gap_45_has_factor_9
Gap-45 equals 45 and is therefore divisible by 9. Researchers deriving D=3 from the Recognition Science synchronization condition cite this to isolate the 3 squared factor inside the 360-period. The proof is a direct term that witnesses the quotient 5 and invokes reflexivity on the definition of gap_45.
claim$9$ divides the integration gap parameter $D^2(D+2)$ evaluated at $D=3$.
background
The DimensionForcing module proves D=3 is forced by requiring that the 8-tick ledger cycle (2^D) synchronizes with the cumulative phase gap whose value at D=3 is 45. Gap-45 is defined as the integer 45, which equals D^2(D+2) at D=3 and supplies the 3^2 factor in lcm(8,45)=360. The module imports PhiForcing and LedgerForcing to combine the arithmetic constraint with topological linking arguments that rule out other dimensions.
proof idea
The term proof supplies the witness 5 for divisibility and applies reflexivity to the equation 9*5=45, using only the constant definition of gap_45.
why it matters in Recognition Science
The result isolates the 3^2 factor required for the synchronization period 360 that appears in the eight-tick octave and D=3 forcing chain. It closes the arithmetic step inside the gap-45/8-tick argument that distinguishes D=3 from other spatial dimensions in the Recognition Science framework.
scope and limits
- Does not derive the numerical value 45 from the J-functional equation.
- Does not prove the physical origin of the 45-tick phase.
- Does not establish uniqueness of D=3 without the surrounding topological and phi-ladder arguments.
formal statement (Lean)
329theorem gap_45_has_factor_9 : 9 ∣ gap_45 := ⟨5, rfl⟩
proof body
Term-mode proof.
330
331/-- The sync period 360 = 8 × 45 / gcd(8,45) = 360. -/