Skip to content

Laplace–Beltrami scoping — do we have it, or add it?

Scope: research note only (no code / rc / ADR). Concertmaster dispatch, 2026-07-18. Verdict (derived, not preference): We HAVE it latently. The shipped weighted graph Laplacian IS the discrete Laplace–Beltrami operator; LB is a weighting of the existing Class-L operator, not a new member. One narrow, optional exposure earns consideration (a mass-normalization constructor) — everything else already exists.

1. The attested math relationship (MPM)

The graph Laplacian srmech ships is, mathematically, a discrete Laplace–Beltrami (LB) operator in two standard, attested senses:

  • Point-cloud sense — a graph Laplacian built with heat-kernel edge weights w_ij = exp(−‖x_i−x_j‖²/(4t)) on points sampling a manifold converges (pointwise, as t→0, n→∞) to the LB operator Δ. Belkin & Niyogi (2003) established the eigenmaps construction; the rigorous pointwise convergence-to-Δ is the follow-up Belkin & Niyogi, "Towards a Theoretical Foundation for Laplacian-Based Manifold Methods," COLT 2005 / JCSS 74(8):1289–1308 (2008).
  • Density-free variant — Coifman & Lafon, "Diffusion Maps," ACHA 21(1):5–30 (2006): the anisotropic normalization family (parameter α; α=1) recovers Δ independent of sampling density. The α-renormalization plays the role of the mass matrix (see §4). This is a reweighting, expressible with existing ops.
  • Mesh sense — the cotangent-weighted Laplacian is the STANDARD discrete LB on a triangulated manifold, and it is STILL a weighted graph Laplacian: edge weight w_ij = ½(cot α_ij + cot β_ij), exact trig of the two angles opposite edge ij (closed-form, Class-N-native). Pinkall & Polthier, "Computing Discrete Minimal Surfaces and Their Conjugates," Experimental Mathematics 2(1):15–36 (1993) (the DDG-standard cotangent reference; earlier origin MacNeal 1949 / Duffin 1959; the Voronoi mass matrix is Meyer–Desbrun–Schröder–Barr 2003).
  • Heat kernel — e^{−tΔ} on a manifold is governed by Δ = LB. srmech's EPH propagate already implements e^{−zL}; hand it a metric-weighted L and it is the manifold heat propagator (§3).

Ground-proof (computational provenance). Simplest witness: cycle graph C_n, unit-weight combinatorial Laplacian, closed-form spectrum λ_k = 2(1−cos(2πk/n)) (the form the heat_trace docstring already states). With mesh h = 2π/n, the rescaled discrete-LB eigenvalues λ_k/h² → k² = the LB spectrum of S¹ (−d²/dθ²):

n λ₁/h² λ₂/h² λ₃/h² target k²
6 0.912 2.736 3.648 1 / 4 / 9
96 0.99964 3.9943 8.971 1 / 4 / 9
384 0.99998 3.9996 8.998 1 / 4 / 9

docs/srmech/notes/ gen-script is inline in the dispatch transcript (stdlib math.cos scratch check; jacobi_eigvals reproduces the same closed form). This is an attested mathematical FACT about the discrete↔continuous form, NOT a framework-validation claim (epistemic ceiling holds).

2. Existing Class-L surface audit (introspect-before-assert)

  • Does the shipped Laplacian take edge weights? YES. dense_laplacian(n, edges, weights=None), dense_adjacency, normalized_laplacian, signed_laplacian, klein4_gain_laplacian, magnetic_laplacian all accept a per-edge weights: Optional[Iterable[float]] (parallel edges accumulate additively). Weights may be negative (signed-metric leg — the dissolved "Class O"). So a caller supplies cotangent / heat-kernel weights today, no new operator.
  • C parity: srmech_graph_dense_laplacian(n, …, const double *weights, …) (NULL → unit), "No N cap" on the builder (srmech.h §1018-1028). A bare-C / microcontroller host builds a weighted (cotangent) Laplacian and spectrally partitions it with srmech_jacobi_eigvals / srmech_fiedler_* — JPL-clean, no abs(), byte-parity.
  • Does EPH already do manifold heat flow? YES (up to the mass matrix, §4). propagate(L, u0, z) = e^{−zL}·u0 (C peer srmech_eph_propagate). Hand it a cotangent-weighted L: z real → manifold thermal diffusion, z imaginary → coherent quantum walk on the manifold. heat_trace(L, t) = Tr(e^{−tL}) = Σ_k e^{−tλ_k} (C peer srmech_heat_trace) is the manifold heat trace / spectral-θ.
  • qm layer: no standalone metric/manifold LB operator (uses Class-L mat_matmul / mat_hermitian_eigendecompose; potentials.py has only a 1-D finite-difference grid Laplacian). No duplication risk.

3–4. Derived recommendation

(a) Already-have-latently — PRIMARY. LB is not a new Class-L member; it is the existing weighted dense_laplacian under metric-derived weights, and its heat kernel / heat trace are propagate / heat_trace. This is the discrete-first stance ([[feedback_continuous_number_line_pedagogical_obstacle]]): the discrete weighted Laplacian IS the substrate-native object; "Laplace–Beltrami" is its continuous-limit name. The cot weights are exact Class-N trig; the t→0 limit is the Class-N asymptotic ring. Do NOT reach for a continuous PDE operator — the content is already carried.

Duality worth naming (notebook, not code): graph-Laplacian (discrete) ↔ Laplace–Beltrami (continuous) is a genuine convergence duality across the discrete↔ continuous seam — the same operator, two projections, not two operators. Name it; don't mint an operator for it.

(b) The one genuine gap — falsification-worthy. The FEM/DEC LB is Δ = M⁻¹L_c, where L_c = cotangent stiffness matrix (= dense_laplacian(cot-weights), HAVE it) and M = mass matrix (diagonal Voronoi/barycentric areas). srmech's normalized_laplacian normalizes by degree D^{−1/2}, not by an arbitrary node mass M. So the exact LB spectrum (generalized eigenproblem L_c v = λ M v) and exact manifold heat kernel e^{−t M⁻¹L_c} need the mass-weighting M. Today it is expressible but non-ergonomic: build M^{−1/2} L_c M^{−1/2} via mat_matmul with a diagonal M^{−1/2}, then reuse the existing eigensolve / propagate. Falsifier: if a caller CANNOT reproduce a published sphere/CP² LB spectrum from mesh geometry using only shipped ops → the gap is real and © is warranted. Current status: reproducible, just verbose.

© Optional exposure IF the mesh/point-cloud LB spectrum becomes a live use-case — a WEIGHTING/NORMALIZATION constructor, NOT a PDE solver. Closed-form, C-native shape: 1. cotangent_weights(triangles, positions) → weights — turns triangle geometry into the exact per-edge ½(cot α + cot β) (Class-N trig; closed-form). Feeds the EXISTING dense_laplacian. 2. mass_normalized_laplacian(n, edges, weights, masses) → M^{−1/2}(D−W)M^{−1/2} — a one-line generalization of normalized_laplacian from degree-D to arbitrary diagonal M. C peer is the existing srmech_graph_dense_laplacian + a diagonal scale.

Both satisfy the standing bar: closed-form, C-native (a bare-C host builds + spectrally partitions), JPL-clean, never abs(), byte-parity. Neither is a GPU mesh-FEA solve (that is the OUT side of the CAD-ban refinement). This is scoping only — recommend (a) now; hold © behind a real LB-spectrum-from-mesh request (conductor decision).

5. MFO + EPH ties (route per MFO-vs-srmech split rule)

  • Strongest tie — MFO. MFO notebook §III.1 already names the Laplace–Beltrami operator: sphere Sⁿ eigenvalues l(l+n−1), CP² Fubini–Study, and the §V heat-kernel spectral-dimension diagnostic d_S = −2 d log K(σ)/d log σ (Weyl's law). LB IS the metric-tensor Laplacian — this is squarely the Metric Field Ontology. The finding strengthens/formalizes the metric-substrate story: srmech's weighted heat_trace + propagate are the computational surface for exactly the §III.1/§V analytic diagnostics — a (cotangent-)weighted Laplacian makes the MFO manifold heat trace computable on the substrate-native discrete object. Per the split rule, the tooling finding lands as this srmech note; a one-line cross-ref strengthens the MFO §III.1/§V metric-substrate narrative (conductor to place).
  • EPH. propagate/eph_harvest already ARE the manifold heat/quantum propagator e^{−zL}; the arg(z) coherence dial names the thermal↔coherent manifold-flow continuum a two-op split cannot express. LB adds no new EPH op — it adds metric weights to the L EPH already consumes.

6. Sources (status)

Ref Attested Status
Belkin & Niyogi, Laplacian Eigenmaps…, Neural Computation 15(6):1373–1396 (2003) author+title+venue+pages ✓ MIT-Press DOI paywalled → not the anchor; author-hosted OA PDF exists ([[feedback_paywalled_doi_cannot_be_attested]])
Belkin & Niyogi, Towards a Theoretical Foundation…, COLT 2005 / JCSS 74(8):1289–1308 (2008) title+venue ✓ the convergence-to-Δ proof; author preprint OA
Coifman & Lafon, Diffusion Maps, ACHA 21(1):5–30 (2006) author+title+venue+pages ✓ Elsevier DOI paywalled → author-hosted OA PDF (Yale/Weizmann) is the anchor
Pinkall & Polthier, Computing Discrete Minimal Surfaces…, Exp. Math. 2(1):15–36 (1993) author+title+venue+issue+pages ✓ Project Euclid OA (clean anchor)

Content-level claims (the α=1 density-free LB recovery; cotangent = discrete LB; Voronoi mass = MDSB 2003) are attested at metadata level here and stated from standard DDG knowledge; a full PDF extraction is the follow-up if any is promoted into notebook prose ([[feedback_pdf_extraction_citation_discipline]]).

7. ALU cost — is the FEM form cheaper, or is having both useful?

Cotangent weight needs NO trig library (the ALU win). For the two edge vectors u, v at the vertex opposite edge ij: cot θ = (u·v)/|u×v| — dot over cross. Numerator u·v is exactly integer/rational in the vertex coordinates. In 2D u×v is the scalar cross → cot is fully rational (Class-N exact; the signed-area sign is a Class-K pin-slot, never abs()). In 3D the only irrationality is one algebraic √ per triangle: |u×v| = √(|u|²|v|² − (u·v)²) (Lagrange) = 2·Area — NO transcendental, NO cos/sin/atan. Equivalent closed form cot γ = (b²+c²−a²)/(4·Area) from squared edge lengths. Ground-proof (this dispatch): coords (0,0,0),(4,1,2),(1,5,3) give u·v=15, |u×v|²=510, all three forms = 0.6642111642; 2D (0,0),(4,1),(1,5) → cot = 9/19 exact rational. So the cotangent stiffness Laplacian L_c is a coordinate-arithmetic build, cheaper than any angle-via-atan2 construction.

Op-count of the two normalizations (n nodes, |E| edges; both share the SAME L build):

normalization operator ALU cost
degree-norm D^{−1/2} L D^{−1/2}, D = weighted row-sum D: O(|E|) adds → diag → n recip-sqrt → O(|E|) scale. diagonal, no solve.
FEM mass-norm — LUMPED M^{−1} L_c, M = diag Voronoi areas (½·cross, rational) M: O(|E|) cross/adds → diag → n recip → O(|E|) scale. diagonal, no solve — SAME class.
FEM mass-norm — CONSISTENT M^{−1} L_c, M = sparse SPD non-diagonal needs a linear solve / Cholesky of M (or inner solve per matvec). strictly more expensive.

Verdict (op counts, not hand-waving): FEM is NOT cheaper. Lumped-mass FEM normalization is the same cost class as degree normalization (both diagonal reciprocal- scalings over the shared L); consistent-mass FEM is strictly more expensive (a solve). The cotangent weight build is cheaper than a trig-angle build, but that is weight construction, not normalization.

Both forms ARE useful — different operators for different substrate questions, ONE α-family. They are the endpoints of the Coifman–Lafon diffusion-maps α-family (Diffusion Maps, ACHA 21(1):5–30, 2006):

  • α=0 — connectivity-normalization (degree D): clustering / diffusion / cut — what fiedler_sparse / recursive_cut / the genome partitioner consume.
  • α=1 — metric-normalization (mass M): geometrically-faithful manifold spectrum (curvature, MFO §III.1/§V), recovered independent of sampling density — the true LB.
  • (α=½ = Fokker–Planck, the intermediate.)

So "different and useful" = YES, and it is ONE one-parameter (α) Class-L family, not two rival operators — the α exponent selects sampling-density-independence. This reframes the §1 duality: not discrete-vs-continuous rivals, but connectivity- normalization vs metric-normalization within one Class-L family, both already reachable from the shipped weighted dense_laplacian.

Fermata (conductor): whether © ships at all, and if so whether the note-level LB↔ graph-Laplacian duality lands in the srmech notebook, the MFO §III.1 cross-ref, or both. (Both notebook edits placed 2026-07-18 per conductor greenlight — srmech §3.5.1, MFO §III.1 cross-ref.)