The weighted plank estimate has the wrong lemma number
Where the issue occurs
Section 1.1, paragraph introducing the three-dimensional weighted plank input · p. 4
The three-dimensional estimate used in our reduction is the weighted full-time plank bound of OpenAI [24, Lemma 2.3], recorded as Lemma 2.15.
This first citation should point to Lemma 3.3 in reference [24].
Section 2.5, proof of Lemma 2.15, first sentence · p. 18
This is the indexed weighted formulation of OpenAI [24, Lemma 2.3].
The same incorrect external lemma number is repeated here.
Section 5.1, proof of Lemma 5.1, weighted full-trace application · p. 72
Apply Lemma 2.15, namely the weighted full-trace form of OpenAI [24, Lemma 2.3].
This third citation also needs Lemma 3.3.
References, entry [24], final note · p. 175
Lemma 2.3.
The bibliography note repeats the same incorrect locator.
Issue & evidence
The cited September 23, 2026 preprint states the weighted plank estimate as Lemma 3.3 on page 20; item 2.3 is instead the critical-exponent definition. The incorrect locator appears in the four passages quoted above. The intended lemma's statement matches the weighted plank input.
Proposed repair by OSVerify
Replace Lemma~2.3 with Lemma~3.3 in the three citations to ThreeDimensional and its bibliography note. This corrects the reference without altering the stated weighted plank estimate.
Related context
BackgroundSection 2.5, proof of Lemma 2.15 (Weighted full-time plank estimate)
This is the paper’s own formulation of the imported estimate. The reported defect is the locator in reference [24], not the stated weighted estimate.
The Jacobian sentence reverses the coordinate maps
Where the issue occurs
Section 4.3, proof of Proposition 4.7, sentence immediately before Equation (4.20) · p. 56
The map to unit base coordinates u of Qs has Jacobian qs Js; denote this map by baseQ.
This sentence attaches the inverse map’s Jacobian to the parent-to-unit map. The displayed density estimate (4.20) already has the correct factor.
Issue & evidence
The Jacobian sentence preceding (4.20) reverses the coordinate maps. The parent-to-unit coordinate map base_Q has absolute Jacobian (q_sJ_s)⁻¹; q_sJ_s belongs to its inverse. The displayed pushforward density estimate uses the correct change-of-variables factor.
Proposed repair by OSVerify
Replace the Jacobian sentence with: “Writing u = base_Q(t,y) for unit base coordinates, the volume elements satisfy dt dy = q_sJ_s du.” Keep (4.20) and the following time-normalization formulas unchanged; their change-of-variables factors are already correct.
Related context
BackgroundSection 2.3, Definition 2.8 and Equation (2.7); Section 2.4, proof of Lemma 2.13
Defines the spatial chart determinant and the inverse affine change of coordinates used to check the direction of the Jacobian.
BackgroundSection 2.4, Equation (2.15)
Displays the corresponding density transformation, which agrees with the factor used later.
Downstream useSection 4.3, Equation (4.20) and following time normalization
These formulas remain unchanged under the proposed sentence correction.
