Foundation Eight Tick

The Eight-Tick Structure

The eight-tick structure is the fundamental discrete clock of Recognition Science. It is a cycle of eight phases, spaced by π/4 radians: 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, and 7π/4. The module establishes that this cycle is closed under its own operation: advancing by one tick wraps around after eight steps, and the eighth power of each phase's complex exponential is exactly 1.

The structure's physical force comes from two specific phases. The phase at tick 4 is the half-cycle, and its complex exponential is -1. The phase at tick 0 is the identity, and its complex exponential is 1. The theorem spin_statistics_key packages these two facts together. Because exchanging two identical particles multiplies their combined phase by the exchange phase, a factor of -1 means the wavefunction is antisymmetric, which is the fermion sign, and a factor of 1 means it is symmetric, the boson sign. This is how the eight-tick clock derives spin-statistics: the signs are not assumed, they are forced by the cycle's geometry.

The eight phases are the eighth roots of unity, and two further theorems follow from that fact. The entire cycle is generated by a single step: every phase is a power of the phase at tick 1, so the structure is exactly the cyclic group ℤ/8ℤ. And the sum of all eight phases is zero, which is the algebraic basis for vacuum fluctuation cancellation. The module also defines a fundamental frequency, 8 divided by the constant tau0, tying the discrete clock to the framework's continuous constants.

THEOREM phase · IndisputableMonolith/Foundation/EightTick.lean

THEOREM phase_eighth_power_is_one · IndisputableMonolith/Foundation/EightTick.lean

THEOREM spin_statistics_key · IndisputableMonolith/Foundation/EightTick.lean

THEOREM eight_tick_generates_Z8 · IndisputableMonolith/Foundation/EightTick.lean

THEOREM sum_8_phases_eq_zero · IndisputableMonolith/Foundation/EightTick.lean

What this page does not claim

This answer does not claim that the eight-tick structure is derived from the cost function in this module. This answer does not claim that the spin-statistics sign is a complete derivation of quantum statistics. This answer does not claim that the eight-tick structure is the only possible discrete clock.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/EightTick.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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