The qualifier i < N applies to the whole display, leaving P_N,h ⊆ C_N,h unstated.
Section 8.5, proof of Lemma 8.6, first failure-indicator estimate · p. 70
For h ≥ Δ, the first failure indicator is bounded by cᵢ,ₕ − pᵢ,ₕ₋Δ = (pᵢ,ₕ − pᵢ,ₕ₋Δ) + (cᵢ,ₕ − pᵢ,ₕ).
This is the proof step that uses the missing endpoint inclusion; the report does not supply a counterexample to the final numerical conclusion.
Issue & evidence
The bad-density chains lemma restricts its entire nesting display to i<N, leaving P_{N,h}⊆C_{N,h} unstated. Its proof nevertheless uses this inclusion in the log-mismatch indicator bound: without it, c_{N,h}−p_{N,h−Δ} can be negative while the indicator equals one. This identifies a defect in the proof under the written hypotheses, not a counterexample to the final numerical conclusion. The subsequent PRE/CURRENT graph application supplies the missing inclusion.
Proposed repair by OSVerify
Separate the nesting hypothesis into P_{i,h}⊆C_{i,h} for every 1≤i≤N and C_{i,h}⊆P_{i+1,h} for i<N. Together with monotonicity in height, these justify the nonnegative log differences and the temporal nesting of the C sets used in the proof. PRE refinement of CURRENT supplies the first condition at every event, and the linked-event inclusion supplies the second. This endpoint correction requires no change to the chain-length bound used in the linear-side energy argument.
Related context
BackgroundSection 8.4, Equation (22) and the following PRE/CURRENT definitions
PRE refines CURRENT, supplying the within-event inclusion in the actual application.
Downstream useSection 8.6, proof of Proposition 8.9, linked-event chain and application of Lemma 8.6
The graph application supplies both required inclusions, including the endpoint condition.
The predictor identity omits a conditional-expectation term
Where the issue occurs
Section 5.2, proof of Lemma 5.3, predictor-moment identity and its increment · p. 32
If aₕ,ₚ is the predictor coefficient on a depth-h cell, orthogonality to Φₚ gives μₕ = εaₕ,ₚ. Consequently μₕ₊₁ − μₕ is √ε times the change of the p-th private coordinate of PₕFₐ. This identity continues to hold after arbitrary further column insertions.
These unnumbered identities omit the conditional-expectation term for general augmented inputs. Equation (7) is the estimate being proved, not the incorrect identity.
Issue & evidence
The predictor identity in the quoted passage from the proof of Lemma 5.3 omits a term for the general augmented inputs admitted in this section. Orthogonality gives μ_h=εa_{h,p}−√ε E_h[(F_a)_p], so its increment also includes −√ε(E_{h+1}−E_h)[(F_a)_p]. For example, F_a=Φ_p has zero residual and predictor coefficient one, contradicting μ_h=εa_{h,p}. The stated identity is valid for scalar embeddings, and this counterexample does not disprove estimate (7).
Proposed repair by OSVerify
Set f_p=(F_a)_p and use μ_h=√ε[(P_hF_a)_p−E_hf_p]. The projection increments and conditional-expectation increments each have a sum of squared L²(J) norms at most ||F_a||²₂. Consequently (∑_h||μ_{h+1}−μ_h||²₂)^(1/2)≤2√ε||F_a||₂ and ||μ_H||₂≤2√ε||F_a||₂. In the existing proof, these bounds retain the 2^(−K_0), K_0√ε, and √ε contributions with adjusted absolute constants. Thus this correction preserves the parameter dependence needed for (7) and its high-truncation application; for scalar embeddings the added term vanishes.
The identity is valid for scalar embeddings; this distinguishes the supported special case from the disputed general use.
Downstream useSection 5.2, remainder of the proof of Lemma 5.3, completing the proof of (7)
The added term must be carried through the smooth-bump, coarse-mean and curvature-difference estimates; the proposed repair preserves their parameter dependence.
Downstream useSection 5.4, proof of Proposition 5.6
Uses the delayed-predictor estimate in the high-truncation argument.
The first-failing-role convention uses the wrong order
Where the issue occurs
Section 9.1, paragraph assigning each failed trial to its first failing role · p. 78
For a fixed assignment the conditions in each role form a product mask on its modes: that role must fail at the trials assigned to it, and must succeed at the earlier roles’ required positions.
The last clause reverses the success condition for the stated first-failing-role convention.
Issue & evidence
The quoted passage in Section 9.1 contain a local convention mismatch. Assignment to the first failing role requires each role to succeed at trials assigned to later roles. Requiring success at trials assigned to earlier roles instead describes assignment to the last failing role. Both conventions partition failed trials, so this mismatch does not itself invalidate the subsequent summed estimates.
Proposed repair by OSVerify
Keep the first-failing-role convention and replace “the earlier roles' required positions” with “the trials assigned to later roles.” Explicitly, if a(t) is the assigned role, role j must fail when a(t)=j and succeed when a(t)>j, with no condition when a(t)<j. Retain success in every core role at the final height. These corrected masks still select whole minimal-component core subtotals by predicate coherence. For each fixed assignment, the assigned failure counts n_j remain unchanged, and the upper bounds in Section 9.2 still follow by discarding the additional success conditions.
