Physics Anchor Policy Certified Anchor Identity From Cert

Certified anchor identities

In particle physics, a residue is the leftover difference between a measured or computed quantity and a reference value. The Recognition Science framework models particle masses and charges as sitting on a ledger, a discrete record of events, and it predicts a closed-form value gap(Z) for each species. The theorem anchor_identity_from_cert states: if an external certificate provides, for every species, an interval of possible residues and an interval of possible gaps, and if those intervals are valid, then the actual residue lies within twice the certified tolerance of the predicted gap. The proof is a direct reuse of a generic lemma; it adds no new physics assumptions.

The theorem does not compute the residue or the gap from first principles. It takes the intervals as inputs, supplied by an external computation such as a Python audit, and it verifies the inequality |residue − gap| ≤ 2·eps for every species. A second theorem, equalZ_residue_from_cert, follows from the same inputs: if two species share the same value of Z, their residues must lie within the same certified band. Both results are tagged THEOREM, meaning they are proved in the machine-checked library of formal theorems with no unproved assumptions beyond the library's standard axioms.

What the declaration does not claim is just as important. It does not implement the Standard-Model renormalization-group kernels or integration in Lean; it only makes the dependency on external bounds explicit and machine-checkable. It does not assert that the predicted gap is correct, only that the residue is close to it under the certified intervals. And it does not derive the intervals themselves, which remain the responsibility of the external computation. In plain terms: the theorem is a bridge, not a source. It turns a trusted table of bounds into a formal guarantee of closeness, and it leaves the physics of the table outside the formal proof.

/-- If an external certificate bounds the per-species residues at the anchor, then every species'
residue is close to the closed-form display `gap(Z)` (inequality form). -/
theorem anchor_identity_from_cert
    (C : AnchorCert Species)
    (hC : Valid Z Fgap C)
    (resAtAnchor : Species → ℝ)
    (hres : ∀ f, memI (C.Ires f) (resAtAnchor f)) :
    ∀ f : Species, |resAtAnchor f - Fgap (Z f)| ≤ 2 * C.eps (Z f) := by
  -- Directly reuse the generic lemma.
  simpa [Species, Z, Fgap] using
    (anchorIdentity_cert (Z := Z) (Fgap := Fgap) (C := C) hC resAtAnchor hres)

/-- Equal-Z degeneracy bound from a certificate: if two species share the same Z, their residues
must lie within the certified band. -/
theorem equalZ_residue_from_cert
    (C : AnchorCert Species)
    (hC : Valid Z Fgap C)
    (resAtAnchor : Species → ℝ)
    (hres : ∀ f, memI (C.Ires f) (resAtAnchor f))
    {f g : Species} (hZ : Z f = Z g) :
    |resAtAnchor f - resAtAnchor g| ≤ 2 * C.eps (Z f) := by
  simpa [Species, Z, Fgap] using
    (equalZ_residue_of_cert (Z := Z) (Fgap := Fgap) (C := C) hC resAtAnchor hres hZ)

What this page does not claim

The theorem does not implement the Standard-Model renormalization-group kernels or integration in Lean. The theorem does not assert the predicted gap is correct, only that the residue is close under certified intervals. The theorem does not derive the intervals themselves; they come from an external computation.

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/Physics/AnchorPolicyCertified.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.

Derived articles

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