Surplus Structure in the Temporal Parameter: Consequences of the Mass-Shell Constraint and the Nref Substitution

The temporal parameter t ∈ ℝ carries three properties that no experiment has confirmed as features of physical reality: negative extension, loop-admitting topology, and reversal symmetry. We identify these as surplus structure in the sense of Weatherall, applying Gisin's observation on ℝ-modelling to the temporal parameter specifically. Its operational content is N_ref: the accumulated state-transition count of a reference system, non-negative and monotonically non-decreasing. The caesium-133 hyperfine oscillator is one realisation; the framework is characterised axiomatically (N1–N4, Section 4) and is independent of the choice of clock.

We establish two theorems. Theorem 1: restricting the Wheeler–DeWitt scalar-field clock to its operationally grounded domain φ ∈ [0, ∞) halves the minisuperspace solution space, excluding independently contracting universes. Theorem 2: within the Page–Wootters framework, the process-accumulation arrow (a spectral property of the clock operator) and the thermodynamic arrow (a boundary condition on the constraint surface) are formally independent.

The Bondi k-calculus then derives the full Lorentz transformation, including the relativity of simultaneity from three operational inputs: transition-count ratios between inertial observers, the relativity principle, and a finite signal speed c. No prior notion of time, metric, or spacetime is assumed; 1/γ is a theorem rather than an input. Closed timelike curves, parameter reversal in T-symmetry, and the block universe as the default interpretation of the formalism are shown to depend on the surplus structure of ℝ and to lack independent empirical support