IndisputableMonolith.Foundation.ArrowOfTime
The ArrowOfTime module defines the arrow of time in Recognition Science as TemporalSequence, a sequence of R-hat steps that accumulate Berry phase. It introduces zAtStep values that grow forward and shrink in reverse, inducing the isBefore relation together with entropyFromZ and its monotonicity. Researchers modeling fundamental time asymmetry cite these to ground the thermodynamic arrow in the phi-ladder. The module consists entirely of definitions and elementary properties.
claimLet $T$ be a temporal sequence of R-hat steps. For each step $n$ let $z_n$ denote the accumulated Berry phase given by zAtStep. The relation $m$ isBefore $n$ holds precisely when $z_m < z_n$. The function entropyFromZ maps $z$ to an entropy value that is monotone non-decreasing along $T$.
background
The module operates in the Foundation domain and introduces TemporalSequence as a sequence of R-hat steps with accumulated Berry phase at each step. It defines zAtStep to assign the phase value at each step, z_nonneg to record non-negativity, forward_accumulates and reverse_subtracts to capture directional accumulation, and z_absolute_immune_to_reversal to protect the absolute value under reversal. The isBefore relation is equipped with before_transitive, before_irrefl and before_asymm, while entropyFromZ supplies the entropy function whose monotonicity is recorded by entropy_monotone.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the core definitions for the arrow of time that feed into higher-level results on entropy increase and temporal ordering within the Recognition Science framework. It fills the chain step that links Berry phase accumulation to the thermodynamic arrow, drawing on the J-uniqueness and eight-tick octave of the unified forcing chain. No open questions are addressed; the module serves as a base layer.
scope and limits
- Does not derive numerical values for constants such as alpha or G.
- Does not model specific physical processes such as particle creation.
- Does not address the connection to spatial dimensions D=3.
- Does not include proofs of existence for temporal sequences.