Parameter note: meeting the prediction-error and original-catalog budgets together

Corollary 5.2 on p. 31 of H uses B as the error exponent, whereas its application in Proposition 5.6 on p. 35 uses the words log B. Here we choose the error exponent from p. 31 directly and calculate the error and catalog size step by step. This is a verifiable quantitative completion; we do not assume that the authors intended an unstated change of parameters. It continues the prediction catalog in Section 7 of the main text and supplies the parameter calculations for Section 14.

1. The two parameter conventions in the source

H, p. 31 · H, p. 35

Corollary 5.2 on p. 31 of H says “For every B ≥ 1”, followed by the error bound

\mathrm e^{-B}\operatorname{poly}\bigl(\log(2D_{\mathrm{high}}),1+L\bigr) \|1_JF_a\|_2\|1_JF_b\|_2,

and the condition

\log D_{\mathrm{pred}} \le\operatorname{poly}\bigl(1+B,\log(2H_{\mathrm{low}})\bigr). \tag{P1}

The application of Corollary 5.2 on p. 35 of H reads:

“using a precision log B that is a sufficiently large fixed polynomial in k.”

Both links point to the original pages of the October 5, 2026 version. The difference might involve another name for the reciprocal error. The source does not explicitly describe such a change here, so we cannot decide the authors’ intent.

If B on p. 35 is read literally as the same variable as B on p. 31, and log B is chosen as a positive power of k, then B grows exponentially. The upper bound in (P1) no longer ensures that log D_pred is a fixed polynomial in k. This is a failure of the supplied estimate to imply the target budget, not a counterexample proving that the actual catalog is too large.

2. Choosing the error exponent B directly

H, p. 31 · H, p. 35

Suppose the old catalog already satisfies

\log D_{\mathrm{old}}\le C_o k^{d_o}, \quad H_{\mathrm{low}}\le D_{\mathrm{old}}, \quad k\ge1.

Write all old-label count and coefficient-sum costs as $D_{\mathrm{old}}^{p_o}$ , where $p_o$ is fixed before B is chosen. Bound the fixed-degree polynomial in Corollary 5.2 by an expression of the form

C(1+K_0)^{p_L}\bigl(1+\log(2D_{\mathrm{high}})\bigr)^{p_h}.

At this point, $K_0,D_{\mathrm{high}}$ have not yet been chosen. We will later show that their logarithmic budgets are fixed polynomials.

Choose

B=\left\lceil8k+2(p_o+1)\log(2D_{\mathrm{old}})\right\rceil. \tag{P2}

Then

\mathrm e^{-B}D_{\mathrm{old}}^{p_o}\le\mathrm e^{-8k}. \tag{P3}

Formula (P2) explicitly chooses the error exponent B from p. 31, not its logarithm.

Suppose the right side of (P1) is bounded by

C_p(1+B+\log(2H_{\mathrm{low}}))^{d_p}

and put $d_0=\max(1,d_o)$ and $d_p\ge1$ . Then

B\le Ck^{d_0},\quad \log D_{\mathrm{pred}}\le Ck^{d_0d_p}. \tag{P4}

Thus (P2) makes the prediction error sufficiently small while keeping the required catalog-size bound.

3. Explicit choices of the delay and private weight

H, p. 32 · H, p. 33 · H, p. 35

Suppose the known cost in Lemma 5.3 also includes $D_{\mathrm{pred}}^{p_d}$ , with $p_d$ already fixed. Put

Q_0=1+k+\log(2D_{\mathrm{old}})+\log(2D_{\mathrm{pred}}), \quad C_*=p_o+p_d+20.

Choose

K_0=\left\lceil\frac{C_*Q_0}{\log2}\right\rceil, \quad\epsilon=\exp(-2C_*Q_0). \tag{P5}

Then both

2^{-K_0}\le\mathrm e^{-C_*Q_0}, \quad\sqrt\epsilon=\mathrm e^{-C_*Q_0}.

hold. Apart from the fixed polynomial factors left for later, the delayed error is therefore bounded by

\begin{aligned} &D_{\mathrm{old}}^{p_o}D_{\mathrm{pred}}^{p_d} \bigl(2^{-K_0}+(1+K_0)\sqrt\epsilon\bigr)\\ &\hspace{1em}\le (2+K_0)\mathrm e^{-20Q_0} \end{aligned} \tag{P6}

Here K₀ and log(1/ε) both have fixed polynomial size and are determined before the two high-slot accuracies.

4. Absorbing the fixed polynomial factors from the later high catalog

H, p. 35 · H, p. 36

Let $c_{\mathrm{det}}$ be the fixed exponent in local detection. Once Q₀ is fixed, the negative logarithm of the first high-slot accuracy can be chosen with

p_a\le C Q_0^{r_a},

where $r_a$ is a fixed integer large enough to cover the old costs. Detection and the cap on coarse-cell requests imply

\log N_a+\log H_a\le C Q_0^{r_ac_{\mathrm{det}}}.

The physical supremum S_a of the first high-slot projection on a coarse cell satisfies

\log S_a \le\frac{K_0\log2}{2} +\frac12\log N_a^{\mathrm{all}} +\frac12\log(1/\epsilon) \le C Q_0^{\max(1,r_ac_{\mathrm{det}})}.

Next choose the second high-slot accuracy with

p_b\le C Q_0^{r_b\max(1,r_ac_{\mathrm{det}})},

where the fixed integer $r_b$ covers the first high-slot supremum and the old budget. There is then a fixed E such that

\log D_{\mathrm{high}}\le C Q_0^E, \quad E\ge\max\bigl(1,r_ac_{\mathrm{det}}, r_bc_{\mathrm{det}}\max(1,r_ac_{\mathrm{det}})\bigr). \tag{P7}

Only a fixed number of compositions occurs. The second high-slot accuracy does not change the first high-slot catalog.

By (P4), $Q_0\le Ck^{d_0d_p}$ . Hence all factors left over in (P3) and (P6), namely

C(1+K_0)^p\bigl(1+\log(2D_{\mathrm{high}})\bigr)^p

are at most $Ck^T$ , where T is also fixed after the two high-slot choices. Increase the starting k so that $Ck^T\le\mathrm e^k$ . The prediction error is then at most $\mathrm e^{-7k}|R|$ , and the delayed error at most $\mathrm e^{-19k}|R|$ . After absorbing the fixed number of replacement terms and the root-length packing constants, the total is still much smaller than $C\mathrm e^{-k}|R|$ .

Only the starting k is allowed to depend on the final fixed exponent T. We do not return to increase the chosen B or K₀, or to change the earlier catalogs. The order is therefore noncircular.

5. The n₀ budget under the same choices

H, p. 35 · H, p. 36 · H, p. 37 · Return to the original-label budget

The number of births on a path satisfies

N_{\mathrm{base}} \le N_{\mathrm{old}}+2N_{\mathrm{old}}^2D_{\mathrm{pred}}+N_a+N_b.

By (P4) and (P7),

\log(2+N_{\mathrm{base}})\le C Q_0^E \le Ck^{d_0d_pE}.

The logarithms of the inverse private weights, coefficient sums, and coarse-cell supremum bounds satisfy fixed polynomial bounds of the same kind. After increasing a fixed exponent C₀, we can take

1\le n_0\le D_{\mathrm{base}}, \quad \mathrm e^k\le D_{\mathrm{base}}\le\exp(k^{C_0}).

Thus the concrete choice in (P2) meets both the absolute-error budget and the original n₀ budget.

6. Scope of this quantitative completion

H, p. 31 · H, p. 35 · H, p. 36 · H, p. 37

Formulas (P2)–(P7) are a verifiable quantitative completion. The direct choice yields both the required absolute error and the logarithmic original-catalog budget, without adding a mathematical assumption or changing detection or the catalog construction. If B on the two pages is read literally as the same variable, the bounds in the original application do not establish the target budget. The choices above supply that step explicitly.

These calculations neither establish the sufficiency of the same-variable literal reading of p. 35 nor show that the original proposition is false. Identifying the logarithm of a reciprocal-error parameter on p. 35 with B on p. 31 is one possible renaming, but it is not stated there. Our conclusion is therefore limited to the quantitative choices given here. They supply the original budget for Section 16 of the main text. The actual later histories, the whole helper basis, and final global absorption still require separate treatment.