A Pre-Geometric Framework from Distinction and Reference: Derived Valence, Emergent Causal Structure, and a Closure Definition of Matter

This record extends the Project Nova framework (v7.1-A constraint-based selection principle for the Standard Model gauge group, II: an anchored substrate, a channel-graded operator, and the parity chain) with seven developments obtained under the project’s standing epistemic conventions (claim registry; pre-registered falsification clauses; adverse results reported at full weight). (1) A Derived Valence Theorem: a bound primitive carries exactly 2+ 1 = 3 relations, making the role-alphabet closure unconditional and the metric link-graph one-dimensional by theorem, so physical space cannot be the link-graph. (2) A derived tetrahedral cap: K4 forms dynamically, K5 is impossible (zero formed in 23,651 attempts at the largest scale). (3) A cap-exemption theorem: reference-sector co-participation is not valence-capped and is the unique sector admitting dimension above one; the reference network of saturated tetrahedra is connected at every scale tested, and under recency rooting it is one-dimensional and formation-ordered — a time-like, not space-like, structure. (4) Retirement of ad hoc fading: expression capacity (four dyadic face-channels per completed tetrahedron) yields an emergent growth frontier: the past closes by exhaustion, not decay. (5) Under localityconstrained gluing the resulting complex exhibits the first scale-stable dimensionality in the project’s history, dH → 2.1–2.2 with ds ≈ 1.4, identified with the branched-polymer universality class of Euclidean dynamical triangulations. (6) An adopted interpretation (I-3) — primitive as unexpressed force potential, reference as null causal field, matter as the expressed state — with a formally verified expression predicate: E(x) = 1 iff x belongs to a closure, exact ledger–audit agreement at two hierarchy levels, an unexpressed (dark-sector analog) fraction near one half at untuned economics, and a falling closed-cycle length spectrum. (7) A first causal-synthesis measurement: constraining gluing by the derived genealogy ordering yields dH = 2.47 → 2.81 → 2.95 with a decelerating diameter exponent (4.05 → 3.63) — the first result consistent with the pre-registered dimension three, explicitly not claimed as established; the discriminating larger run is pre-registered.