RemarkRemark: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Source corrections for three-dimensional EIT rates

Remark

Kaltenbacher's 2024 paper treats the three-dimensional unit ball with background conductivity $σ_{b g} \equiv 1$ , quadratic penalty $\overset{ˇ}{R} (σ) = \frac{1}{2} ∥ σ ∥_{H^{s_{1}} (Ω)}^{2}$ , $s_{1} \in (5 / 2, 7 / 2)$ , and truth $σ^{†} \in B_{2, \infty}^{s_{2}} (Ω)$ for $s_{2} \in (s_{1}, 2 s_{1} - 1)$ . Its Theorem 10 reports a logarithmic VSC exponent

$ν_{p a p e r} : = \frac{(s_{2} - s_{1}) (2 s_{1} - 3)}{s_{1} - 1},$

with $b = 3 / 4$ , and Corollary 11 reports a conductivity-norm exponent $ν_{p a p e r} / p = ν_{p a p e r} / 2$ for the joint, split, and frozen-Newton methods. The source fixes $p = 2$ and separately uses $α ≍ δ^{\tilde{p}}$ with $0 < \tilde{p} \leq 2$ ; $\tilde{p}$ is not the denominator in its displayed norm exponent. These printed formulas are source assertions; the locally proved EIT conclusions use the corrected VSC exponent and parameter qualifications below.

The cited Weidling Theorem 6.15 at $p = 2$ gives denominator $s_{2} - 1$ in the corresponding logarithmic exponent. In addition, the quadratic Bregman distance is one half the squared $H^{s_{1}}$ norm error, so a Bregman rate with logarithmic exponent $ν$ yields a norm exponent $ν / 2$ ; this quadratic norm conversion agrees with the divisor $p = 2$ in Corollary 11, while the VSC exponent and parameter qualifications still require the separate comparisons below. The sufficient parameter coupling in the abstract rate theorem requires $α H (c / α) ≍ δ^{2}$ for a fixed $c > 0$ . For a logarithmic index with $H (q) ≍ (\log q)^{- ν}$ , no pure-power choice $α ≍ δ^{\tilde{p}}$ with $0 < \tilde{p} \leq 2$ satisfies this two-sided coupling. Kaltenbacher's Remark 1 separately asserts that pure powers still give the same logarithmic rate; that separate assertion needs an independent derivation and is not ruled out solely by failure of the sufficient coupling. The local joint and frozen EIT theorems assume the Axiom of Choice, Countable Choice and Dependent Choice, use the corrected exponent $ν = (s_{2} - s_{1}) (2 s_{1} - 3) / (s_{2} - 1)$ , retain the entire stated Sobolev/Besov truth range, and prove the conductivity norm exponent $ν / 2$ . They give balanced parameters, every strict pure power $0 < \tilde{p} < 2$ , and the nominal endpoint $\tilde{p} = 2$ with an explicit stronger actual-noise condition. The universal abstract endpoint assertion under arbitrary deterministic noise bounded only by $δ$ is false, as shown by the exact quadratic counterexample in the endpoint lemma; that example does not claim to be an EIT-specific counterexample.

The local application uses mean-zero Neumann currents and boundary-mean-zero Neumann states. Its data space is the centered sequence $(N_{σ} - N_{1}) Y_{ℓ k}$ in $ℓ^{2} (L^{2} (S^{2}))$ , or equivalently the corresponding affine translate for absolute voltages. This centering is necessary: at the background conductivity $1$ , the uncentered voltage columns have norms $1 / ℓ$ with multiplicity $2 ℓ + 1$ , so their squared sum diverges. The centered Neumann-map differentiability, $H^{1}$ logarithmic stability, exact intrinsic Sobolev/Besov source inequality, augmented factorization and inverse, frozen Taylor estimate, and regularizer/penalty compatibility now have local proofs. The stability proof retains the Fourier modulation in the CGO remainder and integrates its negative-Sobolev norm before absorption; it does not use the invalid unmodulated pointwise inference in the printed Weidling proof.

Remark references

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, Example 2, equations (1.18)–(1.24), and Section 3, Theorem 10 and Corollary 11

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Chapter 6, Theorems 6.10–6.15, printed pp. 177–185

Published conductivity Dirichlet-to-Neumann interface ↗

Finite-dimensional elliptic state-map differentiability

The full collared EIT class satisfies the centered range and source interfaces

A logarithmic endpoint rate with explicit effective noise and tolerance control

Centered Neumann voltage sequences on the three-dimensional ball

The Axiom of Choice

The Axiom of Countable Choice ( ${A C}_{ω}$ )

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain

Facts & Assumptions

Given: The source assertions, comparison formulas, and quadratic regularizer are those stated in the Remark.

[F1]

[Kaltenbacher, Theorem 10]: the paper gives exponent $ν_{p a p e r} = (s_{2} - s_{1}) (2 s_{1} - 3) / (s_{1} - 1)$ and cites Weidling Theorem 6.15 with $p = 2$ .

[F2]

Weidling, Theorem 6.15: its proof sets $s_{1} = 3 / p + 1 + τ$ , $s_{2} = s$ , and $μ = 2 (s_{2} - s_{1}) / (s_{2} - 1)$ .

[F3]

Bregman distances for convex regularizers: $D_{R}^{ξ^{†}} (x, x^{†}) = R (x) - R (x^{†}) - ⟨ ξ^{†}, x - x^{†} ⟩$ .

[F4]

Joint regularization rates under differential range invariance: the generic joint rate theorem assumes $c_{-} δ^{p} \leq α ρ_{δ} \leq c_{+} δ^{p}$ and $H (1 / (2 C_{*} α)) \leq C_{H} ρ_{δ}$ , where $ρ_{δ} = H (1 / (C_{*} α))$ .

[F5]

Kaltenbacher, Theorem 10 and Corollary 11 (printed p. 24): the stated VSC uses $\overset{ˇ}{R} (σ) = \frac{1}{2} ∥ σ ∥_{H^{s_{1}}}^{2}$ . Corollary 11 fixes $p = 2$ and reports the norm exponent $ν_{p a p e r} / p = ν_{p a p e r} / 2$ , separately from the parameter choice $α ≍ δ^{\tilde{p}}$ , $0 < \tilde{p} \leq 2$ .

[F6]

Kaltenbacher, equation (1.32b): the logarithmic index is $ψ (t) = C (- \log (\min {t, t_{0}}))^{- ν_{p a p e r}}$ for constants $C, t_{0} > 0$ .

[A1]

The local EIT applications retain the Axiom of Choice, Countable Choice and Dependent Choice at their actual suppliers; the source algebra in steps 1.1–1.3 needs none of these additional assumptions.

[F7]

The full collared EIT class satisfies the centered range and source interfaces supplies the full EIT source/range/penalty interface with the corrected exponent $ν$ and exact quadratic Bregman distance. Its local stability and function-space source suppliers are fully proved.

[F8]

A logarithmic endpoint rate with explicit effective noise and tolerance control proves the explicit effective-noise endpoint and refutes the universal abstract deterministic-noise endpoint by an exact $ℓ^{2}$ minimizer.

[F9]

Centered Neumann voltage sequences on the three-dimensional ball proves the centered sequence map and computes the divergent uncentered background norm.

Verification

1.1F1F2algebra

Substituting $p = 2$ into [F2] gives $s_{1} = 5 / 2 + τ$ and $μ = 2 (s_{2} - s_{1}) / (s_{2} - 1)$ . The logarithmic VSC exponent contributed by the source Sobolev gap is therefore $(s_{2} - s_{1}) (2 s_{1} - 3) / (s_{2} - 1)$ . This differs from [F1]'s denominator $s_{1} - 1$ because the allowed range has $s_{2} > s_{1}$ ; the cited theorem does not directly yield the exponent printed in Kaltenbacher's Theorem 10.

1.2F3F5algebra

For $R (σ) = \frac{1}{2} ∥ σ ∥_{H^{s_{1}}}^{2}$ , the Hilbert-space subgradient is $ξ^{†} = σ^{†}$ under the $H^{s_{1}}$ inner product. Expanding the square gives $D_{R}^{ξ^{†}} (σ, σ^{†}) = \frac{1}{2} ∥ σ - σ^{†} ∥_{H^{s_{1}}}^{2}$ . Therefore $D_{R}^{ξ^{†}} = O ((\log (1 / δ))^{- ν})$ implies only $∥ σ - σ^{†} ∥_{H^{s_{1}}} = O ((\log (1 / δ))^{- ν / 2})$ by taking square roots; it does not imply exponent $ν$ .

1.3F4F5F6algebra

By [F6], $ψ (t) ≍ (\log (1 / t))^{- ν_{p a p e r}}$ for sufficiently small positive $t$ , and $ψ$ is bounded because of the truncation at $t_{0}$ . For $q$ large, if $0 < t < q^{- 1 / 2}$ then $ψ (t) \leq C 2^{ν_{p a p e r}} (\log q)^{- ν_{p a p e r}}$ ; if $t \geq q^{- 1 / 2}$ then $ψ (t) - q t \leq ψ (t) - \sqrt{q} < 0$ . Thus $H (q) \leq C^{'} (\log q)^{- ν_{p a p e r}}$ . Conversely, choosing $t_{q} = q^{- 1} (\log q)^{- ν_{p a p e r} - 1}$ gives $\log (1 / t_{q}) \sim \log q$ , $ψ (t_{q}) \geq c^{'} (\log q)^{- ν_{p a p e r}}$ , and $q t_{q} = (\log q)^{- ν_{p a p e r} - 1}$ , hence $H (q) \geq c^{''} (\log q)^{- ν_{p a p e r}}$ for large $q$ . Therefore $H (q) ≍ (\log q)^{- ν_{p a p e r}}$ . For $α ≍ δ^{\tilde{p}}$ , $0 < \tilde{p} \leq 2$ , this gives $α H (c / α) ≍ δ^{\tilde{p}} (\log (1 / δ))^{- ν_{p a p e r}}$ . If $\tilde{p} = 2$ , the ratio to $δ^{2}$ tends to zero; if $\tilde{p} < 2$ , that ratio tends to infinity. Hence no such pure-power choice satisfies the two-sided sufficient coupling in Theorem 3; this does not decide the separate assertion in Remark 1.

2.1givenA1F4F7F8F9step 1.1step 1.2step 1.3∎

The local interface [F7] verifies the full admitted EIT hypotheses, with the corrected VSC exponent from step 1.1 and the exact quadratic norm conversion from step 1.2. The two local rate theorems therefore give their explicit EIT conclusions from proved suppliers. Step 1.3 explains why strict pure powers require their own scalar argument instead of the sufficient two-sided balanced bracket. At the nominal endpoint, [F8] retains a valid rate under the stated actual-noise/tolerance condition and gives a concrete counterexample to the unrestricted abstract endpoint. Finally [F9] establishes the centered observation space and its necessary affine-data interpretation. These are proved source corrections preserving the full truth regularity classes, rather than unsupported imports of the printed formulas.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

40 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, Section 3, Theorem 10 and Corollary 11 (standard reference, not scraped)
Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Chapter 6, Theorems 6.10–6.15 (standard reference, not scraped)
Christian Clason, Regularization of Inverse Problems, Chapter 10, Theorem 10.7 (standard reference, not scraped)