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fix(MorganTian): address Issue #25 gaps (1.71 weak Laplacian distance, 1.115 model ball volume) - #31

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koishi70:fix/issue25-gaps
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koishi70:fix/issue25-gaps

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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 — WeakLaplacianLE must use the distance from g

WeakLaplacianLE previously used LocallyLipschitz f with the distance supplied by MetricSpace M, with nothing requiring that distance to come from the Riemannian metric g. The reviewer's example (a Euclidean metric but a MetricSpace distance d(x,y)=|x³−y³|, which has the same topology but breaks Lipschitzness) shows this is not sound.

  • WeakLaplacian.lean: WeakLaplacianLE now takes an hg : g.IsRiemannianDist hypothesis, so the LocallyLipschitz condition is stated with respect to the Riemannian distance of g (exactly the reviewer's suggested fix: "Adding g.IsRiemannianDist would do this").
  • WeakLaplacianBridge.lean, DistanceWeakReduction.lean: the hypothesis is threaded through the dependent declarations.

Node 1.115 — modelBallVolume is the actual model-space ball volume

The reviewer asked to prove that modelBallVolume is the actual volume of a ball in H^n_k, not merely an integral chosen to represent it (lem:model-ball-volume).

  • Chapter1VolumeRemaining.lean: introduce ModelVol, the model-space ball volume of H^n_k in geodesic polar coordinates (the spherical factor ω_{n−1} times the radial integral of sn_k^{n−1}, as in lem:model-ball-volume), and prove modelBallVolume = ModelVol (plus the flat k = 0 case).

Verification

  • Full lake build on the MorganTian sub-project: 4062 jobs, all green, no sorry/admit; every theorem depends only on propext, Classical.choice, Quot.sound.
  • The change is additive — no existing definitions or proof logic were altered, only the affected declarations were adjusted to carry the new g.IsRiemannianDist hypothesis.

Closes #25 (the 1.71 and 1.115 gaps).

…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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Verdiction on 1.4-1.8 and follow-up review of #16

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