timescale_gap_log10
The declaration defines the base-10 logarithm of the biological-to-Planck timescale ratio as the rational 43. Researchers deriving decoherence thresholds from the Gap-45 boundary would cite this constant when calibrating environmental coupling in quantum systems. The definition is a direct numerical assignment drawn from the approximate evaluation log10(1 / 5.4e-44).
claimLet $Δ = log_{10}(τ_{bio} / τ_{Planck})$. Then $Δ := 43$.
background
In the QFT.Decoherence module the Gap-45 threshold separates the quantum regime (coherent superposition preserved) from the classical regime (entanglement with the environment). The module states the decoherence time formula $τ_{decoherence} ≈ τ_0 × φ^{-N}$, where τ_0 is the fundamental tick, N counts coupled environmental modes, and φ is the golden ratio. Upstream structures include the discrete φ-tier scaling of nuclear densities from NucleosynthesisTiers.of and the J-cost calibration from PhiForcingDerived.of, both of which anchor the φ-ladder used to relate Planck and biological scales.
proof idea
This is a direct definition that assigns the constant 43 to timescale_gap_log10. It functions as a one-line numerical anchor for the downstream gap_range theorem, which simply unfolds the definition and applies norm_num to verify the bounds 43 ≤ Δ < 45.
why it matters in Recognition Science
The definition supplies the concrete value for the logarithmic gap in the Gap-45 derivation and is used by the gap_range theorem that confirms the 43-45 order window. It operationalizes the module's core claim that systems exceeding ~10^45 operations per characteristic time decohere, linking to the eight-tick octave (T7) and the φ-ladder mass formula. The declaration closes the empirical calibration step between Planck and biological timescales, supporting the patent claims on Gap-45-based error correction listed in the module documentation.
scope and limits
- Does not derive the numerical value from an explicit first-principles calculation inside this module.
- Does not specify the exact biological timescale chosen for τ_bio.
- Does not quantify dependence on the number of environmental modes N.
Lean usage
theorem gap_range : 43 ≤ timescale_gap_log10 ∧ timescale_gap_log10 < 45 := by unfold timescale_gap_log10; constructor <;> norm_numformal statement (Lean)
73def timescale_gap_log10 : ℚ := 43 -- Approximate value
proof body
Definition body.
74
75/-- **THEOREM**: The gap is between 43 and 45 orders of magnitude. -/