IndisputableMonolith.Cosmology.InflatonPotentialFromJCost
The module defines the inflaton potential on the recognition manifold by setting V equal to the J-cost evaluated at one plus the dimensionless displacement from the canonical reference rung. Cosmologists working inside Recognition Science cite these objects when constructing early-universe models consistent with the J-functional and the recognition composition law. The module consists of a sequence of definitions and certificates that establish non-negativity, vacuum value, reciprocal symmetry, and quadratic behavior near the origin.
claim$V(φ_∞) = J(1 + φ_∞)$, where $φ_∞$ is the dimensionless displacement from the canonical reference rung and $J$ is the recognition cost function.
background
The module sits in the cosmology domain and imports the RS time quantum τ₀ = 1 tick from Constants together with the J-cost function from the Cost module. Its central object is the inflaton potential defined directly as the J-cost of one plus the inflaton displacement, with φ_inf treated as a dimensionless coordinate on the recognition manifold. Supporting definitions introduce the vacuum value, non-negativity, positivity away from vacuum, reciprocal symmetry, and the quadratic approximation at the origin, ending with an explicit certification of the potential.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the inflaton potential used in subsequent cosmological derivations inside Recognition Science, linking the J-cost directly to the forcing-chain steps T5 (J-uniqueness) and T6 (phi fixed point). It provides the concrete potential form required for models that respect the eight-tick octave and the emergence of D = 3 spatial dimensions.
scope and limits
- Does not derive the full inflationary dynamics or slow-roll parameters.
- Does not assign numerical values to the inflaton mass or vacuum energy.
- Does not connect the potential to specific particle spectra or the alpha band.
- Does not address quantum fluctuations or reheating.