arithmetic_before_time
arithmetic_before_time asserts that the arithmeticObject stage precedes the timeTick stage in the pre-temporal forcing order. Researchers sequencing the dependency chain from distinction to time would cite this result when establishing that arithmetic structure is required before temporal ticks. The proof is a one-line decision procedure that invokes the decidable instance of the Before relation on concrete ranks.
claimIn the pre-temporal forcing order, the arithmetic stage precedes the time-tick stage: $rank(arithmeticObject) < rank(timeTick)$.
background
The Pre-Temporal Forcing Order module records a dependency hierarchy among stages that must precede physical time. Stage is the inductive type listing distinction, recognitionInterface, singleValuedPredicate, symmetricComparison, compositionConsistency, rcl, jCost, arithmeticObject, and timeTick. Before is the relation defined by rank a < rank b, equipped with a decidable instance via Nat.decLt.
proof idea
The proof is a one-line wrapper that applies the decidable instance of Before, reducing directly to Nat.decLt on the ranks of arithmeticObject and timeTick.
why it matters in Recognition Science
This theorem occupies the arithmetic_before_time slot in the sibling chain that runs from jCost_before_arithmetic through time_before_spacetime. It supplies a required link in the pre-temporal forcing order that underpins the Recognition Composition Law and the later emergence of the eight-tick octave and D = 3. The result closes one dependency before time is introduced.
scope and limits
- Does not compute or expose the numerical rank values assigned to stages.
- Does not derive any property of timeTick itself.
- Does not address dependencies after the introduction of time.
- Does not connect directly to J-cost, RCL, or spatial dimensions.
formal statement (Lean)
97theorem arithmetic_before_time :
98 Before Stage.arithmeticObject Stage.timeTick := by
proof body
Decided by rfl or decide.
99 decide
100