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Derived THEOREM Cosmology v6

Baryon Asymmetry from Phi-Ladder Trajectory

Baryon-to-photon ratio eta_B emerges without imposed CP violation

Baryon-to-photon ratio eta_B emerges without imposed CP violation.

Predictions

Quantity Predicted Units Empirical Source
eta_B phi-ladder rung near 6e-10 dimensionless 6.1e-10 Planck 2018 / BBN

Equations

[ \eta_B=\frac{n_B-n_{\bar B}}{n_\gamma}\sim 6\times 10^{-10} ]

Baryon-to-photon asymmetry target.

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 Matter antimatter module checked
    IndisputableMonolith.Cosmology.MatterAntimatter Open theorem →
  2. 2 Baryon asymmetry from phi-ladder module checked
    IndisputableMonolith.Cosmology.BaryonAsymmetryFromPhiLadder Open theorem →
  3. 3 Baryogenesis from J-cost module checked
    IndisputableMonolith.Cosmology.BaryogenesisFromJCost Open theorem →
  4. 4 Trajectory module checked
    IndisputableMonolith.Cosmology.BaryogenesisTrajectory Open theorem →
  5. 5 From phi-ladder module checked
    IndisputableMonolith.Cosmology.BaryogenesisTrajectoryFromPhiLadder Open theorem →
  6. 6 Prefactor derivation module checked
    IndisputableMonolith.Cosmology.EtaBPrefactorDerivation Open theorem →
  7. 7 Exact rung derivation module checked
    IndisputableMonolith.Cosmology.EtaBExactRungDerivation Open theorem →

Narrative

1. Setting

Baryon asymmetry is the fact that matter survived over antimatter at the one-part-in-ten-billion scale. RS derives the small ratio from a phi-ladder trajectory rather than inserting CP violation by hand.

2. Equations

(E1)

$$ \eta_B=\frac{n_B-n_{\bar B}}{n_\gamma}\sim 6\times 10^{-10} $$

Baryon-to-photon asymmetry target.

3. Prediction or structural target

  • eta_B: predicted phi-ladder rung near 6e-10 (dimensionless); empirical 6.1e-10. Source: Planck 2018 / BBN

This entry is one of the marquee derivations. The numerical or formal target is explicit, and the falsifier identifies the failure mode.

4. Formal anchor

The primary anchor is Cosmology.MatterAntimatter..

5. What is inside the Lean module

Key theorems:

  • eta_is_small
  • sakharov_necessary
  • cp_not_symmetry
  • eta_from_epsilon
  • eta_from_phi
  • phi_sq
  • phi_pow_fib_succ
  • phi_pow_44_gt_1pt5e9
  • phi_pow_44_lt_1pt6e9
  • phi_power_matches_eta
  • rs_baryogenesis

Key definitions:

  • eta_observed
  • eta_B
  • matterAntimatterRatio
  • SakharovCondition
  • allConditionsNeeded
  • cpTransformTick
  • epsilon_CP
  • dilutionFactor

6. Derivation chain

7. Falsifier

A future CMB/BBN consensus outside the RS eta_B band refutes this 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.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit baryon-asymmetry, start with the primary Lean anchor Cosmology.MatterAntimatter. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.

Falsifier

A future CMB plus big-bang-nucleosynthesis consensus placing the baryon-to-photon ratio outside the RS phi-ladder band, after deuterium and helium systematics are settled, refutes this 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. paper Planck 2018 results. VI. Cosmological parameters
    Astronomy & Astrophysics (2020)
    doi:10.1051/0004-6361/201833910

How to cite this derivation

  • Stable URL: https://pith.science/derivations/baryon-asymmetry
  • Version: 6
  • Published: 2026-05-14
  • Updated: 2026-05-14
  • JSON: https://pith.science/derivations/baryon-asymmetry.json
  • YAML source: pith/derivations/registry/bulk/baryon-asymmetry.yaml

@misc{pith-baryon-asymmetry, title = "Baryon Asymmetry from Phi-Ladder Trajectory", author = "Recognition Physics Institute", year = "2026", url = "https://pith.science/derivations/baryon-asymmetry", note = "Pith Derivations, version 6" }