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| Médium: | Recurso digital |
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Zenodo
2026
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| On-line přístup: | https://doi.org/10.5281/zenodo.19977963 |
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- <p>This upload contains Version v1.0 of the theorem note</p> <p>**Endpoint Inverse Stability and First-Cluster Gram Observability for Entropy–Spectral Ricci-Flow Limits.**</p> <p>The paper proves an endpoint inverse-stability theorem for tame three-dimensional spectral-geometric limits. In a fixed harmonic endpoint gauge, the endpoint metric is controlled, modulo diffeomorphism, scale, and first-cluster orthogonal gauge freedoms, by a finite low-mode observation package consisting of retained heat coefficients, first-cluster Riesz-projector data, and a Hilbert-Schmidt high-mode tail ledger.</p> <p>The central mechanism is</p> <p>$$<br>\text{retained heat data}<br>+<br>\text{first-cluster projector data}<br>+<br>\text{HS-tail control}<br>\Longrightarrow<br>\text{endpoint metric control}<br>\Longrightarrow<br>\text{constant-curvature endpoint}.<br>$$</p> <p>The main results establish:</p> <p>- first-cluster projector stability via mass-to-rank stabilization, Mosco/norm-resolvent convergence, and Riesz projector stability;<br>- endpoint harmonic-slice inverse observability, in which heat coefficients detect the scalar/trace block, first-cluster projector variation detects the retained anisotropic block, and the HS-tail ledger excludes high-frequency leakage;<br>- first-cluster Gram observability, giving calibration of the tensor</p> <p>$$<br>G_\ast=\sum_i \nabla Y_i^\ast\otimes\nabla Y_i^\ast<br>$$</p> <p>against the endpoint metric;<br>- a Bochner error ledger converting Gram calibration into traceless-Ricci decay;<br>- a harmonic-gauge Schauder closure leading to a constant-curvature endpoint.</p> <p>A retained first-cluster nondegeneracy chart is used to localize the finite-dimensional anisotropic block. The note includes a round-model verification showing that this chart is nonempty and natural in the model case. This is a local endpoint nondegeneracy hypothesis, not a global inverse-spectral rigidity assertion.</p> <p>The paper is extracted from a broader entropy-spectral Ricci-flow program. It does **not** assert global inverse spectral rigidity, does **not** prove global Ricci-flow existence from arbitrary initial data, and does **not** claim an unconditional proof of the Poincaré conjecture. Its role is to isolate and prove the endpoint inverse-stability component: once tame endpoint compactness, low spectral stabilization, scalar normalization, and tail control are available, the low-mode spectral package forces constant-curvature endpoint geometry.</p> <p>Version v1.0 incorporates the final endpoint-compactness wording correction, separating weak \(W^{2,p}\) convergence from strong \(C^{1,\alpha'}\) convergence and clarifying that preliminary endpoint compactness is distinct from the later smooth convergence and constant-curvature conclusion.</p>