Abstract
Does a relation-first theory of complementary roles imply more than one timelike dimension? At the primitive level no time count is defined: C = [a ⊣ b] supplies neither a real carrier nor a metric. Exact real-form countermodels show that one complex determinant carrier admits Lorentzian and split signatures, while two complementary null roles with nonzero cross-pairing span a hyperbolic plane of signature (1,1). After explicitly adopting a finite Euclidean Jordan bridge and a causal-carrier bridge, the condition that a primitive role and its full complement are both primitive forces rank two. The nonclassical simple descendants are spin factors J_m = ℝe ⊕ V_m, whose canonical complement fixes ℝe and yields η(x,y) = g_+(x,y^#), with sig(η) = (1,m). Thus the declared causal carrier has one timelike completion direction and an m-dimensional spacelike distinction sector. Reciprocal branches are opposite boosts on one future hyperboloid. Temporal transport, quantitative duration, clock phase, and the thermodynamic arrow remain distinct. Under a separate metrological order-unit bridge, a local positive order unit supplies a unique inverse-scale compensator in the stated class, but common rescaling leaves the absolute second undetermined. Ambient two-time descriptions are comparison frameworks, not extra observable clock coordinates. As a separately owned companion consequence, the paper Complementarity Before Electromagnetism (public record CEM-P1) shows that fixing the field-form degree to r_F = 2 and supplying an invertible complement-odd sector exchange S_T balances the induced two-form sectors only at m = 3, so that the total spin-factor carrier dimension is four. Time alone does not supply those exterior-square selectors. The balance calculation neither replaces the conditional Time-owned form η nor selects an additional or preferred Hodge-defining metric, orientation, constitutive law, or Hodge operator. The result is conditional and does not derive spacetime from bare complementarity.
Where it sits in the release
CFT-P1 opens the time arc. It asks whether relation-first complementarity forces more than one timelike dimension, and answers conditionally: a supplied rank-two carrier admits a single timelike direction and a relational notion of duration. The absolute scale of that duration is not derived — it is named as an open boundary.
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