IsBosonic
The bosonic classification for ledger states requires that the minimal recognition cycle length is a multiple of 4 ticks excluding exactly 4 ticks. Researchers deriving commutation relations from the eight-tick structure would cite this predicate when separating integer-spin cases from half-integer ones. It is implemented as a direct propositional definition on the natural number period.
claimA ledger state with minimal recognition cycle of length $p$ is bosonic when $p$ is divisible by 4 but $p$ is not equal to 4.
background
The SpinStatistics module derives the spin-statistics theorem from the eight-tick ledger. The fundamental time quantum is the tick, defined as the constant 1 in RS-native units. Phase accumulation proceeds in steps of $kπ/4$ for $k=0$ to 7, as given by the phase definition over the finite set of 8 ticks.
proof idea
This is a definition that directly encodes the bosonic condition as the conjunction of period modulo 4 equals zero and period not equal to 4. No lemmas or tactics are invoked; the predicate stands alone as the base for later results on integer-spin commutation.
why it matters in Recognition Science
This definition supplies the integer-spin case that feeds spin_integer_commutes and the full spin_statistics_theorem in the module. It implements the 8-tick octave classification (T7) by excluding the 4-tick fermionic period, thereby forcing symmetry under exchange for states that return to themselves after a 2π rotation.
scope and limits
- Does not compute the explicit phase factor acquired under rotation.
- Does not define the exchange operator or its action on states.
- Does not extend to non-integer periods or continuous spin values.
- Does not invoke the J-cost function or Recognition Composition Law.
formal statement (Lean)
36def IsBosonic (period : ℕ) : Prop := period % 4 = 0 ∧ period ≠ 4
proof body
Definition body.
37
38/-- The phase accumulated under a 2π rotation (4-tick advance). -/