Here is an explanation of the Lean theorem threeSixty_is_tick_gap from the module IndisputableMonolith.CrossDomain.CardinalitySpectrum.
1. What the Declaration Says in Plain English
The theorem states a simple arithmetic identity: the natural number 360 is exactly equal to the product of two primitive Recognition Science (RS) constants: 8 (the "tick") and 45 (the "gap").
2. Why it Matters in Recognition Science
In RS, fundamental structures and domain sizes are not random; they are constructed from a small set of primitive integers like $D_{spatial} = 3$ and $D_{config} = 5$. This module defines a structural "cardinality spectrum" of sizes that frequently appear. 360 is culturally recognized as the number of degrees in a "full turn." This theorem provides an RS-native decomposition for 360 as exactly tick × gap. Instead of being an arbitrary human convention, the cardinality 360 emerges cleanly from multiplying the basic eightTick (representing a full $2^3$ spatial cadence) by gap45 (representing $D_{spatial}^2 \times D_{config}$).
3. How to Read the Formal Statement
theorem threeSixty_is_tick_gap : (360 : ℕ) = eightTick * gap45 := by decide
theorem threeSixty_is_tick_gap: The identifier for this verified fact.(360 : ℕ): Specifies that 360 is evaluated as a natural number (non-negative integer).= eightTick * gap45: Asserts equality with the product of the two RS generator definitions.:= by decide: Instructs the Lean compiler to prove the statement automatically by performing basic arithmetic evaluation, since no complex logic is required.
4. Visible Dependencies and Certificates
Within the supplied source, the theorem structurally depends on:
This exact theorem is subsequently used to populate the full_turn field in the CardinalitySpectrumCert structure. The instantiated certificate cardinalitySpectrumCert serves as a formal proof that 360 is a valid, cleanly decomposing member of the RS numerical spectrum.
5. What this Declaration Does Not Prove
This declaration does not prove anything about continuous geometry, continuous physical rotations, $SO(3)$ symmetry groups, or angular momentum. It is strictly an integer arithmetic identity that relates the integer 360 to discrete RS components. The label "full turn" acts as a conceptual mapping rather than a geometric proof of angular variables.