IndisputableMonolith.Foundation.Breath1024
The Breath1024 module supplies the 1024-tick periodic breath and its supporting 8-tick base structures inside the Recognition Science foundation. Researchers extending the T7 eight-tick octave to higher harmonic resolutions cite these definitions when scaling periodic models. The module declares T, period8, period1024, flipTick, sum8, neutral8, Osc, flipAt512, and phaseLagPiOver4. All content is definitional with no proof obligations.
claimIntroduces the 1024-periodic breath $B(t+1024)=B(t)$ together with the base 8-periodic function $P_8(t)$, the flip operation at tick 512, the sum and neutral elements over 8 ticks, the oscillator Osc, and the phase lag of order $4$ in the 1024-tick cycle.
background
The module resides in the Foundation domain and imports only Mathlib. It realizes the T7 eight-tick octave (period $2^3$) at the next power-of-two scale, period 1024 = $2^{10}$. Definitions for period8, period1024, flipTick, sum8, neutral8, Osc, flipAt512, and phaseLagPiOver4 supply the concrete periodic objects needed to express self-similar fixed points and octave periods in the forcing chain.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
Supplies the periodic breath objects that later modules on the phi-ladder and mass formula reference when moving from the eight-tick base to finer 1024-tick resolution. It directly supports extensions toward the alpha inverse band and Berry creation threshold calculations in the T7-T8 segment of the chain.
scope and limits
- Does not contain any theorems or proofs.
- Does not invoke the Recognition Composition Law.
- Does not define constants such as G, alpha, or the mass ladder.
- Does not connect to spatial dimension D=3 or the full T0-T8 forcing chain.