IndisputableMonolith.Cosmology.Inflation
Recognition Science takes the inflaton potential to be the J-cost itself. The module defines that potential, records its minimum and positivity, and builds slow-roll parameters, e-foldings, and the classical horizon, flatness, and monopole resolutions from it. Cosmologists in the RS program cite it when early-universe dynamics must sit on the forced cost rather than an ad-hoc V. Arguments are mostly direct calculus and inequalities on J.
claimThe inflaton potential is identified with the J-cost: $V(\phi)=J(\phi)$ where $J(x)=\frac{x+x^{-1}}{2}-1$. The module records that $V$ has a global minimum at $\phi=1$ and is positive elsewhere; defines the slow-roll parameters $\varepsilon$ and $\eta$; shows slow roll at large field values; defines the e-folding integral and a sixty-e-fold threshold; asserts that the horizon, flatness, and monopole problems are solved under this $V$; and supplies a power-spectrum formula.
background
Recognition Science forces a unique nonnegative cost $J$ by the Recognition Composition Law and the T5 uniqueness step: $J(x)=\frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$). In ordinary inflation one postulates a scalar potential by hand. Here the same $J$ is promoted to the inflaton potential, so early-universe dynamics inherit the forced cost rather than an extra free function.
The module lives in the Cosmology domain. It imports RS constants (including the native time quantum) and the Cost library that defines $J$. Sibling names cover the potential, its minimum at unity, positivity, slow-roll $\varepsilon$ and $\eta$, the large-field slow-roll regime, e-foldings and a sixty-e-fold claim, the three classic problem resolutions, and a power spectrum.
proof idea
Definition-led module, not a single deep theorem. The potential is set equal to $J$. Minimum-at-one and positivity are elementary properties of $J$ (critical point of $x+x^{-1}$, nonnegativity from AM-GM or the cosh form). Slow-roll parameters are the standard $\varepsilon=\frac12(V'/V)^2$ and $\eta=V''/V$ specialized to this $V$. Large-field slow roll, the e-fold integral, the sixty-e-fold threshold, and the horizon/flatness/monopole claims are evaluations of those expressions in the usual slow-roll regime. The power spectrum is the curvature-perturbation formula under the same $V$.
why it matters in Recognition Science
Closes the gap between the T5-forced cost and inflationary cosmology: no separate inflaton potential is introduced. The classical horizon, flatness, and monopole resolutions then sit on the same $J$ that generates the rest of the RS ladder (phi fixed point, eight-tick structure, $D=3$). The graph currently lists no downstream consumers, so this module is a leaf that later CMB, reheating, or spectral-index developments would import. It keeps inflation inside RS-native units ($c=1$, cost-shaped $V$) rather than grafting an external slow-roll model onto the monolith.
scope and limits
- Does not derive reheating temperature, exit mechanism, or post-inflationary matching.
- Does not compute non-Gaussianity or higher-order spectra.
- Does not prove a Planck-level match for $n_s$ beyond defining the power spectrum.
- Does not replace FLRW or Einstein dynamics; assumes standard slow-roll kinematics.
- Does not treat quantum-gravity corrections to $V$ or multifield extensions.
- Does not link e-foldings to the eight-tick octave by a proved identity.
depends on (2)
declarations in this module (24)
-
def
inflatonPotential -
theorem
potential_min_at_one -
theorem
potential_positive -
def
slowRollEpsilon -
def
slowRollEta -
theorem
slow_roll_at_large_phi -
def
eFoldings -
theorem
sixty_efolds -
theorem
horizon_problem_solved -
theorem
flatness_problem_solved -
theorem
monopole_problem_solved -
def
powerSpectrum -
def
spectralIndex -
theorem
nearly_scale_invariant -
def
tensorScalarRatio -
theorem
small_tensor_modes -
structure
Reheating -
theorem
efficient_reheating -
theorem
inflation_is_cost_relaxation -
structure
InflationPredictions -
def
rsPredictions -
def
planckMeasurements -
structure
InflationFalsifier -
def
experimentalStatus