Encyclopedia Gravity Gravity Analysis Regge Ttcontinuum Certificate Spike Tet Block0 Eq

ARTICLE 2 claims 2 theorems

Gravity Analysis Regge Ttcontinuum Certificate Spike Tet Block0 Eq

A machine-checked proof that a large symbolic expression, part of a gravity calculation, collapses to a simpler polynomial, and the limits of what that collapse means.

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The first block's identity

The declaration tetBlock0_eq is a theorem in the framework's machine-checked library of formal theorems. It states that a specific function, tetBlock0, which is defined as a literal transcription of a 36-term sum, is equal to a much longer polynomial expression. The function takes nine variables named E00 through E22, which represent the components of a 3x3 matrix, and three variables x0, x1, x2, which represent a vector. It also takes three more variables, s2, s3, and p, which are meant to stand for the square root of 2, the square root of 3, and pi, respectively.

The theorem proves that this function, despite its complex definition, is exactly equal to the expanded polynomial shown in the proof. This is a purely algebraic identity. It is proved by unfolding the definition of the function and then using the ring tactic, which is a standard method in the library for verifying polynomial equalities. The proof does not rely on any physical assumptions or on the specific values of the variables. It holds for any real numbers assigned to those fourteen inputs.

The practical importance of this theorem is that it is a step toward a larger result. The library uses this identity to show that the sum of six such blocks, each a literal transcription of a similar 36-term sum, collapses to a much simpler expression: negative one quarter times the squared length of the vector times the squared Frobenius norm of the matrix. That collapse happens only under certain conditions, which are that the matrix is symmetric, has zero trace, and that the vector is in the null space of the matrix. The theorem about the first block, tetBlock0_eq, is the first step in proving that larger collapse.

The theorem does not claim anything about the physical meaning of the variables or the expression. It does not say that this expression represents a physical quantity, nor does it say that the collapse has any consequence for a theory of gravity. It is a statement about a formal object, a polynomial, and its equality to another polynomial. The proof is a computation, not a derivation of a physical law.

In Recognition Science, this is part of an effort to build a library of formal results. The value here is in the verification: a reader can trust that the algebraic identity is correct, because a machine has checked it. The theorem does not, by itself, tell a reader anything about the world. It is a tool, a certified step in a longer argument.

THEOREM tetBlock0_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
tetBlock0_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean:105 
/-- Tet 0 collapses to a pure-QQ polynomial (every s2/s3/p entry of G
carries a literal (0)^2 midpoint factor). -/
theorem tetBlock0_eq (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ) :
    tetBlock0 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p
    =
    (-E00^2*x2^2/32 - E00*E01*x2^2/16 - E00*E02*x2^2/32 - E00*E10*x2^2/16 - E00*E11*x0*x1/16 - E00*E11*x0*x2/16 - E00*E11*x1^2/32 - E00*E11*x1*x2/16 + E00*E11*x2^2/32 - E00*E12*x0*x1/16 + E00*E12*x0*x2/16 - E00*E12*x1^2/32 - E00*E12*x1*x2/16 - E00*E20*x2^2/32 - E00*E21*x0*x1/16 + E00*E21*x0*x2/16 - E00*E21*x1^2/32 - E00*E21*x1*x2/16 + E00*E22*x0^2/32 + E00*E22*x0*x1/8 + E00*E22*x0*x2/8 + 3*E00*E22*x1^2/16 + E00*E22*x1*x2/8 + E00*E22*x2^2/32 - E01^2*x2^2/32 - E01*E02*x2^2/32 - E01*E10*x2^2/16 + E01*E11*x0^2/32 + E01*E11*x2^2/16 + E01*E12*x0^2/32 + E01*E12*x0*x2/8 + E01*E12*x2^2/32 - E01*E20*x2^2/32 + E01*E21*x0^2/32 + E01*E21*x0*x2/8 + E01*E21*x2^2/32 - E01*E22*x0*x1/16 + E01*E22*x0*x2/16 - E01*E22*x1^2/32 - E01*E22*x1*x2/16 - E02*E10*x2^2/32 + E02*E11*x0^2/32 - E02*E11*x0*x2/8 + E02*E11*x2^2/32 - E02*E12*x0^2/32 - E02*E21*x0^2/32 - E02*E22*x0^2/32 - E10^2*x2^2/32 + E10*E11*x0^2/32 + E10*E11*x2^2/16 + E10*E12*x0^2/32 + E10*E12*x0*x2/8 + E10*E12*x2^2/32 - E10*E20*x2^2/32 + E10*E21*x0^2/32 + E10*E21*x0*x2/8 + E10*E21*x2^2/32 - E10*E22*x0*x1/16 + E10*E22*x0*x2/16 - E10*E22*x1^2/32 - E10*E22*x1*x2/16 + E11^2*x0^2/32 + E11^2*x2^2/32 + E11*E12*x0^2/16 + E11*E12*x2^2/32 + E11*E20*x0^2/32 - E11*E20*x0*x2/8 + E11*E20*x2^2/32 + E11*E21*x0^2/16 + E11*E21*x2^2/32 + E11*E22*x0^2/32 - E11*E22*x0*x1/16 - E11*E22*x0*x2/16 - E11*E22*x1^2/32 - E11*E22*x1*x2/16 - E12^2*x0^2/32 - E12*E20*x0^2/32 - E12*E21*x0^2/16 - E12*E22*x0^2/16 - E20*E21*x0^2/32 - E20*E22*x0^2/32 - E21^2*x0^2/32 - E21*E22*x0^2/16 - E22^2*x0^2/32) := by
  unfold tetBlock0
  ring

set_option maxHeartbeats 1600000 in
THEOREM tetBlock0_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
tetBlock0_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean:105 
/-- Tet 0 collapses to a pure-QQ polynomial (every s2/s3/p entry of G
carries a literal (0)^2 midpoint factor). -/
theorem tetBlock0_eq (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ) :
    tetBlock0 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p
    =
    (-E00^2*x2^2/32 - E00*E01*x2^2/16 - E00*E02*x2^2/32 - E00*E10*x2^2/16 - E00*E11*x0*x1/16 - E00*E11*x0*x2/16 - E00*E11*x1^2/32 - E00*E11*x1*x2/16 + E00*E11*x2^2/32 - E00*E12*x0*x1/16 + E00*E12*x0*x2/16 - E00*E12*x1^2/32 - E00*E12*x1*x2/16 - E00*E20*x2^2/32 - E00*E21*x0*x1/16 + E00*E21*x0*x2/16 - E00*E21*x1^2/32 - E00*E21*x1*x2/16 + E00*E22*x0^2/32 + E00*E22*x0*x1/8 + E00*E22*x0*x2/8 + 3*E00*E22*x1^2/16 + E00*E22*x1*x2/8 + E00*E22*x2^2/32 - E01^2*x2^2/32 - E01*E02*x2^2/32 - E01*E10*x2^2/16 + E01*E11*x0^2/32 + E01*E11*x2^2/16 + E01*E12*x0^2/32 + E01*E12*x0*x2/8 + E01*E12*x2^2/32 - E01*E20*x2^2/32 + E01*E21*x0^2/32 + E01*E21*x0*x2/8 + E01*E21*x2^2/32 - E01*E22*x0*x1/16 + E01*E22*x0*x2/16 - E01*E22*x1^2/32 - E01*E22*x1*x2/16 - E02*E10*x2^2/32 + E02*E11*x0^2/32 - E02*E11*x0*x2/8 + E02*E11*x2^2/32 - E02*E12*x0^2/32 - E02*E21*x0^2/32 - E02*E22*x0^2/32 - E10^2*x2^2/32 + E10*E11*x0^2/32 + E10*E11*x2^2/16 + E10*E12*x0^2/32 + E10*E12*x0*x2/8 + E10*E12*x2^2/32 - E10*E20*x2^2/32 + E10*E21*x0^2/32 + E10*E21*x0*x2/8 + E10*E21*x2^2/32 - E10*E22*x0*x1/16 + E10*E22*x0*x2/16 - E10*E22*x1^2/32 - E10*E22*x1*x2/16 + E11^2*x0^2/32 + E11^2*x2^2/32 + E11*E12*x0^2/16 + E11*E12*x2^2/32 + E11*E20*x0^2/32 - E11*E20*x0*x2/8 + E11*E20*x2^2/32 + E11*E21*x0^2/16 + E11*E21*x2^2/32 + E11*E22*x0^2/32 - E11*E22*x0*x1/16 - E11*E22*x0*x2/16 - E11*E22*x1^2/32 - E11*E22*x1*x2/16 - E12^2*x0^2/32 - E12*E20*x0^2/32 - E12*E21*x0^2/16 - E12*E22*x0^2/16 - E20*E21*x0^2/32 - E20*E22*x0^2/32 - E21^2*x0^2/32 - E21*E22*x0^2/16 - E22^2*x0^2/32) := by
  unfold tetBlock0
  ring

set_option maxHeartbeats 1600000 in

What this page does not claim

This theorem does not assign any physical meaning to the variables E or x. This theorem does not state that the expression represents a physical quantity. This theorem does not derive a physical law; it only verifies an algebraic identity.

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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/Analysis/ReggeTTContinuumCertificateSpike.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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