time_kernel_dimensionless
plain-language theorem explainer
The time ratio function in the ILG gravitational model is invariant under common positive rescaling of its two time arguments. Modelers constructing dimensionless kernels from dynamical times would cite this result to confirm scale independence. The proof is a one-line wrapper that invokes the rescaling lemma directly on the supplied positive factor c.
Claim. Let $w$ be the time ratio function. For any real $c>0$ and times $T,τ$, one has $w(cT,cτ)=w(T,τ)$.
background
The TruthCore.TimeKernel module examines time kernels inside the ILG parameterization of gravity. The function w_time_ratio takes two dynamical times and returns their ratio; the upstream Tdyn definition supplies the orbital period $T_{dyn}(v,r)=2πr/v$ constructed from velocity and radius scales. Related T abbreviations in Breath1024 and Gap45 supply fundamental periods and triangular numbers used to index discrete time steps in the broader framework.
proof idea
The proof is a one-line wrapper that applies the rescaling lemma w_time_ratio_rescale instantiated with the positive constant c, Tdyn set to the first argument T, and τ0 set to the second argument τ.
why it matters
The result establishes that the time kernel ratio is dimensionless, a prerequisite for its use in scale-invariant constructions within Recognition Science. It supports the self-similar fixed-point structure of dynamical times and feeds into higher-level kernel definitions even though no immediate downstream theorems are recorded. The invariance aligns with the overall forcing chain that derives spatial and temporal structure from a single functional equation.
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