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Derived THEOREM Cosmology v5

Inflation from the J-Cost Inflaton

Inflation parameters emerge from a J-cost potential without ad hoc tuning

Inflation parameters emerge from a J-cost potential without ad hoc tuning.

Predictions

Quantity Predicted Units Empirical Source
Spectral index n_s RS bracket dimensionless 0.9649 ± 0.0042 Planck 2018
Tensor-to-scalar r RS bracket dimensionless < 0.036 BICEP/Keck 2021

Equations

[ 2^D=8,\qquad D=3,\qquad \Omega_\Lambda^{\mathrm{RS}}=\frac{11}{16}\ \text{where applicable} ]

Shared cosmological forcing frame.

Derivation chain (Lean anchors)

Each row links to the corresponding Lean 4 declaration in the Recognition Science canon. A resolved anchor has a green check; an unresolved anchor flags a registry/canon mismatch.

  1. 1 Inflation module module checked
    IndisputableMonolith.Cosmology.Inflation Open theorem →
  2. 2 Inflation from J-cost module checked
    IndisputableMonolith.Cosmology.CosmicInflationFromJCost Open theorem →
  3. 3 Inflaton potential from J-cost module checked
    IndisputableMonolith.Cosmology.InflatonPotentialFromJCost Open theorem →
  4. 4 Inflaton potential structural module checked
    IndisputableMonolith.Cosmology.InflatonPotentialStructural Open theorem →
  5. 5 Spectral index module checked
    IndisputableMonolith.Cosmology.InflationSpectralIndexFromJCost Open theorem →
  6. 6 Models from config dim module checked
    IndisputableMonolith.Cosmology.InflationModelsFromConfigDim Open theorem →
  7. 7 Reheat temperature module checked
    IndisputableMonolith.Cosmology.InflationReheatTemperature Open theorem →
  8. 8 Gravity J-cost inflaton module checked
    IndisputableMonolith.Gravity.JCostInflaton Open theorem →

Narrative

1. Setting

Inflation from the J-Cost Inflaton is anchored in Cosmology.Inflation. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ 2^D=8,\qquad D=3,\qquad \Omega_\Lambda^{\mathrm{RS}}=\frac{11}{16}\ \text{where applicable} $$

Shared cosmological forcing frame.

3. Prediction or structural target

  • Spectral index n_s: predicted RS bracket (dimensionless); empirical 0.9649 ± 0.0042. Source: Planck 2018
  • Tensor-to-scalar r: predicted RS bracket (dimensionless); empirical < 0.036. Source: BICEP/Keck 2021

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Cosmology.Inflation..

5. What is inside the Lean module

Key theorems:

  • potential_min_at_one
  • potential_positive
  • slow_roll_at_large_phi
  • sixty_efolds
  • horizon_problem_solved
  • flatness_problem_solved
  • monopole_problem_solved
  • nearly_scale_invariant
  • small_tensor_modes
  • efficient_reheating
  • inflation_is_cost_relaxation

Key definitions:

  • inflatonPotential
  • slowRollEpsilon
  • slowRollEta
  • eFoldings
  • powerSpectrum
  • spectralIndex
  • tensorScalarRatio
  • Reheating

6. Derivation chain

7. Falsifier

A future joint cosmological fit that lands outside the stated RS band, after accounting for curvature and calibration systematics, refutes this cosmological derivation.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

11. Why this belongs in the derivations corpus

The corpus is organized around load-bearing consequences, not around file names. This entry is included because Cosmology.Inflation contributes a reusable theorem or definitional bridge that other pages can cite. Keeping the page public gives readers a stable URL, a JSON record, and a direct path into the Lean theorem page. If the entry becomes redundant with a stronger derivation later, the current slug should be retired rather than silently rewritten; the replacement should absorb its anchors and preserve the audit history.

Falsifier

A future joint cosmological fit that lands outside the stated RS band, after accounting for curvature and calibration systematics, refutes this cosmological derivation.

References

  1. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  2. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  3. empirical Planck 2018
    Empirical reference for prediction: Spectral index n_s
  4. empirical BICEP/Keck 2021
    Empirical reference for prediction: Tensor-to-scalar r

How to cite this derivation

  • Stable URL: https://pith.science/derivations/inflation-from-rs
  • Version: 5
  • Published: 2026-05-14
  • Updated: 2026-05-15
  • JSON: https://pith.science/derivations/inflation-from-rs.json
  • YAML source: pith/derivations/registry/bulk/inflation-from-rs.yaml

@misc{pith-inflation-from-rs, title = "Inflation from the J-Cost Inflaton", author = "Recognition Physics Institute", year = "2026", url = "https://pith.science/derivations/inflation-from-rs", note = "Pith Derivations, version 5" }