module module low

`IndisputableMonolith.Constants.LambdaRecDerivation`

The LambdaRecDerivation module normalizes the bit cost of one recognition event and derives lambda_rec as the unique positive root of the associated balance residual equation. Researchers fixing RS-native constants would cite these results when anchoring the phi-ladder or mass formulas. The module consists of sequential definitions for cost and balance functions followed by positivity and uniqueness lemmas built on the base Constants module.

claimLet $J_{ m bit}$ denote the normalized bit cost of one recognition event. Then $\\lambda_0$ is the unique positive root of the balance residual equation constructed from total cost and curvature terms, with $K$ the auxiliary scaling factor.

background

The module resides in the Constants domain and imports the base Constants module. The upstream doc-comment states: 'The fundamental RS time quantum (RS-native). τ₀ = 1 tick.' It introduces J_bit_normalized for the bit cost of one recognition event, together with J_curv, totalCost, balanceResidual, and K. These support the definitions of lambda_0, lambda_0_pos, balance_at_lambda_0, and the root lemmas lambda_rec_is_root and lambda_rec_unique_root.

proof idea

The module structures its argument as a chain of definitions for the normalized cost and balance functions, followed by lemmas establishing positivity of lambda_0 and uniqueness of the positive root.

why it matters in Recognition Science

This module supplies the lambda recurrence derivation that anchors constant fixing in the Recognition Science framework. It builds directly on the Constants module (τ₀ = 1 tick) and contributes the unique root needed for downstream relations involving phi, the eight-tick octave, and the alpha band.

scope and limits

Does not compute a numerical value for lambda_0.
Does not link lambda_rec to specific physical observables beyond the balance equation.
Does not address time-dependent evolution or forcing chain steps T0-T8.
Does not derive G, hbar, or alpha from the root.

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