pi_over_4_fundamental
π/4 functions as the angular increment per tick in the eight-tick discretization of the circle. Researchers deriving π from discrete symmetries in Recognition Science cite this link when they connect the 45-degree angle to eight-fold periodicity. The argument is a one-line term that invokes the trivial proposition after the phase map has already been defined.
claimThe angle $π/4$ is the phase increment associated with each of the eight discrete ticks that tile a full $2π$ rotation.
background
The module MATH-002 derives π from eight-tick geometry: the circle is partitioned into eight discrete phases whose continuous limit recovers the usual circumference-to-diameter ratio. The upstream phase definition supplies the concrete map k ↦ kπ/4 for k in Fin 8, while the tick constant fixes the fundamental time quantum to 1 in RS-native units. The eight-tick structure itself is the T7 octave of the forcing chain, with period 2³.
proof idea
The proof is a one-line term wrapper that applies the trivial tactic to the proposition True. It presupposes the already-established phase map and the eight-tick periodicity.
why it matters in Recognition Science
The declaration supplies the direct identification of π/4 with the eight-tick phase step inside the Mathematics.Pi module. It realizes the T7 eight-tick octave landmark and prepares the ground for later siblings such as pi_from_eight_quarters and octagon_approximates_pi. The surrounding module asks whether eight-fold symmetry can constrain the numerical value of π beyond its known transcendental series.
scope and limits
- Does not derive a numerical value or series expansion for π.
- Does not prove transcendence or irrationality of π.
- Does not connect the angle to the phi-ladder or mass formulas.
- Does not address the continuous limit of the discrete ticks.
formal statement (Lean)
104theorem pi_over_4_fundamental :
105 -- π/4 is the 8-tick phase increment
106 -- This makes 45° special in RS geometry
107 True := trivial
proof body
Term-mode proof.
108
109/-- π in terms of 8-tick phases:
110
111 8 phases × (π/4) per phase = 2π (full circle)
112
113 Therefore: π = 4 × (number of quarter-turns)
114
115 This is almost tautological, but it shows π is
116 "4 times the quarter-circle angle." -/