives

$\langle B_e^{\rm conn} \rangle = 1 - \frac{1}{2\pi}.$

CONDITIONAL on residual $P_{\rm BFV}^{\rm sec}$, with four explicit assumptions: hard-core matching sectors, uniform sector measure, sectorwise connected generator, normalized delta contribution.

Test 4 — Hadronic 6! slot multiplier. The canonical Fubini–Study volume of $\mathbb{P}(\wedge^2 \mathbf{4}) \cong \mathbb{CP}^5$ is $\pi^5/5!$. The Lenz identification $6\pi^5 = 6! \cdot \mathrm{Vol}_{FS}(\mathbb{P}(\wedge^2 \mathbf{4}))$ requires the $6!$ multiplier. The obstruction theorem (Paper #33) proved this is not derivable from canonical $SU(4)$-equivariant data: $W(SU(4)) \cong S_4$ has induced action on the six pair labels of order $24$, not $720$; the $P_7$ wall gives a $3+3$ split but not an ordered six-slot frame.

We formalize the central open conjecture:

Definition (Six-slot physical resolution). A canonical six-slot physical resolution is a finite labeled resolution $\pi_H: \widetilde{H}_{\rm phys} \to \mathbb{P}(\wedge^2 \mathbf{4})$ (not necessarily a topological covering space — since $\mathbb{CP}^5$ is simply connected — but behaving as a degree-$6!$ cover for trace purposes), equipped with a slot-labeling map $\lambda: \widetilde{H}_{\rm phys} \to \{\text{ordered frames of the six pair channels}\}$ satisfying: (i) pre-$P_7$-wall, natural w.r.t. the $SU(4)$ representation structure and its Weyl/normalizer action on the six pair channels; (ii) post-$P_7$-wall, natural w.r.t. the induced $SU(3)_C \times U(1)_{B-L}$ decomposition of $\wedge^2 \mathbf{4} = \wedge^2 C \oplus (\ell \wedge C)$; (iii) basis-independent.

Conjecture (Hadronic slot trace). $\mathrm{Tr}_{\rm num}$ derives the hadronic $6!$ multiplier only if it is realized as a labeled-resolution trace over a canonical six-slot physical resolution.

OPEN. Either such a natural object exists, or it does not. This is the sharpest test of τCG, falsifiable, and the central cliffhanger.

Test 5 — Pure-spinor polarization. The compatible polarization $W_+ = (\ell \wedge C) \oplus (\mathbf{2}_L \otimes r_+) \subset V_{10,\mathbb{C}}$ from the compatible-polarization note has stabilizer intersection

$SU(5)_{W_+} \cap G_{\rm PS} \simeq S(U(3) \times U(2)),$

with Lie algebra $\mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y$ and hypercharge $Y = T_{3R} + (B-L)/2$. The sector selector outputs $\mathrm{Sel}_{\rm phys}(W_+) = G_{\rm SM}$ (locally; globally $S(U(3) \times U(2))$ up to standard finite-center quotient). CONDITIONAL on residual $X_{\rm wall-pol}$ from the pure-spinor condensation obstruction note.

In this note $\mathrm{Sel}_{\rm phys}$ is only tested on the single sector $W_+$; a full τCG theory must define its action on morphisms, sector refinements, and competing polarizations.

Minimal extension discipline

τCG should not be extended by aesthetic preference. The rule:

$\boxed{\text{No new structure is added unless it closes a named residual.}}$

In particular: supertwistors only if they derive $\mathcal{D}_{\rm spin}$; higher-form gauge fields only if they construct $\mathfrak{T}_{\rm phys}$; fully extended factorization algebras only if they produce missing sector projectors; Kaluza–Klein corner modes only if they derive $P_6$ and the spin-1 fifth-force coupling; extra Spin(10) Higgs representations only if they arise from existing τCG data.

This matches the anti-evasion guards of the obstruction trilogy and the 2026-05-01 framework closure verdict.

Seven failure modes

F1 (trace-selector package not unique); F2 (hadronic six-slot resolution may not exist canonically); F3 (electron sectorwise trace requires corner BV–BFV machinery, literature gap K4); F4 (pure-spinor condensate requires action-level machinery beyond $\mathbf{16}+\mathbf{10}$ template); F5 (look-elsewhere expansion forbidden).

F6 (functoriality failure). $\mathfrak{T}_{\rm phys}$ may fail to extend to a genuine functor on any natural category if no morphism action, composition law, or compatibility/naturality condition can be defined. In that case τCG remains a useful trace-language heuristic and specification checklist, not a category-theoretic successor datum. This is the most important formal risk and the reason Definition 1 is called a specification datum (or pre-datum) rather than a functor.

F7 (uniform-measure ambiguity). Even when a finite labeled resolution exists, the physical measure on its sheets may not be uniform. Then $\mathrm{Tr}_{\rm num}(X) = d_X \cdot \mathrm{Vol}(X)$ is not forced, and a non-trivial Radon–Nikodym multiplicity density is required.

Related work

τCG is distinct from contemporary geometric-QCD and twistor-string programs such as Migdal’s planar Makeenko–Migdal loop-equation framework (arXiv:2511.13688, arXiv:2602.21129, arXiv:2605.02373). τCG is not a confining-string construction or a derivation of planar QCD; it is a trace/measure-selection specification for the existing TCG residual ledger. The two programs share twistor-flavored vocabulary but operate in different theoretical regimes.

Status table and verdict

Verdict: partial positive — unifying language at the trace-selector level, no derivation, no active-ledger change. Active TCG/FPA 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 (Paper #25) and the compatible-polarization note (Paper #28): a partial-positive specification note that names what successor theory must construct, without claiming the construction has been performed.

The strongest thesis

τCG is not yet a theory. It is a successor-specification datum whose central task is to construct the physical trace-selector package $\mathfrak{T}_{\rm phys} = (\mathrm{Tr}_{\rm num}, \mathrm{Sel}_{\rm phys})$. Its first candidate, the labeled-resolution trace, passes the bulk and hard-core boundary tests, conditionally organizes the electron and pure-spinor sectors, and leaves the hadronic degree-$6!$ six-slot resolution as the key open construction.

τCG names the common missing object; it does not yet build it.

That distinction matters. After two independent fresh-session reviews (both NEEDS_MAJOR initially, both moving to NEEDS_MINOR then READY after revisions), the paper’s claim level is now what it should be: a specification of what successor theory must do, with one concrete falsifiable conjecture (the six-slot resolution) as the central open construction.

The paper, τCG: A Trace-Selector Package Specification for the Twistor Configuration Geometry Residual Classification, is on Zenodo (DOI 10.5281/zenodo.20262057; CC-BY-4.0). Thirteen pages, one definition + one conjecture + one proposition at the cliffhanger, seven failure modes, sixteen references. Refinement trail: GPT initial mLTCFT draft → Claude house-style rewrite (rebrand to τCG with Greek τ) → two independent GPT NEEDS_MAJOR reviews (functor terminology overpromising, codomain type mismatch, etc.) → Claude application of 8 major revisions (split into trace-selector package, pre-datum framing, hadronic conjecture formalized) → two further independent GPT NEEDS_MINOR reviews → Claude application of 6 precision edits + 3 minor tweaks → READY.

Specification, not derivation. The framework names the missing trace-selector object. Successor theory must build it.