Encyclopedia Gravity Gravity Analysis Regge Ttcontinuum Certificate Spike Tet Block1 Eq

ARTICLE 3 claims 3 theorems

Gravity Analysis Regge Ttcontinuum Certificate Spike Tet Block1 Eq

A single algebraic lemma in a machine-checked library rewrites one piece of a gravitational sum; it proves a transcription, not a physical law.

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The tet block identity

In the Recognition Science framework, the declaration tetBlock1_eq is a formal identity about a specific algebraic expression. The expression, called tetBlock1, is one of six building blocks in a larger sum that the framework uses to study a continuum limit in a gravity analysis. The declaration states that this block, which is defined in terms of nine variables E00 through E22 and three coordinates x0, x1, x2, can be rewritten as a long polynomial in those variables. The proof is a direct expansion: the declaration unfolds the definition and applies the algebraic simplification rule known as ring, which verifies that the two forms are equal for all real-number inputs.

The content of the identity is purely formal. The original expression is a literal transcription of 36 terms of the form (1/2) * G_fg * c_(d(tau,f)) * c_(d(tau,g)) * (x.(m_g - m_f))^2, where the symbols G, c, and m stand for quantities in the framework's gravity model. The declaration proves that this block, after expansion, equals a specific polynomial with no dependence on the parameters s2, s3, and p, which the framework treats as free variables representing sqrt 2, sqrt 3, and pi. This collapse is the key step: it shows that those three parameters cancel out of this block entirely.

What the declaration does not claim is any physical content. It does not assert that the expression represents a force, an energy, or any measurable quantity. It does not say that the framework's gravity model is correct, that the continuum limit exists, or that the sum of all six blocks has any particular value. The declaration is a lemma about algebraic equality, nothing more. Its role is to prepare the ground for a larger theorem, tt_continuum_certificate, which uses this and five sibling identities to show that the full sum equals a simple expression under certain constraints, but that larger result is a separate claim with its own conditions.

The practical consequence is that a reader can trust the transcription. If the framework's analysis depends on this block, the declaration guarantees that the symbolic manipulation is correct: the expanded polynomial is exactly what the original expression evaluates to, for every choice of the real variables. This is the kind of bookkeeping that a machine-checked library of formal theorems exists to provide, a mechanical verification that a human reader would otherwise have to redo by hand.

THEOREM tetBlock1_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
tetBlock1_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean:155 
/-- Tet 1 collapses to a pure-QQ polynomial (every s2/s3/p entry of G
carries a literal (0)^2 midpoint factor). -/
theorem tetBlock1_eq (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ) :
    tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p
    =
    (-E00^2*x1^2/32 - E00*E01*x1^2/32 - E00*E02*x1^2/16 - E00*E10*x1^2/32 + E00*E11*x0^2/32 + E00*E11*x0*x1/8 + E00*E11*x0*x2/8 + E00*E11*x1^2/32 + E00*E11*x1*x2/8 + 3*E00*E11*x2^2/16 + E00*E12*x0*x1/16 - E00*E12*x0*x2/16 - E00*E12*x1*x2/16 - E00*E12*x2^2/32 - E00*E20*x1^2/16 + E00*E21*x0*x1/16 - E00*E21*x0*x2/16 - E00*E21*x1*x2/16 - E00*E21*x2^2/32 - E00*E22*x0*x1/16 - E00*E22*x0*x2/16 + E00*E22*x1^2/32 - E00*E22*x1*x2/16 - E00*E22*x2^2/32 - E01*E02*x1^2/32 - E01*E11*x0^2/32 - E01*E12*x0^2/32 - E01*E20*x1^2/32 - E01*E21*x0^2/32 + E01*E22*x0^2/32 - E01*E22*x0*x1/8 + E01*E22*x1^2/32 - E02^2*x1^2/32 - E02*E10*x1^2/32 + E02*E11*x0*x1/16 - E02*E11*x0*x2/16 - E02*E11*x1*x2/16 - E02*E11*x2^2/32 + E02*E12*x0^2/32 + E02*E12*x0*x1/8 + E02*E12*x1^2/32 - E02*E20*x1^2/16 + E02*E21*x0^2/32 + E02*E21*x0*x1/8 + E02*E21*x1^2/32 + E02*E22*x0^2/32 + E02*E22*x1^2/16 - E10*E11*x0^2/32 - E10*E12*x0^2/32 - E10*E20*x1^2/32 - E10*E21*x0^2/32 + E10*E22*x0^2/32 - E10*E22*x0*x1/8 + E10*E22*x1^2/32 - E11^2*x0^2/32 - E11*E12*x0^2/16 + E11*E20*x0*x1/16 - E11*E20*x0*x2/16 - E11*E20*x1*x2/16 - E11*E20*x2^2/32 - E11*E21*x0^2/16 + E11*E22*x0^2/32 - E11*E22*x0*x1/16 - E11*E22*x0*x2/16 - E11*E22*x1*x2/16 - E11*E22*x2^2/32 - E12^2*x0^2/32 + E12*E20*x0^2/32 + E12*E20*x0*x1/8 + E12*E20*x1^2/32 - E12*E21*x0^2/16 + E12*E22*x0^2/16 + E12*E22*x1^2/32 - E20^2*x1^2/32 + E20*E21*x0^2/32 + E20*E21*x0*x1/8 + E20*E21*x1^2/32 + E20*E22*x0^2/32 + E20*E22*x1^2/16 - E21^2*x0^2/32 + E21*E22*x0^2/16 + E21*E22*x1^2/32 + E22^2*x0^2/32 + E22^2*x1^2/32) := by
  unfold tetBlock1
  ring

set_option maxHeartbeats 1600000 in
THEOREM tetBlock1_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
tetBlock1_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean:155 
/-- Tet 1 collapses to a pure-QQ polynomial (every s2/s3/p entry of G
carries a literal (0)^2 midpoint factor). -/
theorem tetBlock1_eq (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ) :
    tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p
    =
    (-E00^2*x1^2/32 - E00*E01*x1^2/32 - E00*E02*x1^2/16 - E00*E10*x1^2/32 + E00*E11*x0^2/32 + E00*E11*x0*x1/8 + E00*E11*x0*x2/8 + E00*E11*x1^2/32 + E00*E11*x1*x2/8 + 3*E00*E11*x2^2/16 + E00*E12*x0*x1/16 - E00*E12*x0*x2/16 - E00*E12*x1*x2/16 - E00*E12*x2^2/32 - E00*E20*x1^2/16 + E00*E21*x0*x1/16 - E00*E21*x0*x2/16 - E00*E21*x1*x2/16 - E00*E21*x2^2/32 - E00*E22*x0*x1/16 - E00*E22*x0*x2/16 + E00*E22*x1^2/32 - E00*E22*x1*x2/16 - E00*E22*x2^2/32 - E01*E02*x1^2/32 - E01*E11*x0^2/32 - E01*E12*x0^2/32 - E01*E20*x1^2/32 - E01*E21*x0^2/32 + E01*E22*x0^2/32 - E01*E22*x0*x1/8 + E01*E22*x1^2/32 - E02^2*x1^2/32 - E02*E10*x1^2/32 + E02*E11*x0*x1/16 - E02*E11*x0*x2/16 - E02*E11*x1*x2/16 - E02*E11*x2^2/32 + E02*E12*x0^2/32 + E02*E12*x0*x1/8 + E02*E12*x1^2/32 - E02*E20*x1^2/16 + E02*E21*x0^2/32 + E02*E21*x0*x1/8 + E02*E21*x1^2/32 + E02*E22*x0^2/32 + E02*E22*x1^2/16 - E10*E11*x0^2/32 - E10*E12*x0^2/32 - E10*E20*x1^2/32 - E10*E21*x0^2/32 + E10*E22*x0^2/32 - E10*E22*x0*x1/8 + E10*E22*x1^2/32 - E11^2*x0^2/32 - E11*E12*x0^2/16 + E11*E20*x0*x1/16 - E11*E20*x0*x2/16 - E11*E20*x1*x2/16 - E11*E20*x2^2/32 - E11*E21*x0^2/16 + E11*E22*x0^2/32 - E11*E22*x0*x1/16 - E11*E22*x0*x2/16 - E11*E22*x1*x2/16 - E11*E22*x2^2/32 - E12^2*x0^2/32 + E12*E20*x0^2/32 + E12*E20*x0*x1/8 + E12*E20*x1^2/32 - E12*E21*x0^2/16 + E12*E22*x0^2/16 + E12*E22*x1^2/32 - E20^2*x1^2/32 + E20*E21*x0^2/32 + E20*E21*x0*x1/8 + E20*E21*x1^2/32 + E20*E22*x0^2/32 + E20*E22*x1^2/16 - E21^2*x0^2/32 + E21*E22*x0^2/16 + E21*E22*x1^2/32 + E22^2*x0^2/32 + E22^2*x1^2/32) := by
  unfold tetBlock1
  ring

set_option maxHeartbeats 1600000 in
THEOREM tetBlock1_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
tetBlock1_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean:155 
/-- Tet 1 collapses to a pure-QQ polynomial (every s2/s3/p entry of G
carries a literal (0)^2 midpoint factor). -/
theorem tetBlock1_eq (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ) :
    tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p
    =
    (-E00^2*x1^2/32 - E00*E01*x1^2/32 - E00*E02*x1^2/16 - E00*E10*x1^2/32 + E00*E11*x0^2/32 + E00*E11*x0*x1/8 + E00*E11*x0*x2/8 + E00*E11*x1^2/32 + E00*E11*x1*x2/8 + 3*E00*E11*x2^2/16 + E00*E12*x0*x1/16 - E00*E12*x0*x2/16 - E00*E12*x1*x2/16 - E00*E12*x2^2/32 - E00*E20*x1^2/16 + E00*E21*x0*x1/16 - E00*E21*x0*x2/16 - E00*E21*x1*x2/16 - E00*E21*x2^2/32 - E00*E22*x0*x1/16 - E00*E22*x0*x2/16 + E00*E22*x1^2/32 - E00*E22*x1*x2/16 - E00*E22*x2^2/32 - E01*E02*x1^2/32 - E01*E11*x0^2/32 - E01*E12*x0^2/32 - E01*E20*x1^2/32 - E01*E21*x0^2/32 + E01*E22*x0^2/32 - E01*E22*x0*x1/8 + E01*E22*x1^2/32 - E02^2*x1^2/32 - E02*E10*x1^2/32 + E02*E11*x0*x1/16 - E02*E11*x0*x2/16 - E02*E11*x1*x2/16 - E02*E11*x2^2/32 + E02*E12*x0^2/32 + E02*E12*x0*x1/8 + E02*E12*x1^2/32 - E02*E20*x1^2/16 + E02*E21*x0^2/32 + E02*E21*x0*x1/8 + E02*E21*x1^2/32 + E02*E22*x0^2/32 + E02*E22*x1^2/16 - E10*E11*x0^2/32 - E10*E12*x0^2/32 - E10*E20*x1^2/32 - E10*E21*x0^2/32 + E10*E22*x0^2/32 - E10*E22*x0*x1/8 + E10*E22*x1^2/32 - E11^2*x0^2/32 - E11*E12*x0^2/16 + E11*E20*x0*x1/16 - E11*E20*x0*x2/16 - E11*E20*x1*x2/16 - E11*E20*x2^2/32 - E11*E21*x0^2/16 + E11*E22*x0^2/32 - E11*E22*x0*x1/16 - E11*E22*x0*x2/16 - E11*E22*x1*x2/16 - E11*E22*x2^2/32 - E12^2*x0^2/32 + E12*E20*x0^2/32 + E12*E20*x0*x1/8 + E12*E20*x1^2/32 - E12*E21*x0^2/16 + E12*E22*x0^2/16 + E12*E22*x1^2/32 - E20^2*x1^2/32 + E20*E21*x0^2/32 + E20*E21*x0*x1/8 + E20*E21*x1^2/32 + E20*E22*x0^2/32 + E20*E22*x1^2/16 - E21^2*x0^2/32 + E21*E22*x0^2/16 + E21*E22*x1^2/32 + E22^2*x0^2/32 + E22^2*x1^2/32) := by
  unfold tetBlock1
  ring

set_option maxHeartbeats 1600000 in

What this page does not claim

The declaration does not prove any physical law or property of gravity. The declaration does not establish that the continuum limit exists or is unique. The declaration does not assign a physical meaning to the variables E, x, s2, s3, or p.

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/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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red:

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