physical_eight_tick
The declaration confirms that the eight-tick ledger cycle equals eight when the spatial dimension is fixed at three. Researchers closing the Recognition Science derivation of spacetime geometry would cite this equality when linking the topological forcing argument to the octave period. The proof reduces directly to reflexivity on the definition of the eight-tick function evaluated at the physical dimension constant.
claimThe eight-tick period forced by the physical spatial dimension satisfies $2^D = 8$, where $D$ is the constant three.
background
The Dimension Forcing module shows that spatial dimension is fixed at three by a combination of topological linking and synchronization requirements. EightTickFromDimension is the function that sends a dimension value D to the natural number 2^D, encoding the fundamental period of a simplicial ledger cycle. D_physical is the constant three, declared to be the RS-compatible spatial dimension.
proof idea
The proof is a one-line wrapper that applies reflexivity to the definition of EightTickFromDimension at D_physical. Because the definition expands directly to 2 raised to the power of the dimension argument and the physical dimension equals three, the equality to eight follows by arithmetic evaluation.
why it matters in Recognition Science
This equality realizes the T7 eight-tick octave (period 2^3) and T8 D=3 landmarks inside the forcing chain. It supplies the numerical anchor for downstream synchronization statements such as gap_45 and the lcm condition that produces the 360-degree rotation period. The primary justification remains the Alexander duality argument that nontrivial circle linking occurs only in three dimensions, independent of the cost-algebra or kinship-graph results.
scope and limits
- Does not derive D=3 from the cost functional equation alone.
- Does not extend the eight-tick count to non-Euclidean or higher-dimensional ledgers.
- Does not compute explicit mass or coupling values from the period.
- Does not replace the Alexander duality axiom with a purely algebraic derivation.
formal statement (Lean)
399theorem physical_eight_tick : EightTickFromDimension D_physical = 8 := rfl
proof body
Term-mode proof.
400
401/-- **WHY D = 3**
402
403 The dimension is not a free parameter. It is forced by:
404
405 1. **Alexander duality (PRIMARY, proved from axiom)**:
406 `SphereAdmitsCircleLinking D ↔ D = 3`, proved from the reduced
407 cohomology of S¹ axiom in `AlexanderDuality.lean`. Independent of T7.
408 H̃₁(Sᴰ \ S¹) ≅ H̃^{D-2}(S¹), nontrivial iff D = 3.
409
410 2. **Clifford algebra (CHARACTERIZATION)**: Cl₃ ≅ M₂(ℂ) gives
411 2-component complex spinors — the unique structure for spin-½.
412 (See `CliffordBridge.cl3_iso_m2c`)
413
414 3. **Spin group (CHARACTERIZATION)**: Spin(3) ≅ SU(2) is the simplest
415 non-abelian compact Lie group (gauge structure for weak interactions).
416
417 4. **Bott periodicity (CONSEQUENCE)**: Period 8 = 2³ = 2^D follows
418 from D = 3, linking Clifford periodicity to dimension.
419
420 5. **Gap-45 (CONSEQUENCE)**: lcm(8, 45) = 360 = 2³ × 3² × 5 follows
421 from the 8-tick = 2^3 derived from D = 3.
422
423 The Alexander duality argument is the logically primary route.
424 Items 2–5 are consequences or characterizations, not premises. -/