Energy counterexample

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Statement and scope

Proved. There are finite systems of distinct singleton moduli, with the original activation rule, and cutoffs $Q\to\infty$ such that

$$\mathcal E:=\sum_q e_q\log q\longrightarrow2\log2, \qquad H_B(Q)-\beta H_Q\sim(\log Q)^{1/21}\longrightarrow\infty. \tag{1}$$

Here $B$ is the delayed excluded union, $\beta$ is the full periodic union measure, and $e_q$ are the full first-kill measures in increasing modulus order. All the moduli are at most $Q$.

Consequently there is no absolute $C$ for which

$$H_B(X)\le\beta H_{\lfloor X\rfloor} +C\left(1+\sum_{q\le X}e_q\log q\right) \tag{2}$$

holds for every finite distinct-modulus singleton system. This accounts for the entire preparatory sieve, not just the late first-kill cells. The construction also refutes a uniform additive $O(\log\log X)$ error and the proposed rank-sensitive error $C_r=\exp(o(r))$.

This is not an infinite counterexample to Problem 25. Its normalized harmonic excess tends to zero. A uniform additive $o(\log X)$ estimate and the constant first-kill entropy bound remain open.

The rank-filtered refinement below increases the attainable lower-bound exponent to every $\gamma<(7\,2^{1/7}-3\,3^{1/3}-3)/4=0.100469\ldots$, still with energy tending to $2\log2$. It does not contradict the weaker square-root energy target.

A sparse divisor-residue channel

Let $L=2^k$ and $Q>L$ be integers. For each odd residue $r\in\{1,\ldots,L-1\}$, put

$$D_r=\{q\in\mathbb N:L<q\le Q,\ q\equiv r\pmod L\}, \qquad b_r=\mu\left(\bigcup_{q\in D_r}q\widehat{\mathbb Z}\right).$$

We claim that some such $r$ satisfies

$$b_r\ll L^{-1/2}(\log(2Q))^{\sqrt2-1}. \tag{3}$$

The implied constant is absolute.

For every odd prime $p\le Q$, independently choose a nonnegative integer $V_p$ with

$$\Pr(V_p=v)=(1-1/p)p^{-v}, \qquad M=\prod_{\substack{p\le Q\\p\ne2}}p^{V_p}.$$

These are precisely the finite-dimensional Haar valuation laws. Every $q\in D_r$ is odd and has all prime factors at most $Q$, so

$$b_r=\Pr(\exists q\in D_r:q\mid M).$$

For each realization of $M$, at most $\min(\varphi(L),\tau(M))$ unit residues modulo $L$ occur among its divisors in $(L,Q]$. Here $\tau$ is the number of positive divisors. Consequently

$$\frac1{\varphi(L)}\sum_{r\text{ odd}\bmod L}b_r \le\mathbb E\min\left(1,\frac{\tau(M)}{\varphi(L)}\right) \le\varphi(L)^{-1/2}\mathbb E\sqrt{\tau(M)}. \tag{4}$$

The divisor moment factors over the independent valuations. For $p\ge3$,

$$(1-1/p)\sum_{v\ge0}\sqrt{v+1}\,p^{-v} =1+\frac{\sqrt2-1}{p}+O(p^{-2}).$$

The remainder is bounded uniformly in $p$: the terms $v\ge2$ have sum $O(p^{-2})$. Comparison with the local factors of $\prod_{p\le Q}(1-1/p)^{-(\sqrt2-1)}$ therefore gives

$$\mathbb E\sqrt{\tau(M)} \ll\prod_{p\le Q}(1-1/p)^{-(\sqrt2-1)} \ll(\log(2Q))^{\sqrt2-1}. \tag{5}$$

The last bound is the elementary Euler-product estimate proved in coprime entropy. Equations (4)--(5), together with $\varphi(L)=L/2$, prove (3). No distribution theorem for primes in arithmetic progressions is used.

From now on choose

$$Q=\left\lfloor\exp(L^{21/20})\right\rfloor$$

and choose the least odd $r$ minimizing $b_r$. This is a finite, deterministic choice: each $b_r$ is the exact proportion of covered residues in a finite common period. The minimum satisfies (3). Since $\sqrt2-1<5/12$, we obtain

$$b_r\ll L^{-1/2+(21/20)(5/12)}=L^{-1/16}. \tag{6}$$

The finite congruence system

First install the $k$ classes

$$C_j=[2^{j-1}-1]_{2^j},\qquad1\le j\le k. \tag{7}$$

These are disjoint full cylinders: membership is equivalent to $v_2(n+1)=j-1$. Their full survivor set is the single fiber $[-1]_L$, of measure $1/L$.

Set

$$\alpha=L-1-r\in[0,L-2],$$

and then install one class

$$[\alpha]_q\qquad(q\in D_r). \tag{8}$$

All new moduli are odd and greater than $L$, whereas the old moduli are the distinct powers of two at most $L$. Thus all moduli are globally distinct and all old classes precede the new ones in increasing modulus order. The residues in both (7) and (8) are normalized; in particular $0\le\alpha<q$ for every new class.

The full union in (8) is the common translate by $\alpha$ of the multiples union generated by $D_r$, hence has measure $b_r$. Because all its moduli are odd, this event is independent of the 2-adic fiber $[-1]_L$. The final full survivor measure is therefore exactly

$$\delta=1-\beta=\frac{1-b_r}{L}. \tag{9}$$

Exact delayed survivors and activation

The positive roots omitted by the delayed versions of the old classes are

$$2^h-1,\qquad1\le h\le k-1.$$

They are not covered by another old class, since the full old cylinders are disjoint. Every new modulus exceeds $L$, so no new class is active at these roots. The remaining old delayed survivors are the positive integers in $[-1]_L$.

The first positive point of this fiber is $L-1$. For every later one, write

$$n=Lt+L-1,\qquad t\ge1.$$

If $n\le Q$, then

$$q=n-\alpha=Lt+r$$

satisfies $L<q\le n\le Q$ and $q\equiv r\pmod L$, so $q\in D_r$. Moreover $n=q+\alpha$ is precisely the first active endpoint of its new class. This checks activation without any change of cutoff or delay convention.

Thus the delayed survivor set $A$ satisfies the exact identity

$$A\cap[1,Q]=\{2^h-1:1\le h\le k\}, \qquad H_A(Q)=\sum_{h=1}^k\frac1{2^h-1}<2. \tag{10}$$

Bounded total first-kill energy

Since the old full cylinders are disjoint, their first-kill measures are $2^{-j}$. Their energy is

$$\mathcal E_{\mathrm{old}} =\log2\sum_{j=1}^k\frac{j}{2^j} =\left(2-\frac{k+2}{L}\right)\log2. \tag{11}$$

The sum of all the new first-kill measures is exactly the increment in full union measure, namely $b_r/L$. All new moduli are at most $Q$, so (6) gives

$$0\le\mathcal E_{\mathrm{new}} \le\frac{b_r}{L}\log Q \ll L^{-1/16+1/20}=L^{-1/80}. \tag{12}$$

Together (11)--(12) prove $\mathcal E\to2\log2$.

On the other hand, (9)--(10) give

$$\begin{aligned} H_B(Q)-\beta H_Q &=\delta H_Q-H_A(Q)\\ &\ge(1-b_r)L^{1/20}-2, \end{aligned} \tag{13}$$

because $H_Q\ge\log(Q+1)>L^{21/20}$. In fact, using $H_Q=\log Q+O(1)$ and $b_r=o(1)$, this excess is

$$(1+o(1))L^{1/20} =(1+o(1))(\log Q)^{1/21}. \tag{14}$$

This proves (1), disproves (2), and disproves a uniform additive $O(\log\log X)$ error.

Consequence for the rank-sensitive target

The later prime-only amplification supersedes the rank-constant obstruction in this section: no finite rank-only additive bound exists, already at rank one. The construction here still has the distinct additional feature of bounded first-kill energy, and the calculations below remain valid.

The proved rank truncation in rank and moment obstructions states that, with

$$R=\left\lceil8\log\log(3Q)\right\rceil,$$

the union $B_{>R}$ of classes whose moduli have $\Omega(q)>R$ has $H_{B_{>R}}(Q)=O(1)$, uniformly over finite singleton families. Explicitly its bound is

$$H_{B_{>R}}(Q) \ll(3/2)^{-R}(\log(3Q))^{5/2}=O(1).$$

Delete those high-rank classes from the construction and write $B_{\le R}$ and $\beta_{\le R}$ for the remaining delayed union and full union measure. Removing classes can only decrease Haar measure, and loses at most $O(1)$ harmonic mass. Consequently

$$H_{B_{\le R}}(Q)-\beta_{\le R}H_Q \ge H_B(Q)-\beta H_Q-O(1) \gg L^{1/20}. \tag{15}$$

Since $R=(42/5)\log L+O(1)$, any universal bounds

$$H_B(X)\le\beta H_{\lfloor X\rfloor}+C_R \quad\text{for all systems with }\Omega(q)\le R$$

must therefore satisfy, along this sequence,

$$C_R\ge\exp\bigl((1/168+o(1))R\bigr). \tag{16}$$

In particular $C_R=\exp(o(R))$ is impossible. This does not contradict the bounded-rank natural-density theorem, which does not assert such a uniform additive error.

Why this does not resolve Problem 25

The normalized excess is

$$\frac{H_B(Q)-\beta H_Q}{H_Q} =\frac{1-b_r}{L}-\frac{H_A(Q)}{H_Q} \sim\frac1L\longrightarrow0.$$

Thus these finite systems do not produce a fixed positive logarithmic excess. Also, combining gadgets while retaining their preparatory classes for unbounded $k$ retains every class in (7). Those classes already have full Haar union measure one; their delayed survivor set is contained in $\{2^h-1:h\ge1\}$ and has natural density zero. Any additional deletions preserve that zero density. This direct combination therefore cannot give the required infinite counterexample.

The spectral conductor-energy inequality remains valid. What fails is its proposed uniform conversion to harmonic discrepancy. Neither a uniform additive $o(\log X)$ bound nor the constant first-kill entropy target is refuted by this construction.

Two linear entropy charges also fail

The same example refutes linear charging of the unnormalized first-kill relative entropy either by the reciprocal sum or by the first-kill energy. This does not refute the constant normalized entropy target.

Let $I$ index the new odd-modulus classes, and put $c_i=H_{E_i}(Q)/H_Q$. Their exact endpoint deletion gives

$$m:=\sum_{i\in I}H_{E_i}(Q) =\frac{\log Q-\log L+O(1)}L\sim\frac{H_Q}{L}, \qquad \sum_{i\in I}e_i=\frac{b_r}{L}.$$

The log-sum inequality, together with $b_r\ll L^{-1/16}$, yields

$$\begin{aligned} H_Q\sum_i c_i\log_+(c_i/e_i) &\ge m\log\frac{mL}{H_Qb_r}\\ &\gg L^{1/20}\log L. \end{aligned} \tag{17}$$

On the other hand, summing along the single residue channel gives

$$\sum_i\frac1{q_i}\asymp 1+\frac{\log Q}{L} \asymp L^{1/20},$$

while (11)--(12) show $\sum_i e_i\log q_i\to2\log2$. Thus neither of these costs can bound (17) up to an absolute linear factor, even after adding an absolute constant. The lower bound after normalization by $H_Q$ is only $\gg(\log L)/L$, which tends to zero; it remains compatible with a universal bound for $\sum_i c_i\log_+(c_i/e_i)$.

Optimizing the divisor moment

The exponent $1/21$ can be slightly improved without changing the construction. For every fixed $0<\theta<1$, replace the square root in (4) by the $\theta$th power. The local factors are

$$(1-1/p)\sum_{v\ge0}(v+1)^\theta p^{-v} =1+(2^\theta-1)/p+O_\theta(p^{-2}),$$

so the same proof gives

$$b_r\ll_\theta L^{-\theta}(\log(2Q))^{2^\theta-1}.$$

Choose $Q=\lfloor\exp(L^a)\rfloor$ with $a>1$. The new energy is then at most a constant times

$$L^{a2^\theta-\theta-1}.$$

It tends to zero if $a<(1+\theta)/2^\theta$, while the harmonic excess is asymptotic to $L^{a-1}$. Elementary differentiation gives

$$\max_{0<\theta<1}\frac{1+\theta}{2^\theta} =\frac{2}{e\log2},\qquad \theta_{\rm max}=\frac1{\log2}-1.$$

Thus for every fixed

$$0<\gamma<1-\frac{e\log2}{2}$$

there are systems with $\mathcal E\to2\log2$ and excess $\asymp(\log Q)^\gamma$. All activation and distinctness checks are unchanged. This remains sublogarithmic and does not give an infinite counterexample. The prepared-block entropy bound also shows that this mechanism's normalized entropy is small, even after optimizing its parameters.

Rank-filtered residue channels

Theorem. For every

$$0<\gamma<\gamma_*:= \frac{7\,2^{1/7}-3\,3^{1/3}-3}{4} =0.100469471\ldots, \tag{18}$$

there are finite distinct-singleton systems with cutoffs $Q\to\infty$ and original activation such that

$$\mathcal E\longrightarrow2\log2,\qquad H_B(Q)-\beta H_Q\ge(\log Q)^\gamma. \tag{19}$$

Only a lower bound for the excess is claimed here, unlike the asymptotic equivalence in (1).

Write $L=2^k$, $H=\log Q$, and fix $c>1$, to be chosen below. Put

$$R=\lfloor c\log H\rfloor,\qquad I(c)=c\log c-c+1.$$

For each odd residue $r\pmod L$, define

$$D_r=\{q:L<q\le Q-L,\ q\equiv r\pmod L,\ \Omega(q)\ge R\}, \quad b_r=\mu(M(D_r)),\quad m_r=\sum_{q\in D_r}\frac1q. \tag{20}$$

We choose $Q$ so that $\log L/\log H$ tends to a fixed positive constant. All estimates below are in this regime.

The rank-sensitive divisor moment

Let $V_p$, for odd primes $p\le Q$, have the Haar geometric laws, and set $M=\prod_{3\le p\le Q}p^{V_p}$. The number $N_R(M)$ of divisors of $M$ with $\Omega\ge R$ satisfies

$$N_R(M)\le2^{-R}\sum_{d\mid M}2^{\Omega(d)} =2^{-R}\prod_{3\le p\le Q}(2^{V_p+1}-1).$$

At most $\min(\varphi(L),N_R(M))$ odd channels in (20) can be represented among its divisors. Therefore

$$\frac1{\varphi(L)}\sum_r b_r \le\varphi(L)^{-1/3}\mathbb E N_R(M)^{1/3} \ll L^{-1/3}H^A,\qquad A=3^{1/3}-1-\frac{c\log2}{3}. \tag{21}$$

For the last estimate, independence and the uniform local expansion give

$$(1-1/p)\sum_{v\ge0}(2^{v+1}-1)^{1/3}p^{-v} =1+\frac{3^{1/3}-1}{p}+O(p^{-2}).$$

The remainder is uniform for $p\ge3$, since $2^{1/3}<3$. The elementary Euler-product bound then gives the power $H^{3^{1/3}-1}$; the factor $2^{-R/3}$ supplies the remaining term in $A$.

Enough channels retain reciprocal mass

Set $Y=(Q-L)^{1/R}$ and

$$S=\sum_{R^2<p\le Y}\frac1p=\log H-O(\log\log H).$$

The last equality follows from $\sum_{p\le x}1/p=\log\log x+O(1)$, itself a consequence of the two-sided elementary Euler-product estimates used earlier. Products of $R$ distinct primes from this interval are odd, at most $Q-L$, and greater than $L$: their minimum logarithm is greater than $2R\log R$, whereas $\log L=O(\log H)$.

Sampling $R$ primes independently with weights $1/(pS)$, the collision probability is at most

$${R\choose2}S^{-2}\sum_{p>R^2}p^{-2}=O(S^{-2})=o(1).$$

Unique factorization therefore gives

$$\sum_r m_r\ge(1-o(1))\frac{S^R}{R!} =H^{1-I(c)+o(1)}. \tag{22}$$

No prime distribution in residue classes is used here.

For definiteness let $s=S^R/(2R!)$, so (22) is at least $s$ for all sufficiently large $Q$. Define $\mathcal G=\{r:m_r\ge s/L\}$. The other channels contribute at most $s/2$, because there are $L/2$ odd channels. Uniformly $m_r\le CH/L$, by summing the progression $r+tL$, $t\ge1$. Hence

$$|\mathcal G|\gg Ls/H=L H^{-I(c)+o(1)}.$$

Choose the least-footprint channel in $\mathcal G$. Equation (21) then proves the simultaneous estimates

$$m_r\ge \frac{H^{1-I(c)+o(1)}}L,\qquad b_r\ll L^{-1/3}H^{A+I(c)+o(1)}. \tag{23}$$

The extra $H^{I(c)+o(1)}$ in the footprint is the cost of restricting to large-mass channels; it is not discarded.

Installing the filtered channel

Use the binary preparation (7), leaving the full fiber $[-1]_L$, and put $\alpha=L-1-r\in[0,L-2]$. Append $[\alpha]_q$ for $q\in D_r$. All new moduli are odd and exceed the old moduli. They are distinct, and every residue is normalized. Each distinct endpoint

$$n=q+\alpha\le Q$$

lies in the old survivor fiber and is active at its own modulus. Its reciprocal mass, summed over $D_r$, is

$$\sum_{q\in D_r}\frac1{q+\alpha}=m_r-O(1/L);$$

indeed $\sum_{q>L,\ q\equiv r\ (L)}\alpha/q^2=O(1/L)$. Other appended congruences may also cover these points, which only helps the union count.

The new union has density $b_r$ on the odd-prime coordinates and is independent of the binary fiber. Thus its incremental measure is $b_r/L$, and its entire first-kill energy is at most

$$\mathcal E_{\rm new}\le b_rH/L. \tag{24}$$

The old energy tends to $2\log2$. Its delayed survivor harmonic mass is at most $H_Q/L+3$: the progression $[-1]_L$ costs at most $H_Q/L+1$, and the inactive roots $2^j-1$ cost less than two. Its harmonic discrepancy is consequently at least $-3$. It follows that the final discrepancy is at least

$$-3+m_r-O(1/L)-\frac{b_rH_Q}{L}. \tag{25}$$

Choose a fixed $\eta>0$ and $Q=\lfloor\exp(L^{1/\eta})\rfloor$. Then $L=H^{\eta+o(1)}$. By (23)--(24), the new energy tends to zero if

$$\eta>\eta_{\rm crit}(c):=\frac34(1+A+I(c)) =\frac{3\,3^{1/3}-c\log2+3I(c)}4. \tag{26}$$

Under this condition, (23) and (25) give an excess lower bound $H^{1-I(c)-\eta+o(1)}-O(1)$. The supremum of its exponent allowed by (26) is

$$1-I(c)-\eta_{\rm crit}(c) =\frac{4-3\,3^{1/3}+c\log2-7I(c)}4.$$

This expression is maximized at $c=2^{1/7}$, since its derivative has the sign of $\log2-7\log c$. Substitution gives (18). For every smaller $\gamma$, choose $\eta$ with strict margins in (26) and $1-I(c)-\eta>\gamma$. This proves (19).

The examples do not refute the square-root energy estimate. For these parameters $\eta>1/2$, while the trivial upper bound for positive excess is

$$(H_B(Q)-\beta H_Q)_+\le(1-\beta)H_Q\le H_Q/L =H^{1-\eta+o(1)}=o(\sqrt H).$$

Nor do these finite systems produce positive normalized logarithmic excess. The general conditioned-tail and square-root estimates remain unresolved.

One fixed family can have finite energy and unbounded additive excess

Proposition. There is one infinite distinct-singleton system whose delayed excluded union $B$, full Haar union measure $\beta$, and increasing-order first-kill masses $e_q$ satisfy

$$d(B)=\beta<\frac14,\qquad \sum_q e_q\log q<\infty,\qquad \limsup_{X\to\infty}(H_B(X)-\beta H_X)=\infty. \tag{27}$$

Its survivor set has natural density $1-\beta>3/4$, so this is not a counterexample to Problem 25. It rules out bounded additive amortization even for one fixed family of finite total energy; it does not rule out the required $o(\log X)$ bound.

The finite input is the proved construction (1): for some absolute $C_*$, every $T>0$ admits a finite singleton family $\mathcal F$, all of whose moduli are at most a cutoff $X$, with

$$\mathcal E(\mathcal F)\le C_*,\qquad D_{\mathcal F}(X):=H_{B_{\mathcal F}}(X) -\beta_{\mathcal F}H_X>T. \tag{28}$$

Only this established finite input is needed below.

The exact affine lift

Let an already installed finite system have full union $U_{\rm old}$ of measure less than one and common period $P$. Choose a surviving residue $b\in[0,P)$, and let $L$ be a multiple of $P$. Replace every class $a\pmod q$ in a fresh finite family by

$$La+b\pmod{Lq}. \tag{29}$$

This residue is normalized, and the lifted full union lies in $[b]_L$, hence avoids $U_{\rm old}$. Its actual delayed union is exactly $LB_{\mathcal F}+b$: since $0\le b<L$, the inequality $Ln+b\ge Lq$ is equivalent to $n\ge q$.

The lifted footprint is $\beta_{\mathcal F}/L$. If all lifted moduli follow the old ones, its first-kill masses are $e_q/L$, so its energy contribution is

$$\frac{\mathcal E(\mathcal F)+\beta_{\mathcal F}\log L}{L} \le\frac{C_*+\log L}{L}. \tag{30}$$

At $Y=LX+b$, with $\theta=b/L\in[0,1)$,

$$0\le \frac1L H_{B_{\mathcal F}}(X) -H_{LB_{\mathcal F}+b}(Y) \le\frac1L\sum_{n\ge1} \left(\frac1n-\frac1{n+1}\right)=\frac1L.$$

Also $H_Y-H_X\le\log(Y/X)\le\log L+1$. Consequently

$$H_{LB_{\mathcal F}+b}(Y)-\frac{\beta_{\mathcal F}}L H_Y \ge\frac{D_{\mathcal F}(X)-\log L-2}{L}. \tag{31}$$

Induction and protection of earlier cutoffs

Suppose the first $j-1$ blocks and increasing cutoffs $Y_i$ have been chosen. Their finite delayed union $B_{<j}$, of footprint $\beta_{<j}$, has a bounded harmonic discrepancy: for a finite constant $C_{j-1}$,

$$H_{B_{<j}}(N)-\beta_{<j}H_N\ge-C_{j-1}\qquad(N\ge1). \tag{32}$$

This follows from periodic partial summation and the finitely many activation exceptions. Use $C_0=0$ for the empty prefix.

Choose $L_j$, a multiple of the old period, sufficiently large that

$$\begin{gathered} L_j>Y_{j-1}\quad(j>1),\qquad L_j^{-1}\le2^{-j-3},\qquad (C_*+\log L_j)/L_j\le2^{-j},\\ L_j^{-1}\le2^{-j}/H_{Y_i}\qquad(1\le i<j). \end{gathered} \tag{33}$$

The old footprint is below $1/4$, so an old surviving residue $b_j$ exists. Apply (28) with

$$D_{\mathcal F_j}(X_j)> L_j(j+C_{j-1}+3)+\log L_j+2,$$

lift by (29), and put $Y_j=L_jX_j+b_j$. Each new modulus is at least $L_j>Y_{j-1}$, whereas all old moduli are at most $Y_{j-1}$. Thus the global moduli are distinct and strictly ordered block by block. Their full block unions are disjoint. Equations (31)--(32) give

$$H_{B_{\le j}}(Y_j)-\beta_{\le j}H_{Y_j}>j+3. \tag{34}$$

In the resulting infinite family,

$$\beta=\sum_j\frac{\beta_{\mathcal F_j}}{L_j}<\frac14, \qquad \sum_qe_q\log q =\sum_j\frac{\mathcal E(\mathcal F_j) +\beta_{\mathcal F_j}\log L_j}{L_j} \le\sum_j2^{-j}<\infty.$$

Future moduli exceed $Y_j$, so none changes deletion below that cutoff. Their future Haar cost is controlled as well:

$$(\beta-\beta_{\le j})H_{Y_j} \le H_{Y_j}\sum_{k>j}L_k^{-1} \le\sum_{k>j}2^{-k}=2^{-j}.$$

Equation (34) therefore gives $H_B(Y_j)-\beta H_{Y_j}>j+3-2^{-j}$.

Finally, the whole delayed block $j$ lies in $\{L_jn+b_j:n\ge1\}$, with at most $N/L_j$ points below $N$. Hence

$$|(B\setminus B_{\le J})\cap[1,N]| \le N\sum_{j>J}L_j^{-1}.$$

Finite-prefix periodicity and this summable tail prove $d(B)=\beta$. This completes (27), including original activation and natural-density existence. Preparations can thus be repeated with summable density and energy cost while producing indefinitely increasing additive spikes. Only a normalized, sublogarithmic estimate could settle the original problem; the present construction neither supplies nor refutes it.