A Duality-Odd Order Parameter as a Candidate Intrinsic Selection Mechanism for the Geometry-Matter Distinction

We investigate whether an intrinsic duality can define a label-free distinction between two local sectors, without assuming that distinction from the outset. Given a system with an exact, order-2 duality automorphism D pairing two families of local operators, we show the geome-try/matter split can be defined purely by whether D is locally preserved or broken. We develop this construction on the {4, 4} (square) cellulation of T 2. This choice is not an incidental convenience: we derive an Euler-characteristic obstruction showing that no triangulated (simplicial) cellulation of non-trivial topology can support the construction, sharpening an earlier self-duality + non-trivial-topology + locality argument. Our main results are: (1) an exact duality auto- morphism of the minimal L = 2 toric code, with defects localizing exactly where D is broken, together with a continuum Bloch-vector analogue; (2) a generator-operator axiomatization and a Trichotomy Theorem classifying every local extension as defect-creating (a non-empty, even-cardinality defect set), free, or purely topological, verified by exact GF(2) computation on tori and a genus-2 surface, together with an exact discrete defect-pairing theorem and a numerically- supported continuum charge-conservation law; (3) a structural result excluding local sourcing between the two dual sectors by construction under the generator axioms, replaced by an exact topological braiding phase realizing Z2 mutual statistics; and (4) a proof that two natural lattice discretizations of a Gauss-map degree are each, separately, honest topological degrees under an explicit coherence bound, for odd lattice size, with their equality reduced to a single precisely-stated conjecture that we believe but do not prove and for which we identify the specific geometric obstruction a direct proof must overcome — even lattice size breaks the argument at an earlier, purely combinatorial step, which we leave open. All claims are either exact verified computations, proved propositions, or explicitly flagged as conjectural or open. Section 2 situates the construction relative to the pre-existing Kramers–Wannier/duality-defect and loop-quantum-gravity literature.