Encyclopedia Gravity Gravity Analysis Regge Ttcontinuum Certificate Spike Tt Continuum Certificate
ARTICLE 4 claims 4 theorems
Gravity Analysis Regge Ttcontinuum Certificate Spike Tt Continuum Certificate
A machine-checked identity shows that a large discrete sum over a tetrahedron collapses to a simple continuous expression, exactly when the data obeys the transverse-traceless conditions.
The TT continuum certificate
The declaration tt_continuum_certificate is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. It proves that a specific, literal 216-term sum, built from a tetrahedron's edge vectors and a symmetric matrix of field values, equals a much simpler expression: -(1/4) times the squared length of the edge vector times the squared Frobenius norm of the matrix. The equality holds only under three conditions: the matrix is symmetric, its trace is zero, and the edge vector is orthogonal to the matrix's rows. These are exactly the transverse-traceless (TT) conditions familiar from gravitational wave analysis, where the perturbation is divergence-free and trace-free.
The proof works by decomposing the 216-term sum into six blocks of 36 terms each. For each block, the framework's kernel re-verifies that all terms involving the symbols s2, s3, and p, which stand for sqrt 2, sqrt 3, and pi, cancel out completely. What remains is a polynomial in the matrix entries and edge coordinates. The final identity is then discharged by an explicit offline cofactor certificate, a precomputed algebraic witness that the polynomial vanishes. The entire chain is checked by the kernel, so the conclusion is not an approximation or a numerical coincidence; it is an exact algebraic identity.
In Recognition Science, this certificate acts as a bridge. It shows that a discrete, combinatorial sum over a tetrahedron, the kind of object the framework's ledger of recognition events produces, converges to a continuous quadratic form under the TT conditions. This is a step toward connecting the framework's discrete geometry to the classical continuum description of gravitational waves. The framework models the discrete sum as a literal transcription of a physical expression, and the theorem proves that this discrete object has a clean continuum limit.
What the certificate does not claim is just as important. It does not prove that the TT conditions are forced by the framework; they are assumptions in the theorem. It does not show that any specific physical system satisfies these conditions, nor does it derive the Einstein field equations or any gravitational wave solution. The certificate is an algebraic identity about a particular sum, not a statement about the dynamics of gravity. The physical interpretation, that this discrete sum represents a gravitational wave energy or action, is a model choice, not a consequence of the theorem.
THEOREM tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
/-- TT continuum certificate: the literal 216-term per-tet sum (s2, s3, p
COMPLETELY FREE, standing for sqrt 2, sqrt 3, pi) equals
-(1/4) * |x|^2 * ||E||_F^2 on the TT constraints. Proof: collapse each
per-tet block by `tetBlock*_eq` (kernel re-verifies the s2/s3/p
cancellation), then discharge the pure-QQ identity with the explicit offline
cofactor certificate via `linear_combination`. -/
theorem tt_continuum_certificate (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ)
(hsym01 : E01 = E10) (hsym02 : E02 = E20) (hsym12 : E12 = E21)
(htr : E00 + E11 + E22 = 0)
(htrans0 : x0 * E00 + x1 * E10 + x2 * E20 = 0)
(htrans1 : x0 * E01 + x1 * E11 + x2 * E21 = 0)
(htrans2 : x0 * E02 + x1 * E12 + x2 * E22 = 0) :
((tetBlock0 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock2 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock3 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock4 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock5 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p))
=
-((1 : ℝ)/4) * (x0^2 + x1^2 + x2^2) * (E00^2 + E01^2 + E02^2 + E10^2 + E11^2 + E12^2 + E20^2 + E21^2 + E22^2) := by
rw [tetBlock0_eq, tetBlock1_eq, tetBlock2_eq, tetBlock3_eq,
tetBlock4_eq, tetBlock5_eq]
linear_combination
(E00*x0*x1/2 - E01*x0^2/4 + E01*x1^2/4 + E01*x2^2/8 + E02*x1*x2/4 - E10*x0^2/4 - E10*x1^2/4 - E10*x2^2/8 - E12*x0*x2/4 - E20*x1*x2/4 - E21*x0*x2/4 + E22*x0*x1/2) * hsym01
+ (E00*x0*x2/2 - E02*x0^2/4 + E02*x1^2/8 + E02*x2^2/4 + E11*x0*x2/2 - E12*x0*x1/4 - E20*x0^2/4 - E20*x1^2/8 - E20*x2^2/4 - E21*x0*x1/4) * hsym02
+ (E00*x1*x2/2 - E02*x0*x1/2 + E11*x1*x2/2 + E12*x0^2/8 - E12*x1^2/4 + E12*x2^2/4 - E21*x0^2/8 - E21*x1^2/4 - E21*x2^2/4) * hsym12
+ (-E00*x0^2/4 + E00*x1^2/4 + E00*x2^2/4 - E01*x0*x1 - E02*x0*x2/2 + E11*x0^2/4 - E11*x1^2/4 + E11*x2^2/4 - E12*x1*x2/2 + E22*x0^2/4 + E22*x1^2/4 + E22*x2^2/4) * htr
+ (E00*x0/2 + E01*x1/2 + E02*x2/2) * htrans0
+ (E01*x0/2 + E11*x1/2 + E12*x2/2) * htrans1
+ (-E00*x2/2 + E02*x0/2 - E11*x2/2 + E12*x1/2) * htrans2
THEOREM tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
/-- TT continuum certificate: the literal 216-term per-tet sum (s2, s3, p
COMPLETELY FREE, standing for sqrt 2, sqrt 3, pi) equals
-(1/4) * |x|^2 * ||E||_F^2 on the TT constraints. Proof: collapse each
per-tet block by `tetBlock*_eq` (kernel re-verifies the s2/s3/p
cancellation), then discharge the pure-QQ identity with the explicit offline
cofactor certificate via `linear_combination`. -/
theorem tt_continuum_certificate (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ)
(hsym01 : E01 = E10) (hsym02 : E02 = E20) (hsym12 : E12 = E21)
(htr : E00 + E11 + E22 = 0)
(htrans0 : x0 * E00 + x1 * E10 + x2 * E20 = 0)
(htrans1 : x0 * E01 + x1 * E11 + x2 * E21 = 0)
(htrans2 : x0 * E02 + x1 * E12 + x2 * E22 = 0) :
((tetBlock0 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock2 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock3 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock4 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock5 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p))
=
-((1 : ℝ)/4) * (x0^2 + x1^2 + x2^2) * (E00^2 + E01^2 + E02^2 + E10^2 + E11^2 + E12^2 + E20^2 + E21^2 + E22^2) := by
rw [tetBlock0_eq, tetBlock1_eq, tetBlock2_eq, tetBlock3_eq,
tetBlock4_eq, tetBlock5_eq]
linear_combination
(E00*x0*x1/2 - E01*x0^2/4 + E01*x1^2/4 + E01*x2^2/8 + E02*x1*x2/4 - E10*x0^2/4 - E10*x1^2/4 - E10*x2^2/8 - E12*x0*x2/4 - E20*x1*x2/4 - E21*x0*x2/4 + E22*x0*x1/2) * hsym01
+ (E00*x0*x2/2 - E02*x0^2/4 + E02*x1^2/8 + E02*x2^2/4 + E11*x0*x2/2 - E12*x0*x1/4 - E20*x0^2/4 - E20*x1^2/8 - E20*x2^2/4 - E21*x0*x1/4) * hsym02
+ (E00*x1*x2/2 - E02*x0*x1/2 + E11*x1*x2/2 + E12*x0^2/8 - E12*x1^2/4 + E12*x2^2/4 - E21*x0^2/8 - E21*x1^2/4 - E21*x2^2/4) * hsym12
+ (-E00*x0^2/4 + E00*x1^2/4 + E00*x2^2/4 - E01*x0*x1 - E02*x0*x2/2 + E11*x0^2/4 - E11*x1^2/4 + E11*x2^2/4 - E12*x1*x2/2 + E22*x0^2/4 + E22*x1^2/4 + E22*x2^2/4) * htr
+ (E00*x0/2 + E01*x1/2 + E02*x2/2) * htrans0
+ (E01*x0/2 + E11*x1/2 + E12*x2/2) * htrans1
+ (-E00*x2/2 + E02*x0/2 - E11*x2/2 + E12*x1/2) * htrans2
THEOREM tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
/-- TT continuum certificate: the literal 216-term per-tet sum (s2, s3, p
COMPLETELY FREE, standing for sqrt 2, sqrt 3, pi) equals
-(1/4) * |x|^2 * ||E||_F^2 on the TT constraints. Proof: collapse each
per-tet block by `tetBlock*_eq` (kernel re-verifies the s2/s3/p
cancellation), then discharge the pure-QQ identity with the explicit offline
cofactor certificate via `linear_combination`. -/
theorem tt_continuum_certificate (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ)
(hsym01 : E01 = E10) (hsym02 : E02 = E20) (hsym12 : E12 = E21)
(htr : E00 + E11 + E22 = 0)
(htrans0 : x0 * E00 + x1 * E10 + x2 * E20 = 0)
(htrans1 : x0 * E01 + x1 * E11 + x2 * E21 = 0)
(htrans2 : x0 * E02 + x1 * E12 + x2 * E22 = 0) :
((tetBlock0 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock2 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock3 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock4 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)
+ (tetBlock5 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p))
=
-((1 : ℝ)/4) * (x0^2 + x1^2 + x2^2) * (E00^2 + E01^2 + E02^2 + E10^2 + E11^2 + E12^2 + E20^2 + E21^2 + E22^2) := by
rw [tetBlock0_eq, tetBlock1_eq, tetBlock2_eq, tetBlock3_eq,
tetBlock4_eq, tetBlock5_eq]
linear_combination
(E00*x0*x1/2 - E01*x0^2/4 + E01*x1^2/4 + E01*x2^2/8 + E02*x1*x2/4 - E10*x0^2/4 - E10*x1^2/4 - E10*x2^2/8 - E12*x0*x2/4 - E20*x1*x2/4 - E21*x0*x2/4 + E22*x0*x1/2) * hsym01
+ (E00*x0*x2/2 - E02*x0^2/4 + E02*x1^2/8 + E02*x2^2/4 + E11*x0*x2/2 - E12*x0*x1/4 - E20*x0^2/4 - E20*x1^2/8 - E20*x2^2/4 - E21*x0*x1/4) * hsym02
+ (E00*x1*x2/2 - E02*x0*x1/2 + E11*x1*x2/2 + E12*x0^2/8 - E12*x1^2/4 + E12*x2^2/4 - E21*x0^2/8 - E21*x1^2/4 - E21*x2^2/4) * hsym12
+ (-E00*x0^2/4 + E00*x1^2/4 + E00*x2^2/4 - E01*x0*x1 - E02*x0*x2/2 + E11*x0^2/4 - E11*x1^2/4 + E11*x2^2/4 - E12*x1*x2/2 + E22*x0^2/4 + E22*x1^2/4 + E22*x2^2/4) * htr
+ (E00*x0/2 + E01*x1/2 + E02*x2/2) * htrans0
+ (E01*x0/2 + E11*x1/2 + E12*x2/2) * htrans1
+ (-E00*x2/2 + E02*x0/2 - E11*x2/2 + E12*x1/2) * htrans2
THEOREM tetBlock0_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
/-- 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
The TT conditions are not derived from the framework; they are assumptions in the theorem. The certificate does not prove any statement about the dynamics of gravity or the existence of gravitational waves. The physical interpretation of the sum as an energy or action is a model choice, not a theorem.
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:
- How does this discrete-to-continuum identity connect to the full Einstein field equations?
- What physical quantity does the sum represent, if any, beyond a formal algebraic object?
- Do the TT conditions arise naturally from the framework's forcing chain, or are they imposed from outside?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
THEOREM tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
/-- TT continuum certificate: the literal 216-term per-tet sum (s2, s3, p COMPLETELY FREE, standing for sqrt 2, sqrt 3, pi) equals -(1/4) * |x|^2 * ||E||_F^2 on the TT constraints. Proof: collapse each per-tet block by `tetBlock*_eq` (kernel re-verifies the s2/s3/p cancellation), then discharge the pure-QQ identity with the explicit offline cofactor certificate via `linear_combination`. -/ theorem tt_continuum_certificate (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ) (hsym01 : E01 = E10) (hsym02 : E02 = E20) (hsym12 : E12 = E21) (htr : E00 + E11 + E22 = 0) (htrans0 : x0 * E00 + x1 * E10 + x2 * E20 = 0) (htrans1 : x0 * E01 + x1 * E11 + x2 * E21 = 0) (htrans2 : x0 * E02 + x1 * E12 + x2 * E22 = 0) : ((tetBlock0 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock2 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock3 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock4 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock5 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)) = -((1 : ℝ)/4) * (x0^2 + x1^2 + x2^2) * (E00^2 + E01^2 + E02^2 + E10^2 + E11^2 + E12^2 + E20^2 + E21^2 + E22^2) := by rw [tetBlock0_eq, tetBlock1_eq, tetBlock2_eq, tetBlock3_eq, tetBlock4_eq, tetBlock5_eq] linear_combination (E00*x0*x1/2 - E01*x0^2/4 + E01*x1^2/4 + E01*x2^2/8 + E02*x1*x2/4 - E10*x0^2/4 - E10*x1^2/4 - E10*x2^2/8 - E12*x0*x2/4 - E20*x1*x2/4 - E21*x0*x2/4 + E22*x0*x1/2) * hsym01 + (E00*x0*x2/2 - E02*x0^2/4 + E02*x1^2/8 + E02*x2^2/4 + E11*x0*x2/2 - E12*x0*x1/4 - E20*x0^2/4 - E20*x1^2/8 - E20*x2^2/4 - E21*x0*x1/4) * hsym02 + (E00*x1*x2/2 - E02*x0*x1/2 + E11*x1*x2/2 + E12*x0^2/8 - E12*x1^2/4 + E12*x2^2/4 - E21*x0^2/8 - E21*x1^2/4 - E21*x2^2/4) * hsym12 + (-E00*x0^2/4 + E00*x1^2/4 + E00*x2^2/4 - E01*x0*x1 - E02*x0*x2/2 + E11*x0^2/4 - E11*x1^2/4 + E11*x2^2/4 - E12*x1*x2/2 + E22*x0^2/4 + E22*x1^2/4 + E22*x2^2/4) * htr + (E00*x0/2 + E01*x1/2 + E02*x2/2) * htrans0 + (E01*x0/2 + E11*x1/2 + E12*x2/2) * htrans1 + (-E00*x2/2 + E02*x0/2 - E11*x2/2 + E12*x1/2) * htrans2The declaration tt_continuum_certificate is a formal theorem in the Recognition Science framework's machine-checked library of formal theorems. tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.leanTHEOREM tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
/-- TT continuum certificate: the literal 216-term per-tet sum (s2, s3, p COMPLETELY FREE, standing for sqrt 2, sqrt 3, pi) equals -(1/4) * |x|^2 * ||E||_F^2 on the TT constraints. Proof: collapse each per-tet block by `tetBlock*_eq` (kernel re-verifies the s2/s3/p cancellation), then discharge the pure-QQ identity with the explicit offline cofactor certificate via `linear_combination`. -/ theorem tt_continuum_certificate (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ) (hsym01 : E01 = E10) (hsym02 : E02 = E20) (hsym12 : E12 = E21) (htr : E00 + E11 + E22 = 0) (htrans0 : x0 * E00 + x1 * E10 + x2 * E20 = 0) (htrans1 : x0 * E01 + x1 * E11 + x2 * E21 = 0) (htrans2 : x0 * E02 + x1 * E12 + x2 * E22 = 0) : ((tetBlock0 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock2 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock3 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock4 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock5 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)) = -((1 : ℝ)/4) * (x0^2 + x1^2 + x2^2) * (E00^2 + E01^2 + E02^2 + E10^2 + E11^2 + E12^2 + E20^2 + E21^2 + E22^2) := by rw [tetBlock0_eq, tetBlock1_eq, tetBlock2_eq, tetBlock3_eq, tetBlock4_eq, tetBlock5_eq] linear_combination (E00*x0*x1/2 - E01*x0^2/4 + E01*x1^2/4 + E01*x2^2/8 + E02*x1*x2/4 - E10*x0^2/4 - E10*x1^2/4 - E10*x2^2/8 - E12*x0*x2/4 - E20*x1*x2/4 - E21*x0*x2/4 + E22*x0*x1/2) * hsym01 + (E00*x0*x2/2 - E02*x0^2/4 + E02*x1^2/8 + E02*x2^2/4 + E11*x0*x2/2 - E12*x0*x1/4 - E20*x0^2/4 - E20*x1^2/8 - E20*x2^2/4 - E21*x0*x1/4) * hsym02 + (E00*x1*x2/2 - E02*x0*x1/2 + E11*x1*x2/2 + E12*x0^2/8 - E12*x1^2/4 + E12*x2^2/4 - E21*x0^2/8 - E21*x1^2/4 - E21*x2^2/4) * hsym12 + (-E00*x0^2/4 + E00*x1^2/4 + E00*x2^2/4 - E01*x0*x1 - E02*x0*x2/2 + E11*x0^2/4 - E11*x1^2/4 + E11*x2^2/4 - E12*x1*x2/2 + E22*x0^2/4 + E22*x1^2/4 + E22*x2^2/4) * htr + (E00*x0/2 + E01*x1/2 + E02*x2/2) * htrans0 + (E01*x0/2 + E11*x1/2 + E12*x2/2) * htrans1 + (-E00*x2/2 + E02*x0/2 - E11*x2/2 + E12*x1/2) * htrans2It proves that a specific, literal 216-term sum, built from a tetrahedron's edge vectors and a symmetric matrix of field values, equals a much simpler expression: -(1/4) times the squared length of the edge vector times the squared Frobenius norm of the matrix. tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.leanTHEOREM tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
/-- TT continuum certificate: the literal 216-term per-tet sum (s2, s3, p COMPLETELY FREE, standing for sqrt 2, sqrt 3, pi) equals -(1/4) * |x|^2 * ||E||_F^2 on the TT constraints. Proof: collapse each per-tet block by `tetBlock*_eq` (kernel re-verifies the s2/s3/p cancellation), then discharge the pure-QQ identity with the explicit offline cofactor certificate via `linear_combination`. -/ theorem tt_continuum_certificate (E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p : ℝ) (hsym01 : E01 = E10) (hsym02 : E02 = E20) (hsym12 : E12 = E21) (htr : E00 + E11 + E22 = 0) (htrans0 : x0 * E00 + x1 * E10 + x2 * E20 = 0) (htrans1 : x0 * E01 + x1 * E11 + x2 * E21 = 0) (htrans2 : x0 * E02 + x1 * E12 + x2 * E22 = 0) : ((tetBlock0 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock1 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock2 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock3 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock4 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p) + (tetBlock5 E00 E01 E02 E10 E11 E12 E20 E21 E22 x0 x1 x2 s2 s3 p)) = -((1 : ℝ)/4) * (x0^2 + x1^2 + x2^2) * (E00^2 + E01^2 + E02^2 + E10^2 + E11^2 + E12^2 + E20^2 + E21^2 + E22^2) := by rw [tetBlock0_eq, tetBlock1_eq, tetBlock2_eq, tetBlock3_eq, tetBlock4_eq, tetBlock5_eq] linear_combination (E00*x0*x1/2 - E01*x0^2/4 + E01*x1^2/4 + E01*x2^2/8 + E02*x1*x2/4 - E10*x0^2/4 - E10*x1^2/4 - E10*x2^2/8 - E12*x0*x2/4 - E20*x1*x2/4 - E21*x0*x2/4 + E22*x0*x1/2) * hsym01 + (E00*x0*x2/2 - E02*x0^2/4 + E02*x1^2/8 + E02*x2^2/4 + E11*x0*x2/2 - E12*x0*x1/4 - E20*x0^2/4 - E20*x1^2/8 - E20*x2^2/4 - E21*x0*x1/4) * hsym02 + (E00*x1*x2/2 - E02*x0*x1/2 + E11*x1*x2/2 + E12*x0^2/8 - E12*x1^2/4 + E12*x2^2/4 - E21*x0^2/8 - E21*x1^2/4 - E21*x2^2/4) * hsym12 + (-E00*x0^2/4 + E00*x1^2/4 + E00*x2^2/4 - E01*x0*x1 - E02*x0*x2/2 + E11*x0^2/4 - E11*x1^2/4 + E11*x2^2/4 - E12*x1*x2/2 + E22*x0^2/4 + E22*x1^2/4 + E22*x2^2/4) * htr + (E00*x0/2 + E01*x1/2 + E02*x2/2) * htrans0 + (E01*x0/2 + E11*x1/2 + E12*x2/2) * htrans1 + (-E00*x2/2 + E02*x0/2 - E11*x2/2 + E12*x1/2) * htrans2The equality holds only under three conditions: the matrix is symmetric, its trace is zero, and the edge vector is orthogonal to the matrix's rows. tt_continuum_certificate · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.leanTHEOREM tetBlock0_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean
/-- 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 inFor each block, the framework's kernel re-verifies that all terms involving the symbols s2, s3, and p, which stand for sqrt 2, sqrt 3, and pi, cancel out completely. tetBlock0_eq · IndisputableMonolith/Gravity/Analysis/ReggeTTContinuumCertificateSpike.lean