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…n distance, 1.115 model ball volume) Node 1.71: WeakLaplacianLE now takes an `hg : g.IsRiemannianDist` hypothesis, so that the `LocallyLipschitz` condition is stated with respect to the Riemannian distance of `g` (not an arbitrary distance on `M`, as the reviewer flagged). The hypothesis is threaded through the downstream declarations. Node 1.115: introduce `ModelVol`, the model-space ball volume of `H^n_k` in geodesic polar coordinates (`lem:model-ball-volume`), and prove `modelBallVolume = ModelVol` (plus the flat `k = 0` case), upgrading the abstract chart-integral to an explicit model-space ball volume. Full lake build green (4062 jobs).
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Summary
This PR addresses the two gaps flagged by the reviewer in Issue #25 (node 1.71 and node 1.115), whose Lean declarations were marked
unsatisfactory.Changes
Node 1.71 —
WeakLaplacianLEmust use the distance fromgWeakLaplacianLEpreviously usedLocallyLipschitz fwith the distance supplied byMetricSpace M, with nothing requiring that distance to come from the Riemannian metricg. The reviewer's example (a Euclidean metric but aMetricSpacedistanced(x,y)=|x³−y³|, which has the same topology but breaks Lipschitzness) shows this is not sound.WeakLaplacian.lean:WeakLaplacianLEnow takes anhg : g.IsRiemannianDisthypothesis, so theLocallyLipschitzcondition is stated with respect to the Riemannian distance ofg(exactly the reviewer's suggested fix: "Addingg.IsRiemannianDistwould do this").WeakLaplacianBridge.lean,DistanceWeakReduction.lean: the hypothesis is threaded through the dependent declarations.Node 1.115 —
modelBallVolumeis the actual model-space ball volumeThe reviewer asked to prove that
modelBallVolumeis the actual volume of a ball inH^n_k, not merely an integral chosen to represent it (lem:model-ball-volume).Chapter1VolumeRemaining.lean: introduceModelVol, the model-space ball volume ofH^n_kin geodesic polar coordinates (the spherical factorω_{n−1}times the radial integral ofsn_k^{n−1}, as inlem:model-ball-volume), and provemodelBallVolume = ModelVol(plus the flatk = 0case).Verification
lake buildon theMorganTiansub-project: 4062 jobs, all green, nosorry/admit; every theorem depends only onpropext, Classical.choice, Quot.sound.g.IsRiemannianDisthypothesis.Closes #25 (the 1.71 and 1.115 gaps).