Gravity Freudenthal Length Chain Endpoint Cert Snorm Zero 0 1

A zero entry in the endpoint table

The declaration snorm_zero_0_1 is a machine-checked proof that a specific entry in a six-by-six table equals zero. The table, called the Schläfli summand norm, assigns a real number to each ordered pair of edges in a Freudenthal tetrahedron, a four-vertex simplex in a five-dimensional space. The entry at row 0, column 1, which corresponds to the pair of edges indexed 0 and 1, evaluates to 0.

This zero is not an accident of rounding or a numerical approximation. The proof, written in the framework's machine-checked library of formal theorems, expands the definition of the norm, simplifies the algebraic expression, and concludes that the result is exactly 0. The same library also proves the values of the other 35 entries in the table, which range from -4 to 6. Together, these entries form a complete certificate for the Schläfli summand norm at this particular tetrahedron.

The zero at row 0, column 1 means that the geometric contribution of the edge pair (0, 1) to a certain curvature-like sum is nil. In the broader context of the Freudenthal length chain, this table feeds into a closed-form expression for a derivative of dihedral angles. A zero entry simplifies that expression, indicating that the corresponding pair of edges does not affect the derivative at this configuration. This is a structural fact about the tetrahedron's geometry, not a statement about any physical measurement.

In Recognition Science, this certificate is part of a larger effort to derive constants and structures from a single cost function. The framework models physical reality as a ledger of recognition events, and from that model it derives quantities like the golden ratio and three spatial dimensions. This particular theorem, however, is a technical lemma about a specific geometric object. It does not, by itself, establish any physical law or cosmological constant. It is a building block, verified to be correct, that supports further derivations.

theorem snorm_zero_0_1 :
    schlaefliPolySummandNorm freudenthalTetSqEdges 0 1 = 0 := by
  rw [schlaefliPolySummandNorm_eq_num_div_den]
  unfold schlaefliPolySummandNum schlaefliPolySummandDen freudenthalTetSqEdges
    DihedralCayleyMenger.oppositeCMVertices
  simp [CofactorPolynomial.cmCofactor3Poly, CofactorPolynomial.cmCofactorPartial]
  norm_num

theorem snorm_zero_0_1 :
    schlaefliPolySummandNorm freudenthalTetSqEdges 0 1 = 0 := by
  rw [schlaefliPolySummandNorm_eq_num_div_den]
  unfold schlaefliPolySummandNum schlaefliPolySummandDen freudenthalTetSqEdges
    DihedralCayleyMenger.oppositeCMVertices
  simp [CofactorPolynomial.cmCofactor3Poly, CofactorPolynomial.cmCofactorPartial]
  norm_num

theorem snorm_5_0 :
    schlaefliPolySummandNorm freudenthalTetSqEdges 5 0 = 2 := by
  rw [schlaefliPolySummandNorm_eq_num_div_den]
  unfold schlaefliPolySummandNum schlaefliPolySummandDen freudenthalTetSqEdges
    DihedralCayleyMenger.oppositeCMVertices
  simp [CofactorPolynomial.cmCofactor3Poly, CofactorPolynomial.cmCofactorPartial]
  norm_num

theorem snorm_4_1 :
    schlaefliPolySummandNorm freudenthalTetSqEdges 4 1 = 4 := by
  rw [schlaefliPolySummandNorm_eq_num_div_den]
  unfold schlaefliPolySummandNum schlaefliPolySummandDen freudenthalTetSqEdges
    DihedralCayleyMenger.oppositeCMVertices
  simp [CofactorPolynomial.cmCofactor3Poly, CofactorPolynomial.cmCofactorPartial]
  norm_num

theorem snorm_2_3 :
    schlaefliPolySummandNorm freudenthalTetSqEdges 2 3 = 6 := by
  rw [schlaefliPolySummandNorm_eq_num_div_den]
  unfold schlaefliPolySummandNum schlaefliPolySummandDen freudenthalTetSqEdges
    DihedralCayleyMenger.oppositeCMVertices
  simp [CofactorPolynomial.cmCofactor3Poly, CofactorPolynomial.cmCofactorPartial]
  norm_num

What this page does not claim

This theorem does not establish any physical law or cosmological constant. The zero at row 0, column 1 does not imply that the edge pair is geometrically insignificant in all contexts, only in this specific sum. The framework's derivation of three spatial dimensions does not depend on this single certificate.

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/Gravity/FreudenthalLengthChainEndpointCert.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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