Foundation Seesaw Mechanism Rs V3 Seesaw Mech Rs V3 Cert

The certified content

The seesaw mechanism is a standard physics proposal for why neutrinos are so light: a heavy right-handed partner pushes the ordinary neutrino mass down through a formula like m_nu ~ m_D^2 / M_R. The Recognition Science declaration SeesawMech_RS_v3Cert does not prove that mechanism. It proves three general facts about a cost function, and the seesaw name is a research note about where the idea was meant to go.

The cost function here is domainCost, defined as Jcost(m / e) for two real numbers m and e. The three proved facts are: the cost is zero when m equals e, the cost is never negative when both inputs are positive, and the number phi minus 3/2 is positive. These are general properties of the cost function, not specific to neutrinos or any physical system. The certificate structure simply bundles these three theorems together, and a proof shows the bundle is inhabited, meaning the facts are consistent.

The physics estimate in the research note is a separate, informal calculation. It plugs in a top quark mass of 173 GeV and a grand unification scale of 2e16 GeV to get a neutrino mass around 1.5e-3 eV, compared with an empirical value near 0.05 eV. That estimate is not part of the formal certificate. The certificate itself never defines m as a neutrino mass or e as an energy scale; it only states facts about the ratio m/e.

What the certificate does establish is a small piece of the framework's cost structure: the cost function vanishes at equality, stays nonnegative, and has a positive threshold constant. These are useful properties for any recognition cost, but they do not connect to the seesaw mechanism. The module shares its body verbatim with 2383 sibling modules, and the docstring itself says the content is stated once in a template and cited from there. The seesaw-specific claim would require a definition of m and e in neutrino terms, which this file does not provide.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty SeesawMech_RS_v3Cert := ⟨cert⟩

What this page does not claim

The seesaw mechanism itself is not proved; the certificate proves only general cost properties. The physics estimate of 1.5e-3 eV is an informal research note, not a formal result. No connection is made between the cost function and any specific particle or energy scale.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/Seesaw_Mechanism_RS_v3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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