module module moderate

`IndisputableMonolith.Constants.LambdaRecDerivation`

The LambdaRecDerivation module derives the recursive lambda constant from normalized bit costs of recognition events and balance residuals. It introduces lambda_0 as the positive root of the balance equation and establishes lambda_rec as the unique self-consistent extension. Researchers working on RS-native constants cite it to close the loop between J-cost normalization and the phi-ladder fixed point.

claimThe module defines $J_{bit}$ as the normalized bit cost of one recognition event, constructs the balance residual equation, and sets $lambda_0$ as its unique positive root together with the recursive extension $lambda_{rec}$ satisfying the root condition.

background

The module sits inside the Constants domain and imports the parent Constants module whose fundamental object is the RS time quantum $tau_0 = 1$ tick. It introduces the normalized bit cost $J_{bit}$ of a single recognition event, the totalCost and balanceResidual functions, and the auxiliary objects $K$ and $lambda_0$ that enforce balance at the base rung of the phi-ladder.

proof idea

This is a definition module, no proofs. It consists of direct definitions for $lambda_0$, $K$, $lambda_{rec}$ together with lemmas that state the root property and uniqueness of the positive solution.

why it matters in Recognition Science

The module supplies the lambda_rec derivation required for consistent evaluation of higher constants in the Recognition Science framework, including the alpha band and mass-ladder formulas. It connects the J-cost normalization directly to the self-similar fixed point that appears in the eight-tick octave construction.

scope and limits

Does not compute numerical values of lambda_rec
Does not extend the derivation to multi-event or higher-dimensional cases
Does not incorporate corrections from quantum field theory
Does not address stability under small perturbations of the balance equation

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (26)

def J_bit_normalized L28
def J_curv L34
def totalCost L37
def balanceResidual L41
def lambda_0 L45
lemma lambda_0_pos L47
lemma lambda_0_sq L53
theorem balance_at_lambda_0 L61
theorem balance_unique_positive_root L67
def K L92
theorem lambda_rec_is_root L95
theorem lambda_rec_unique_root L100
theorem lambda_rec_is_forced L114
def Q3_vertices L144
def Q3_faces L147
def euler_S2 L150
def dihedral_angle L153
def angular_deficit_per_vertex L156
theorem angular_deficit_value L159
theorem total_curvature_gauss_bonnet L164
def kappa_normalized L169
theorem kappa_normalized_eq_one L172
theorem J_curv_derivation L182
theorem balance_determines_lambda L186
structure GDerivationChain L209
theorem G_derivation_chain_complete L217