rs_universal_clock
The eight-tick phase is uniform across all space because it is fixed globally by the ledger structure at initialization rather than determined locally or propagated at finite speed. Cosmologists examining ledger-based alternatives to inflation would cite this when treating homogeneity as a consistency condition enforced by J-cost minimization. The proof is a one-line term-mode reduction that accepts the claim as the trivial proposition True.
claimThe phase of the eight-tick cycle is identical in every spatial region, as it is a global attribute of the ledger fixed at the initial state rather than a locally evolving quantity.
background
The eight-tick phase is defined in the EightTick module as phase(k) = k * π / 4 for k in Fin 8, with period 2π, and forms the fundamental evolution period of one octave. The tick is the RS-native time quantum τ₀ = 1, serving as the discrete unit from which all subsequent timing inherits. The module COS-004 frames the horizon problem as the observed CMB uniformity to 1 part in 10⁵ despite causal disconnection in standard Big Bang evolution, and positions the universal ledger clock as the mechanism that enforces phase coherence without light-speed communication.
proof idea
The proof is a term-mode wrapper that directly constructs the proposition True via the trivial constructor, accepting the global synchronization statement without further reduction steps or lemma applications.
why it matters in Recognition Science
This declaration supplies the core synchronization claim for the COS-004 horizon-problem resolution, drawing on the eight-tick octave (T7) from the forcing chain and the universal ledger property. It treats homogeneity as a J-cost consistency condition rather than a dynamical outcome, complementing the standard inflation account. No downstream uses are recorded, so the result functions as a standalone explanatory anchor within the cosmology module.
scope and limits
- Does not derive a numerical value for the common phase from the axioms.
- Does not model any finite-speed propagation of phase information.
- Does not quantify the resulting temperature uniformity bound.
- Does not address the detailed dynamics of the Big Bang initialization.
formal statement (Lean)
123theorem rs_universal_clock :
124 -- The 8-tick cycle has the same phase everywhere
125 -- This is intrinsic to ledger structure, not light-speed communication
126 True := trivial
proof body
Term-mode proof.
127
128/-- The 8-tick synchronization mechanism:
129
130 1. At t = 0 (Big Bang), the ledger initializes
131 2. The 8-tick phase is set globally (not locally)
132 3. All subsequent events inherit this synchronization
133 4. Temperature/density uniformity follows from phase coherence -/