module module high

`IndisputableMonolith.Cosmology.Inflation`

The Cosmology.Inflation module sets the inflaton potential equal to the J-cost function. Cosmologists modeling early-universe expansion cite these definitions when computing e-foldings and slow-roll parameters from RS primitives. The module proceeds by direct identification of the potential with J, followed by algebraic checks of its minimum, positivity, and slow-roll regime at large field values.

claimThe inflaton potential is $V(\phi) = J(\phi)$, where $J$ is the recognition cost function. Slow-roll parameters are defined as $\epsilon(\phi) = \frac12 (J'(\phi)/J(\phi))^2$ and $\eta(\phi) = J''(\phi)/J(\phi)$. The module also states that sixty e-foldings solve the horizon, flatness, and monopole problems.

background

The module imports the RS-native time quantum $\tau_0 = 1$ tick from Constants and the J-cost definition from Cost. J satisfies the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$. The local setting applies these primitives to cosmology by identifying the inflaton potential with J-cost, then deriving its derivatives and global properties.

Key objects include the potential itself, its minimum at $\phi=1$, positivity for $\phi > 0$, slow-roll parameters, and the e-foldings count. These rest on the phi-ladder and eight-tick octave already fixed in the forcing chain.

proof idea

This is a definition module, no proofs. The structure begins with the direct equating of the potential to J-cost, then applies algebraic identities from the Cost module to establish the minimum location, positivity, and the slow-roll regime for large $\phi$. Subsequent definitions compute the e-foldings integral and verify that it exceeds sixty, which directly yields the three problem solutions.

why it matters in Recognition Science

The module supplies the inflaton potential that feeds the cosmology results on horizon, flatness, and monopole resolution. It extends T5 J-uniqueness and T6 phi fixed point into the inflationary epoch, furnishing the RS-native mechanism for the required expansion. No downstream theorems are listed, so the module stands as the entry point for all inflationary calculations in the framework.

scope and limits

Does not derive the post-inflation reheating temperature.
Does not incorporate multi-field dynamics or non-canonical kinetic terms.
Does not compute the scalar spectral index or tensor-to-scalar ratio.
Does not address the eta problem or supergravity embeddings.

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