Experimental anchoring

Conditionally on the history and $t$, $D_i\in[-1/q,1/q]$. The log moment-generating function of $D_i-S_i$ has value and first derivative zero at the origin; its second derivative is the variance under an exponentially tilted law, at most $(2/q)^2/4=1/q^2$. Integrating twice gives the bounded-range inequality

Iteration and Markov's inequality give $\Prb_t(\sum_i(D_i-S_i)\ge nr)\le \inf_{\theta>0}e^{-\theta nr+n\theta^2/(2q^2)} =e^{-nq^2r^2/2}$. Substitution proves the assertion. Independence of device outputs between rounds was not assumed.

Each joint outcome probability is no larger than an Alice marginal, so Lemma 2.3 bounds it by $2^{-F_{\rm NS}(S_i)}$. By (5),

The function is convex and nondecreasing on the physical domain $[-4,4]$; its extension above that domain is not used in Jensen's inequality. Outside an event of probability $\alpha$, Lemma 3.1 makes the last expression at least the right side of (6). If the empirical lower endpoint exceeds $4$, that event is necessarily in the exceptional set. The quantum proof uses the imported one-round bound of [22], convexity of $F_Q$ on $[-2\sqrt2,2\sqrt2]$, and the same concentration inequality. The fixed-threshold consequence follows by monotonicity. Postselection changes both the conditional law and the exceptional probability; Lemma 3.9 supplies that conversion explicitly.

Average (8) over the earlier outcomes, keeping $T$ and earlier settings fixed. Each next setting pair is still conditionally uniform. The entropy chain rule gives $H(W\mid T)=\sum_i H(W_i\mid T,W_{<i})=2n$. Deterministic data processing through $G_{\rm set}$ gives the remaining inequalities.

Enumerate the sixteen local assignments $(a_0,a_1,b_0,b_1)\in\{0,1\}^4$ and their means $\frac14\sum_{xy}b_{xy}^{a_xb_y}$. The maximum is one, attained by all nondetections. The other eight vertices of the binary no-signalling polytope are the PR boxes [22, App. A.3]; their means are

Their maximum occurs at $(r,s,t)=(0,0,0)$ and equals $4040170031/4000000000$; each of the other seven means is below one. These finite sums can be checked directly with the displayed decimal rationals. Increasing every coefficient by less than $10^{-10}$ changes the expectation under any normalized distribution by less than $10^{-10}$. Monotonicity of the maximum then proves (12). The largest printed local mean other than the all-nondetection strategy is $1-5.25\times10^{-10}$. Raising every uncertain coefficient by $10^{-10}$ still leaves each such mean below one. The all-nondetection coefficients are fixed to exactly one by the published training constraint. Thus the local bound for the original function also follows from this rounding envelope; no unchecked numerical optimization tolerance is needed.

For $pv>1$, the entropy-production expression for $\delta$ is increasing in $m$ wherever its bracket is positive. The actual theorem therefore also holds with the larger $\delta_+$. Its admissible range $pv\le(1+3m_{\rm act}/2)^n$ holds already with the lower endpoint in (12). Numerically stable evaluation is

The intervals in (14) follow by bounding logarithms with $\ln x=2\sum_{j\ge0}((x-1)/(x+1))^{2j+1}/(2j+1)$ after binary range reduction, and bounding the positive exponential and $-\ln(1-u)$ series by their geometric tail bounds. Twenty terms in the range-reduced logarithms and four terms in each of the latter small-argument series suffice for the displayed intervals. Finally $\ell+4\log_2\ell=1064$. The conservative value agrees with the fragment's rounded $1344.9$, but the rounded printed $m$ is not treated as an exact extremal value.

Write $\delta_e=\TV(P_{K\mid e},U_\ell)$. Equality of the $E$ marginals gives $\E\delta_e\le\epsilon$. Always $0\le\delta_e\le1-1/M$, since a point mass is farthest from the uniform law on $M$ points. Apply Audenaert's bound to the diagonal $M$-dimensional density matrices:

The loss function is concave and nondecreasing on $[0,1-1/M]$, since its derivative is $\log_2((M-1)(1-u)/u)\ge0$ there. Jensen and monotonicity give the first inequality of (17); $\log_2(M-1)\le\ell$ gives the second. No uncontrolled $\delta_e>1/2$ exception or auxiliary majorant is needed. Finally, under the ideal law any guess measurable in $E$ succeeds with probability $1/M$. Total variation changes that success probability by at most $\epsilon$, including for the optimal real-law guess.

Apply Lemma 3.6 to the law conditioned on $J=1$, with $E=(T,W,R)$. The numerical Shannon loss is less than $1.06531\times10^{-9}$ bits. The output satisfies $K=\mathrm{Ext}(Z_{\rm run}(W),R)$, so conditioning and deterministic data processing give the entropy chain in (18). They do not require the generally false identity $H(Z_{\rm run}\mid T)=H(Z_{\rm run}\mid T,W,R,J=1)$. Any guess of $Z_{\rm run}$ from $(T,W,R)$ induces a guess of $K$ correct whenever the response guess is correct. Its ideal-law success probability is $2^{-1024}$, so (16) gives (19). Dropping conditioning cannot improve guessing. All distributions in this argument are conditional passing-ensemble laws, not entropies assigned to one realized string.

This is Lemma 3.7 with the published parameters. Keeping its probability space, conditioning and side-information class is essential to the statement.

Let $B$ indicate the good event $P(C\mid E)\le\delta$, and restrict the conditional passing law to $B=1$:

This deletes mass at most $p/\pi$. For every $(z,e)$ that survives, determinism and the good-event definition give $P(Z=z\mid E=e)\le P(C=f(z,e)\mid E=e)\le\delta$. Taking $\mu=P_E$ therefore gives

This proves (23). The deleted submeasure has guessing probability at most its mass, proving (21).

Let $g=\Prb(B=1\mid J=1)\ge1-p/\pi$. The normalized good law has guessing probability at most $\delta/(\pi g)$ and hence Shannon entropy at least $[\log_2(\pi g/\delta)]_+$. Conditioning on $B$ can only reduce average Shannon entropy, and the bad-law entropy is nonnegative. Therefore

The right side is nondecreasing in $g\ge0$, giving (22). No independence between $Z$ and $E$ was used. In particular the lemma is stronger, for postselected posterior-response entropy, than a transfer argument requiring an independent prior response law.

Average (15) over $t$ and apply Lemma 3.9 to $C=Z_{\rm run}(W)$. Since $\pi\ge\kappa$, deletion mass is at most $p/\kappa=\kappa$, and the smooth bound is at least $b_++\log_2\kappa$. The Shannon bound is at least

Here monotonicity in $\pi$ follows directly because both nonnegative factors in (22) increase with $\pi$. The unsmoothed bound is

Numerically $b_++\log_2\kappa$ lies between $1304.97687$ and $1304.97689$; the Shannon correction to it is less than $1.3\times10^{-9}$. These conservative intervals follow from Lemma 3.5; they do not identify $1344.9$ with an unconditional or unsmoothed entropy. Dropping public settings can only reduce guessing and increase Shannon entropy.

At the realized setting word $w$,

The event in (27) is therefore contained in that of (15). This is an exceptional-event bound on a prior atom, not itself a conditional entropy after passing. Corollary 3.10 instead proves a posterior entropy statement without this independence premise.

Source completeness makes $C$ a function of $(T,W)$, so the realized conditional likelihood $P_t(C\mid W)$ equals one almost surely. In the entropy-production theorem choose an error parameter $p'>v^{-1}$ arbitrarily close to $v^{-1}$. Then $p'v>1$, the corresponding $\delta(p')$ is less than one, and the admissible-range condition holds for all sufficiently close choices. Its event inequality consequently bounds the probability of passing by $p'$, for almost every $t$ and hence unconditionally. Let $p'\downarrow v^{-1}$. This is a frequentist bound under the joint model hypotheses, not a posterior probability that a source ontology is false. The physical applicability of B-admissibility is not proved by the test.