module module high

`IndisputableMonolith.Constants.Derivation`

This module derives the relation τ₀² = ℏG/(πc⁵) that connects the Recognition Science time quantum to standard constants. Researchers deriving physical constants from RS primitives would cite it. The module assembles the result from codata definitions for c, ħ, G and the dimensional framework imported from Dimensions.

claim$\tau_0^2 = \frac{\hbar G}{\pi c^5}$

background

Recognition Science fixes c = 1 in native units and defines τ₀ as the fundamental time quantum equal to one tick. The imported Constants module supplies this τ₀ definition while Dimensions supplies the dimensional analysis framework that converts RS primitives into physical constants.

The Derivation module introduces codata objects for c, ħ and G together with positivity and non-zero lemmas, then states the central relation between τ₀ and those constants.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the relation τ₀² = ℏG/(πc⁵) that links the RS time quantum to observable constants, supporting later derivations of the mass ladder and the alpha band inside (137.030, 137.039). It closes the step from the eight-tick octave (T7) and D = 3 to concrete constants.

scope and limits

Does not derive numerical values or approximations for any constant.
Does not incorporate quantum corrections or higher-order terms.
Does not address conversion between RS-native units and other systems beyond the stated relation.

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Constants.Dimensions module_import

declarations in this module (40)

def c_codata L24
def hbar_codata L25
def G_codata L26
lemma c_codata_pos L28
lemma hbar_codata_pos L29
lemma G_codata_pos L30
lemma c_codata_ne_zero L32
lemma hbar_codata_ne_zero L33
lemma G_codata_ne_zero L34
def tau0 L38
lemma tau0_pos L40
lemma tau0_ne_zero L48
lemma inner_pos L50
lemma inner_nonneg L54
theorem tau0_sq_eq L58
def ell0 L71
lemma ell0_pos L73
lemma ell0_ne_zero L74
structure RSUnitSystem L78
def canonicalUnits L87
def c_derived L98
theorem c_derived_eq_codata L100
lemma c_derived_pos L107
def hbar_derived L110
lemma hbar_derived_pos L112
theorem planck_relation_satisfied L119
def G_derived L129
lemma G_derived_pos L131
theorem G_relation_satisfied L138
def planck_length L150
def planck_time L151
def planck_mass L152
lemma planck_length_pos L154
lemma planck_time_pos L158
lemma planck_mass_pos L162
lemma planck_time_inner_nonneg L166
theorem tau0_planck_relation L172
theorem units_self_consistent L227
theorem tau0_matches_foundation L249
def derivation_status L255