Coarse-graining support: Support.CoarseGraining.OriginCubeEllipticRecovery.MuGeVecDot #
Imported from the Apache-2.0 CoarseGraining development at commit
c7ddd76c08ade64fed1b8d2ca51be14dfee8deb4.
Origin-cube elliptic recovery -- lower bound and exact slice equalities #
The long mu_ge_vecDot_openCubeSet_originCube_of_hasOpenCubeEllipticRecoveryData packaging, together with the pure-flux and pure-gradient slice equalities feeding DeterministicCoarseData.
Lower bound Mu against the scalar product on the centered open cube,
packaged directly from deterministic recovery-plus-ellipticity data.
On the centered open cube, the zero-right recovered response-space witness can be split into a primal/adjoint scalar half-pair at the level of averaged block response integrands.
This exposes the existing deterministic pair-half reconstruction directly from
HasOpenCubeEllipticRecoveryData, without yet claiming the sharper exact slice
identity Mu(U; (0,q), a) = ResponseJ(U; 0, q, a).
Sigma-free pure-flux coupling handoff on the centered open cube.
Once a recovered primal/adjoint half-pair is known to satisfy the primal scalar
Euler-Lagrange identity for ResponseJ(U; 0, q0, a), the recovery energy and
zero state-pairing identities identify the pure-flux slice of Mu with the
scalar response value. The remaining bridge is therefore exactly the derivation
of the hPairFirst first-variation clause from recovery data.
Pure-flux exact slice on the centered open cube from recovery data alone.
This removes the temporary explicit first-variation hypothesis from the
sigma-free coupling handoff: recovery admissibility of the \mu minimizer and
the recovered half-pair reconstruction imply the primal scalar Euler-Lagrange
identity needed by the deterministic coupling lemma.
Pure-gradient exact slice on the centered open cube from recovery data alone.
The recovered half-pair gives the scalar first variation at (-p0,0); the
deterministic coupling lemma then uses quadratic homogeneity of ResponseJ to
return the note-facing slice ResponseJ(U; p0, 0, a).
Full mixed response slice on the centered open cube from recovery data alone.
For the recovered field at block datum (-p,q), the average state-pairing is
-p·q. The deterministic half-pair coupling therefore identifies the block
energy with ResponseJ(U;p,q,a) + p·q, which is exactly the mixed-term
identity needed for the full response-side block-quadratic package.
Exact pure-flux slice equality on the centered open cube, packaged from deterministic recovery-plus-ellipticity data together with the current deterministic coarse block/sigma data.
Exact pure-gradient slice equality on the centered open cube, packaged from deterministic recovery-plus-ellipticity data together with the current deterministic coarse block/sigma data.