RDG--Maxwell Bridge 09: sparse macroelement metric escape
Date: 26 October 2026
Status: exact one-macroelement constructive escape from the diagonal-Hodge no-go
Programme: RDG--Maxwell bridge
Evidence: remeshing-retract record
Verifier: remeshing-retract verifier
Result
Bridge 08 proves that no strictly positive diagonal metric can make any of the three minimal simplicial-collapse reconstruction projectors self-adjoint. Bridge 05's compatible-metric construction supplies the smallest currently known escape: couple cells inside the refined macroelement.
For the centre-to-vertex-0 collapse, complete transport to a basis
and define
The certificate verifies exactly, in degrees 0, 1 and 2,
Thus transport is an isometry, restriction is its weighted adjoint, and the chain-compatible reconstruction is energy-orthogonal.
Exact local blocks
On vertices,
Only the collapsed coarse vertex and the new centre vertex couple.
On edges, in the ordering \((01,12,20,03,13,23)\),
There are two off-diagonal edge pairs.
On the three fine faces,
The faces form one coupled block. Positivity follows from \(L_f=(B^{-1})^{\mathsf T}B^{-1}\), so no eigenvalue inference is required.
What “local” now means
The metric is not diagonal, but neither is it globally dense. Every nonzero coupling lies within the single triangle and its stellar refinement. This suggests a hierarchy:
- cell-local: one positive weight per cell—ruled out here;
- macroelement-local: a positive block on cells participating in one geometry move—achieved here;
- globally nonlocal: couplings between unrelated regions—not required by this certificate.
The minimal carrier of energy compatibility is therefore relational: it is a small incidence neighborhood, not an isolated cell.
Remaining assembly gate
One macroelement does not establish a scalable method. Adjacent refined triangles share vertices and edges. Their local blocks must be assembled so that the global matrices remain positive and the global restriction remains the weighted adjoint of transport.
The next certificate must test:
on at least two adjacent refined faces. Failure would show that independently valid local repairs interfere at overlaps. Success would provide a reusable dynamic-geometry finite-element primitive.
Implication for the corpus
The result gives “relation as carrier” a literal energetic meaning. A scalar attached to each object is too weak; the admissible energy lives on a small block of relations. The mirror face is not another object value but the off-diagonal pairing needed to make forward geometry transport and backward restriction adjoint.
Logical boundary
- Exact: the displayed matrices and all weighted-adjoint, self-adjointness and isometry identities.
- Locality achieved: support is confined to one refined macroelement.
- Open: global assembly, uniqueness, conditioning, approximation order and continuum convergence.
- Not claimed: that these blocks are standard Whitney-form mass matrices or that this asymmetric collapse is numerically optimal.