module module moderate

`IndisputableMonolith.Constants.Derivation`

Module that anchors SI CODATA values of $c$, $\hbar$, $G$ and constructs the RS time quantum $\tau_0$ so that $\tau_0^2=\hbar G/(\pi c^5)$. Anyone matching Recognition Science native units to laboratory constants cites it. Content is definitions, positivity/nonzero lemmas, and the dimensional identity for $\tau_0$.

claimThe module records CODATA anchors $c$, $\hbar$, $G>0$, defines the RS time quantum $\tau_0$, and states the key dimensional identity $\tau_0^2=\hbar G/(\pi c^5)$ (with $\tau_0>0$).

background

Recognition Science works in native units where $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$, and the fundamental clock tick is one unit of time. Matching those units to SI requires a bridge through measured $c$, $\hbar$, and $G$.

The parent Constants module treats $\tau_0$ as the RS-native time quantum (one tick). Constants.Dimensions supplies the dimensional-analysis framework that converts RS primitives into SI-dimensioned quantities.

This derivation module sits between those layers: it imports the CODATA numerical anchors, proves the elementary positivity and non-vanishing facts needed for division, and packages the classical combination $\hbar G/(\pi c^5)$ as $\tau_0^2$.

proof idea

Definition-and-lemma module, not a deep proof development. CODATA constants are introduced as positive reals; short lemmas record positivity and $\neq 0$. The time quantum $\tau_0$ is defined (or characterized) so the algebraic identity $\tau_0^2=\hbar G/(\pi c^5)$ holds by dimensional construction from the Dimensions framework. No multi-step tactic proof is required beyond positivity bookkeeping.

why it matters in Recognition Science

Gives the explicit SI clock that corresponds to one RS tick, so every later mass, length, or rate formula written in $\varphi$-ladder units can be converted to meters, seconds, and kilograms. The identity $\tau_0^2=\hbar G/(\pi c^5)$ is the dimensional skeleton behind the RS-native assignments $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$. Downstream constant and phenomenology modules that quote laboratory values depend on this bridge even when the import graph does not yet list them as direct used_by edges.

scope and limits

Does not derive $c$, $\hbar$, or $G$ from $\varphi$; it imports CODATA anchors.
Does not prove the RS-native assignments $c=1$, $\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$.
Does not address experimental uncertainty bands or unit-system conventions beyond the stated identity.
Does not construct masses, $\alpha$, or the eight-tick octave; only the $\tau_0$ bridge.

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