module module high

`IndisputableMonolith.Cosmology.Inflation`

The Cosmology.Inflation module sets the inflaton potential equal to the J-cost function within Recognition Science. Cosmologists deriving slow-roll parameters or e-foldings from the RS forcing chain would cite these definitions. The module consists of a sequence of definitions and short lemmas that translate J-cost properties into standard inflationary quantities.

claimThe inflaton potential is $V(phi) = J(phi)$, where $J$ satisfies the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$. Slow-roll parameters are then $epsilon = (1/2)(J'(phi)/J(phi))^2$ and $eta = J''(phi)/J(phi)$, with the number of e-foldings given by the integral of $dphi / sqrt(2 epsilon)$.

background

Constants supplies the RS-native time quantum tau_0 = 1 tick. Cost defines the J-cost function obeying the Recognition Composition Law and the J-uniqueness relation J(x) = (x + x^{-1})/2 - 1. The module applies these objects directly to cosmology by identifying the inflaton potential with J-cost, thereby importing the phi-ladder and defect structure into inflationary dynamics.

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the inflaton potential used by the sibling declarations sixty_efolds, horizon_problem_solved, flatness_problem_solved and monopole_problem_solved. It thereby links the T5 J-uniqueness step of the forcing chain to concrete cosmological predictions inside the RS framework.

scope and limits

Does not derive the scalar power spectrum amplitude.
Does not compute the tensor-to-scalar ratio.
Does not address reheating or post-inflationary evolution.
Does not compare numerical predictions against CMB data.

depends on (2)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Cost module_import

declarations in this module (24)

def inflatonPotential L48
theorem potential_min_at_one L51
theorem potential_positive L59
def slowRollEpsilon L68
def slowRollEta L77
theorem slow_roll_at_large_phi L85
def eFoldings L94
theorem sixty_efolds L100
theorem horizon_problem_solved L109
theorem flatness_problem_solved L116
theorem monopole_problem_solved L122
def powerSpectrum L130
def spectralIndex L137
theorem nearly_scale_invariant L142
def tensorScalarRatio L148
theorem small_tensor_modes L153
structure Reheating L161
theorem efficient_reheating L169
theorem inflation_is_cost_relaxation L184
structure InflationPredictions L195
def rsPredictions L201
def planckMeasurements L208
structure InflationFalsifier L218
def experimentalStatus L225