time_before_spacetime
Recognition Science places the time-tick stage before the spacetime stage in its pre-temporal forcing order. Researchers tracing the logical dependencies from primitive distinction through J-cost and arithmetic objects to physical spacetime would cite this ordering. The proof is a one-line decision that evaluates the rank comparison directly via the decidable instance on natural numbers.
claimIn the pre-temporal forcing order, the time-tick stage precedes the spacetime stage: $rank(timeTick) < rank(spacetime)$.
background
The module records the dependency order that exists before time in Recognition Science. Physical time is itself a forced object, so the ordering is a forcing order: stage A precedes stage B when B requires A as prior structure. The inductive type Stage enumerates the chain from distinction through recognitionInterface, singleValuedPredicate, symmetricComparison, compositionConsistency, rcl, jCost, arithmeticObject, timeTick to spacetime. The relation Before a b is defined to hold exactly when rank a < rank b, with a decidable instance supplied by Nat.decLt.
proof idea
The proof is a one-line wrapper that invokes the decidable instance for Before. The instance reduces the comparison to Nat.decLt on the ranks of the two stages, which the decide tactic resolves by computation.
why it matters in Recognition Science
This result closes the pre-temporal segment of the forcing chain by confirming spacetime depends on timeTick. It supports the framework claim that physical spacetime emerges after recognition primitives, J-cost, and arithmetic objects, consistent with the later derivation of D=3 and the eight-tick octave. No downstream uses are recorded yet.
scope and limits
- Does not compute explicit numerical ranks for any stage.
- Does not derive the metric or causal structure of spacetime.
- Does not connect the order to the J-uniqueness or phi fixed-point steps.
- Does not address the transition from forcing order to physical time.
formal statement (Lean)
101theorem time_before_spacetime :
102 Before Stage.timeTick Stage.spacetime := by
proof body
Decided by rfl or decide.
103 decide
104