IndisputableMonolith.Mathematics.Euler
The Mathematics.Euler module supplies multiple equivalent characterizations of the base of the natural logarithm e inside the Recognition Science framework, beginning with its expression as a limit. Researchers deriving continuous limits or exponential behavior from the discrete phi-ladder would cite these results. The module structures its content as a collection of lemmas that translate classical definitions of e into RS-native terms using J-cost and self-similarity.
claimThe module centers on $e$ as a limit, together with its series expansion, fixed-point equation, and algebraic relations to powers of the golden ratio $phi$ forced by the J-cost ledger.
background
The module sits inside the Recognition Science derivation of physics from a single functional equation. It imports the fundamental RS time quantum $tau_0 = 1$ tick and the PhiForcing module, whose doc-comment states that phi is forced by self-similarity in a discrete ledger with J-cost. The J-cost structure and the Recognition Composition Law supply the algebraic backbone for all subsequent expressions of e.
proof idea
The module organizes a sequence of sibling declarations: e_as_limit establishes the limit form, e_as_series the summation form, e_fixed_point the fixed-point characterization, and the remaining lemmas (e_over_phi, e_minus_phi, e_times_phi, phi_to_e) supply direct algebraic relations. Each declaration applies the self-similarity results from PhiForcing; the four attempt lemmas explore alternative routes to the same relations.
why it matters in Recognition Science
This module supplies the exponential constant e required to pass from the discrete T5-T8 forcing chain to continuous physics. It feeds the mass formula and the eight-tick octave constructions that appear later in the framework, closing the gap between the phi-ladder and standard analytic functions. No direct used_by edges are recorded yet, but the module is a prerequisite for any RS-native treatment of growth or decay processes.
depends on (2)
declarations in this module (29)
-
theorem
e_as_limit -
theorem
e_as_series -
theorem
e_fixed_point -
def
e_over_phi -
def
e_minus_phi -
def
e_times_phi -
def
e_to_phi -
def
phi_to_e -
def
attempt1 -
def
attempt2 -
def
attempt3 -
def
attempt4 -
def
attempt5 -
def
eContinuedFraction -
def
phiContinuedFraction -
theorem
e_from_normalization -
def
partitionFunctionFormula -
theorem
e_pos -
theorem
e_gt_two -
theorem
phi_lt_two -
theorem
e_gt_phi -
theorem
e_ne_phi -
theorem
e_gt_one -
def
phiVsE -
theorem
euler_phi_connection -
def
rsInterpretation -
theorem
e_is_unique_base -
def
summary -
structure
EulerFalsifier