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Standing. Proof by Plasma AI, included in the complete ratio-chain examination and distinct passing grade of L163 on 2026-10-15. This page has no separate claim tier. That examination checked the supplied proof; the original publications remained unread by the reviewer and grader. The range decomposition is Lemma 2.1 and Section 3 of Gerhard Larcher's [[../library/analysis/larcher_2014_star_discrepancy_sequences_unit_interval/_index|On the star discrepancy of sequences in the unit interval]], arXiv:1407.2094v1, which follows a method of Liardet. The quadratic potential and the four-block integer specialization below are this page's argument; no literature search was made for them. They replace Larcher's optimized minimization over admissible functions (his Lemmas 2.2--2.13), which has not been checked. The bound proved is weaker than Larcher's: its constant $1/(16\log4)\approx0.0451$ is below his $0.0646$ and below the $1/16$ of Theorem 4.1 of Korsky's 2026 preprint. Neither of those constants is verified here.

Statement

For a finite list $z_1,\ldots,z_L\in[0,1)$, repeated entries allowed, put

$$D_j(x)=\#\{i\le j:\ z_i<x\}-jx\qquad(1\le j\le L,\ 0\le x\le1),$$

and let

$$H_L(z_1,\ldots,z_L)=\max_{1\le j\le L}\ \sup_{0\le x\le1}|D_j(x)|$$

be its maximum prefix counting error, the quantity of the Lemma 4.2 reconstruction.

Proposition FP. Every list of length $L\ge1$ in $[0,1)$ satisfies

$$H_L\ \ge\ \frac{\lfloor\log_4L\rfloor}{16}\ \ge\ \frac{\log L}{16\log4}-\frac1{16}.$$

There is no exceptional length and no condition on the entries.

Proof

The range functional. For a list of length $N$ put

$$R_N(x)=\max_{1\le j\le N}D_j(x)-\min_{1\le j\le N}D_j(x),\qquad P_N=\int_0^1R_N(x)\,dx .$$

Since $R_N(x)\le2\max_j|D_j(x)|$, we have $P_N\le2H_N$.

Larcher's range lemma. For nonempty sets $A_0,A_2\subseteq A$ and any real function $\phi$ on the finite set $A$,

$$\max_A\phi-\min_A\phi\ \ge\ \tfrac12\Bigl(\max_{A_0}\phi-\min_{A_0}\phi\Bigr) +\tfrac12\Bigl(\max_{A_2}\phi-\min_{A_2}\phi\Bigr) +\tfrac12\Bigl|\max_{A_2}\phi-\max_{A_0}\phi\Bigr| +\tfrac12\Bigl|\min_{A_2}\phi-\min_{A_0}\phi\Bigr| .$$

Indeed $\max_A\phi$ is at least the larger of the two block maxima, which is their mean plus half their difference, and symmetrically $\min_A\phi$ is at most the mean of the two block minima minus half their difference.

Four blocks. Let $N=4m$ and take the index blocks $A_0=\{1,\ldots,m\}$, $A_1=\{m+1,\ldots,3m\}$ and $A_2=\{3m+1,\ldots,4m\}$. Put

$$f=\max_{j\in A_2}D_j-\max_{j\in A_0}D_j,\qquad g=\min_{j\in A_2}D_j-\min_{j\in A_0}D_j .$$

We show $\int_0^1|f|\ge\frac18$ and $\int_0^1|g|\ge\frac18$ for every list.

Boundary values. $D_j(0)=0$, and $D_j(1)=j-j=0$ because every entry is below $1$. Hence $f$ and $g$ vanish at $0$ and at $1$.

Between coordinates. On an open interval containing no entry, each $D_j$ is affine with slope $-j$. A maximum or a minimum of finitely many affine functions is continuous and piecewise affine, with slope equal to minus an index active there. So $f$ and $g$ are continuous and piecewise affine on such an interval, and off finitely many points $-f'$ and $-g'$ equal $j_2-j_0$ for some $j_2\in A_2$ and $j_0\in A_0$. Hence they lie in $[2m+1,4m-1]$, in particular in $(0,N)$.

Jumps. Each $D_j$ is left-continuous, so $f$ and $g$ are left-continuous. At a coordinate $v$, let $q_b(v)$ be the number of indices in $A_b$ whose entry equals $v$ ($b=0,1,2$). At $v$ every $D_j$ jumps up by $\#\{i\le j:\ z_i=v\}$, which is at least $q_0(v)+q_1(v)$ for $j\in A_2$ and at most $q_0(v)$ for $j\in A_0$. Taking maxima, or minima, of the shifted values gives

$$h(v):=f(v+)-f(v)\ \ge\ q_1(v),\qquad g(v+)-g(v)\ \ge\ q_1(v).$$

All jumps are therefore nonnegative, including those at coordinates with $q_1(v)=0$. A coordinate $v=0$ is included, with its jump from $f(0)=0$ to $f(0+)$; there is no coordinate at $1$.

The potential. Put $F(s)=s|s|/2$, a $C^1$ function with $F'(s)=|s|$. On each interval between consecutive coordinates, and between $0$ or $1$ and the nearest coordinate, $F\circ f$ is continuous and piecewise $C^1$, so the fundamental theorem of calculus applies on it with one-sided limits at its ends. Summing over these intervals, the values at $0$ and $1$ contribute $F(0)=0$, and only the jumps remain:

$$\int_0^1|f(x)|\,\bigl(-f'(x)\bigr)\,dx =\sum_v\Bigl(F\bigl(f(v+)\bigr)-F\bigl(f(v)\bigr)\Bigr) =\sum_v\int_{f(v)}^{f(v)+h(v)}|s|\,ds .$$

An integral of $|s|$ over an interval of length $h\ge0$ is at least $h^2/4$, its value on the interval centered at $0$. Since $h(v)\ge q_1(v)$ and $q_1(v)$ is a nonnegative integer, $h(v)^2\ge q_1(v)^2\ge q_1(v)$. Hence

$$\int_0^1|f|\,(-f')\ \ge\ \frac14\sum_vq_1(v)=\frac{|A_1|}4=\frac m2 .$$

As $0<-f'<4m$ almost everywhere, $\int_0^1|f|\ge(m/2)/(4m)=1/8$. The same argument, with the same jump bound, gives $\int_0^1|g|\ge1/8$.

One level of the recursion. Apply the range lemma at each $x$ with $A=\{1,\ldots,N\}$ and integrate over $x$:

$$P_N\ \ge\ \frac12\int_0^1\Bigl(\max_{A_0}D_j-\min_{A_0}D_j\Bigr) +\frac12\int_0^1\Bigl(\max_{A_2}D_j-\min_{A_2}D_j\Bigr) +\frac12\int_0^1|f|+\frac12\int_0^1|g| .$$

The first integral is $P_m$ of the early list $z_1,\ldots,z_m$. For $j\in A_2$, $D_j=D_{3m}+D'_{j-3m}$, where $D'_k$ is the discrepancy function of the late list $z_{3m+1},\ldots,z_{4m}$; the common term $D_{3m}$ cancels from the range, so the second integral is $P_m$ of the late list. Therefore

$$P_{4m}(z_1,\ldots,z_{4m})\ \ge\ \frac12P_m(z_1,\ldots,z_m) +\frac12P_m(z_{3m+1},\ldots,z_{4m})+\frac18 .$$

Induction. For $N=4^h$ every list satisfies $P_N\ge h/8$: the case $h=0$ is $P_1\ge0$, and the recursion with $m=4^{h-1}$ passes from $h-1$ to $h$ for the early and late sublists alike. Thus $H_{4^h}\ge P_{4^h}/2\ge h/16$.

Every length. Given $L\ge1$, let $h=\lfloor\log_4L\rfloor$. The prefix of length $4^h\le L$ is itself a list in $[0,1)$, and $H_L$ maximizes over more prefixes, so $H_L\ge H_{4^h}\ge h/16\ge(\log_4L-1)/16$. $\square$

Remarks

• Integer rounding. Block lengths are exact integers because only the lengths $4^h$ are split; an arbitrary length loses at most $1/16$ through the prefix restriction. Larcher's text writes block endpoints such as $a^{t-1}$ for real $a\in(3,4)$ and suppresses the rounding; nothing here needs those adjustments.
• Repeated entries. Repetitions enter only through the jump bound $h(v)\ge q_1(v)$, which counts entries with multiplicity. No perturbation to a list of distinct entries is needed, and no count of distinct jump locations is used.
• What is not claimed. The proposition is a weaker relative of Larcher's finite-list estimate with a separate proof. It does not check Larcher's optimized lemmas, his constant $0.0646$, or the bound $\frac1{16}\log L$ for $L\ge L_0$ that Korsky's Theorem 4.1 states.

Role in Problem 1221

In the all-sequence composition, Proposition FP replaces the imported Theorem 4.1 of the ratio argument. Its constant $c_0=1/(16\log4)$ admits any ratio coefficient $\alpha<c_0/6$, for example $\alpha=1/200$ in place of the preprint's $1/100$; the comparison is written out there.