# Overview and publication identity JEREMY RODGERS Sealed or Leaky The Source Tetralemma, Bell-Certified Hidden Information, and Finite-Resource Witnesses Conditional mathematical results and published-data translation Independent Researcher Website: [everythingequation.com](https://everythingequation.com) DOI: [10.5281/zenodo.23077474](https://doi.org/10.5281/zenodo.23077474) Version 3.1 — preprint edition Mathematical revision: 21 September 2026 Publication preparation: 1 October 2026 **Abstract.** *Status: Conditional mathematical results; published-data translation.* For a specified classical initial readout $T$, single outcomes, source-level outcome determinism, setting independence and no-signalling conditional on $T$ imply that a CHSH violation excludes source completeness. The finite source response variable obeys the sharp bounds $H_{\min}(Z\mid T)\ge-\log_2(3/2-S/4)$ and $H(Z\mid T)\ge(S-2)/2$ for $2 ## 1 Question, main result, and scope *Status: Framework and scope.* Shadow Theory distinguishes a source from a nominated readout, or Tier 1. The question here is not whether that interpretation can be assumed, but which independently stated premises force a specified readout to omit source distinctions. The source–readout papers distinguish an exact projected description from reconstruction of all physical source distinctions after declared redundancy has been removed [[14](/sealed-or-leaky/references#bib-R1), [15](/sealed-or-leaky/references#bib-R2), [16](/sealed-or-leaky/references#bib-R3), [17](/sealed-or-leaky/references#bib-R4), [18](/sealed-or-leaky/references#bib-R5), [10](/sealed-or-leaky/references#bib-R6)]. The measurement constitutions supply separate model-relative examples [[11](/sealed-or-leaky/references#bib-RM), [20](/sealed-or-leaky/references#bib-Pilot), [21](/sealed-or-leaky/references#bib-Massive)]; none of their dynamics is used to prove the Bell results. Theorem [5.1](/sealed-or-leaky/operational-equivalence-and-its-restricted-alternatives#op:surrogate) gives the negative boundary: if an alternative class contains the complete-law operational surrogate of each source model, internal experiments cannot discriminate those models from their surrogates. Restricted premises can give positive results without breaching that boundary. #### Main implication. *Status: Proved conditional on the premises of Section [2.1](/sealed-or-leaky/the-source-tetralemma#tet:setting).* For a classical initial readout $T=\tau(\lambda)$, retain precisely (SO), (D), (MI) and (NS) below. A CHSH value $S>2$ implies that $\lambda$ is not a measurable function of $T$. For its finite response table $Z$, $$ H_{\min}(Z\mid T)\ge{f_{\mathrm{NS}}}(S),\qquad H(Z\mid T)\ge g_{\rm NS}(S):=(S-2)/2, \qquad {f_{\mathrm{NS}}}(S):=-\log_2(3/2-S/4). $$ Both bounds are sharp in the no-signalling model class. The quantum guessing bound supplies the additional bound $H(Z\mid T)\ge{f_{\mathrm{Q}}}(S)$ under the separately stated quantum-realizability hypothesis. The maximum of these Shannon bounds may be used; no optimal quantum Shannon tradeoff is claimed. #### What the tetralemma does not say. *Status: Logical correction.* The premise (D) as written already requires definite $\pm1$ outcomes. Therefore the four premises are not logically independent. Theorem [2.7](/sealed-or-leaky/the-source-tetralemma#tet:tetralemma) proves exclusion and the separate necessity of (D), (MI) and (NS) within the single-outcome class. The branching model is retained explicitly as an escape from that class; complete-state unitary determinism is not substituted for (D). The four routes are neither mutually exclusive nor an exhaustive classification of physical theories. #### Published-data anchoring. *Status: Conditional translation; not a reanalysis of trial records.* Section [3](/sealed-or-leaky/experimental-anchoring#exp:section) separates illustrative evaluations at the published CHSH central values [[23](/sealed-or-leaky/references#bib-Hensen15), [26](/sealed-or-leaky/references#bib-Rosenfeld17), [27](/sealed-or-leaky/references#bib-Storz23)] from a genuine finite-sample translation of the entropy-production and soundness theorems in [[28](/sealed-or-leaky/references#bib-Bierhorst18)]. The latter is restricted to classical pre-existing side information obeying that paper's sequential conditions. Passing once does not establish the required lower bound on the probability of passing. In particular, the $1024$ extracted bits do not constitute $1024$ bits of unsmoothed min-entropy of $\lambda$. The rounding-controlled source-response bound of Corollary [3.10](/sealed-or-leaky/experimental-anchoring#exp:rawcor) is stronger than the extracted-string Shannon bound and uses only the published aggregate certificate. #### Attribution and contribution. *Status: Literature context and results proved here.* Deterministic hidden-variable signalling and Bell-certified unpredictability have substantial antecedents [[13](/sealed-or-leaky/references#bib-V02), [5](/sealed-or-leaky/references#bib-CR11), [22](/sealed-or-leaky/references#bib-Pironio10)]. The contributions here are the precise source–readout formulation, the corrected necessity statement, the sharp Shannon refinement, and a postselection-aware finite-source translation with explicit adversary restrictions. No priority is claimed for the elementary probability inequalities or for the imported randomness theorems. The earlier trine, POVM, accessible-conservation, pilot and reweighting results are retained in their declared domains, not offered as additional empirical evidence. ### 1.1 Conventions and levels of conclusion For probability laws, ${\mathrm{TV}}(P,Q)=\sup_A|P(A)-Q(A)|\in[0,1]$; the variation norm of their signed difference is $2{\mathrm{TV}}(P,Q)$. Trace distance is $D(\rho,\sigma)=\frac12\|\rho-\sigma\|_1$. All conditional laws requiring disintegration are on standard Borel spaces, and identities involving a conditioning value hold almost everywhere. Entropies are in bits; $h_2$ denotes binary Shannon entropy. For finite $Z$ and classical $E$, $$ p_{\rm guess}(Z\mid E)={\mathbb E}\max_z{\mathbb P}(Z=z\mid E),\qquad H_{\min}(Z\mid E)=-\log_2 p_{\rm guess}(Z\mid E). $$ This is average conditional min-entropy, not the minimum over side-information values. The distinct deletion-smoothing convention used below is defined in Lemma [3.9](/sealed-or-leaky/experimental-anchoring#exp:tailtransfer). A signed incidence matrix has column $e_q-e_r$ for an edge $r\to q$. Symbols are local to their sections: CHSH $S$ is a scalar, whereas the source set $S$ in Section [5](/sealed-or-leaky/operational-equivalence-and-its-restricted-alternatives#op:section) is a set. | Level | Conclusion and boundary | | --- | --- | | Mathematical implication | A proved implication of specified definitions and premises. | | Constitutive prediction | A consequence of a particular preparation, dynamics and access catalogue. | | Operational incompleteness | An admitted record distinction omitted by a specified readout. | | Published-data translation | An imported finite-sample certificate with its original statistical and adversary assumptions; not independent experimental verification. | | Fundamental ontology | Not uniquely identified in a class closed under complete-law operational surrogates. | Every theorem, proposition, lemma and corollary has a status tag. A conditional theorem is proved as an implication; its physical premises are not thereby established. Explicit limitations are tagged **[Not established]** or **[Scope]**. No unproved statement is promoted by being included in an interpretation. --- # Section 2: The source tetralemma ## 2 The source tetralemma ### 2.1 Setting and premises A Bell experiment realized in a source model is specified as follows. Let $(\Lambda,\mathcal A,\nu)$ be a standard Borel probability space of complete initial source states of everything relevant to the run except the devices that generate the settings. Alice's and Bob's settings $x,y\in\{0,1\}$ are drawn with probabilities $q(x,y)>0$. A *Tier-1 readout* is a measurable map $\tau:\Lambda\to\mathcal T$ into a standard Borel space. Its value $T=\tau(\lambda)$ is the nominated initial classical information. The Tier-1 readout is *source-complete* if $\lambda=G(T)$ almost surely for some measurable $G$. *Status: Stated premises; unchanged from version 3.* The following premises are not derived here. **(SO) Single outcomes.** For each setting pair, Alice's and Bob's outcomes are $\pm1$-valued random variables $A,B$ on the run. Each run has exactly one outcome per party, jointly distributed with $\lambda$. **(D) Determinism.** There are measurable $F_A,F_B:\Lambda\times\{0,1\}^2\to\{\pm1\}$ with $A=F_A(\lambda,x,y)$ and $B=F_B(\lambda,x,y)$ almost surely. Dependence on the distant setting is allowed. **(MI) Measurement independence.** The settings $(x,y)$ are independent of $\lambda$, and hence of $T$. **(NS) Tier-1 no-signalling.** For almost every $t$, the conditional box $P_t(a,b\mid x,y)={\mathbb P}(A=a,B=b\mid x,y,T=t)$ is no-signalling: $\sum_bP_t(a,b\mid x,y)$ does not depend on $y$, and $\sum_aP_t(a,b\mid x,y)$ does not depend on $x$. Realism here means existence of the stipulated source model, not an extra convex-linearity axiom. Write $E_{xy}={\mathbb E}[AB\mid x,y]$ and $S=E_{00}+E_{01}+E_{10}-E_{11}$. Under (D), define the finite response variable $$ Z=\bigl(F_A(\lambda,x,y),F_B(\lambda,x,y)\bigr)_{x,y\in\{0,1\}} \in\{\pm1\}^8. $$ It settles every outcome at fixed settings, but need not contain every source distinction. **Remark 2.1 (The access quantifier).** *Status: Scope.* (NS) is conditional no-signalling for the particular nominated $T$, not merely no-signalling of the unconditioned laboratory marginals. The latter does not imply the former. Calling $T$ all information any observer could ever acquire requires an additional physical access claim, absent here. A conclusion for every admissible $T$ means a separate implication for each $T$ satisfying the premises, not proof that every conceivable observer or global source variable is admissible. An optional hypothesis (QT) requires every $P_t$ to be quantum-realizable with a tensor-product structure; it does not replace classical $T$ by unrestricted quantum side information. **Lemma 2.2 (Outcome determinism already supplies single outcomes).** *Status: Proved.* Within the stated probability model, (D) supplies the single-valued outcomes required by (SO). Consequently there is no model of the other three displayed premises that violates only (SO) while retaining (D) in its stated meaning. **Proof.** For every setting pair, the two measurable functions in (D) take exactly one value in $\{\pm1\}$ at each source state; their pushforward gives the joint outcome law with $\lambda$. Almost-sure qualifications can be imposed simultaneously for the four setting pairs by taking a finite union of null sets. A branching final vector is not either of these scalar outcome functions. Replacing (D) by deterministic evolution of that vector would change the premise. □ ### 2.2 Guessing and hidden information **Lemma 2.3 (No-signalling guessing bound).** *Status: Proved.* For a no-signalling $\pm1$ box with CHSH value $S$, every Alice or Bob marginal has guessing probability at most $3/2-S/4$. For $2\le S\le4$ this is a sharp bound. **Proof.** Put $\alpha_x={\mathbb E}[A\mid x]$ and $\beta_y={\mathbb E}[B\mid y]$. Positivity gives $2P(A\ne B\mid xy)\ge|\alpha_x-\beta_y|$ and $2P(A=B\mid xy)\ge|\alpha_x+\beta_y|$. Let $L=P(A\ne B\mid00)+P(A\ne B\mid01)+P(A\ne B\mid10)+P(A=B\mid11)$, so $S=4-2L$. Writing $u=\alpha_0-\beta_0$, $v=\alpha_0-\beta_1$, $w=\alpha_1-\beta_0$, $z=\alpha_1+\beta_1$ gives $2L\ge|u|+|v|+|w|+|z|$. The combinations $u+v-w+z=2\alpha_0$, $-u+v+w+z=2\alpha_1$, $-u+v-w+z=2\beta_0$ and $u-v-w+z=2\beta_1$ bound all four biases by $L$. Thus the marginal guessing probability is at most $(1+L)/2=3/2-S/4$. Mixing the all-plus local box with a PR box attains this value. This is also the bound in [[22](/sealed-or-leaky/references#bib-Pironio10), Eq. (14)]. □ **Theorem 2.4 (Hidden-information theorem).** *Status: Proved conditional on (SO), (D), (MI), (NS).* Suppose $S>2$. For each $x$ and $y$, $$ \begin{aligned}p_{\rm guess}(A\mid X=x,T)&\le\tfrac32-\tfrac S4<1,\\ H_{\min}(Z\mid T)&\ge{f_{\mathrm{NS}}}(S)>0,\\ H(Z\mid T)&\ge{f_{\mathrm{NS}}}(S)>0. \end{aligned} $$ Equation (1, 2, 3). The same marginal bound holds for Bob. The readout is not source-complete: on a set of readout values of positive probability, its conditional source law assigns positive probability to different response values. Under (QT), for $2 **Proof.** (MI) makes the law of $T$ independent of the settings, so $P=\int P_t\,d{\mathbb P}(t)$ and $S=\int S_t\,d{\mathbb P}(t)$. By (NS) the Alice marginal given $x,t$ is independent of $y$. Lemma [2.3](/sealed-or-leaky/the-source-tetralemma#tet:guess) and averaging prove [(1)](/sealed-or-leaky/the-source-tetralemma#tet:pg). For fixed $x,y$, the random variable $F_A(\lambda,x,y)$ is a coordinate of $Z$. Hence, pointwise in $t$, the largest atom of $Z\mid t$ is no larger than the largest atom of that coordinate. (MI) identifies this coordinate's conditional law with the observed law at $x,y,t$. Averaging gives [(2)](/sealed-or-leaky/the-source-tetralemma#tet:min); Shannon entropy dominates average conditional min-entropy, by its pointwise version and Jensen's inequality. If $\lambda=G(T)$ almost surely, then $Z$ is a function of $T$ and its conditional entropy is zero, a contradiction. Regular conditional source laws are concentrated on $\tau^{-1}(t)$ almost everywhere. A nondegenerate finite conditional response distribution therefore supplies different responses in those fibres, as in Theorem [6.1](/sealed-or-leaky/deterministic-completions-require-unresolved-information#op:information). For (QT), the single-box bound $G_Q$ is imported from [[22](/sealed-or-leaky/references#bib-Pironio10), Eq. (6)]. Extend it by $1$ for $s\le2$ on the quantum CHSH domain. The extension is concave: at $s=2$ its right derivative is $-1/4$, no larger than the left derivative $0$. Jensen gives ${\mathbb E} G_Q(S_t)\le G_Q(S)$, and the preceding coordinate argument applies unchanged. □ **Lemma 2.5 (Sharp Shannon refinement and simultaneous sharpness).** *Status: Proved conditional on (SO), (D), (MI), (NS).* For $2\le S\le4$, $$ H(Z\mid T)\ge g_{\rm NS}(S):=(S-2)/2\ge{f_{\mathrm{NS}}}(S). $$ Equation (4). Both the first bound and $H_{\min}(Z\mid T)\ge{f_{\mathrm{NS}}}(S)$ are simultaneously sharp over this no-signalling class with classical $T$. **Proof.** Concavity of binary entropy on each half of $[0,1]$ gives $h_2(p)\ge2\min(p,1-p)$. Lemma [2.3](/sealed-or-leaky/the-source-tetralemma#tet:guess) implies, at each $t$, $$ H(A\mid x,T=t)\ge (S_t-2)_+/2. $$ Since $u\mapsto(u-2)_+$ is convex, averaging and the deterministic-coordinate inequality give $H(Z\mid T)\ge(S-2)/2$. The comparison with ${f_{\mathrm{NS}}}$ follows because a convex function lies below its endpoint chord: ${f_{\mathrm{NS}}}(2)=0$, ${f_{\mathrm{NS}}}(4)=1$. For sharpness let $r=(S-2)/2$, let $C_0$ be a Bernoulli flag of mean $r$, and let $R_0$ be zero when $C_0=0$ and a fair bit when $C_0=1$. Take $\lambda=(C_0,R_0)$, $T=C_0$, independent settings, and all-plus responses in the first case. In the second case use $A=(-1)^{R_0}$ and $B=(-1)^{R_0+xy}$. The conditional boxes are local deterministic and PR, respectively, hence both no-signalling. Their average CHSH value is $2+2r$. The response tables in the PR branch are distinct and equiprobable, so $H(Z\mid T)=r$ and $p_{\rm guess}(Z\mid T)=1-r/2=3/2-S/4$. This is an extremal mathematical construction, not a physical PR-box claim or quantum-class sharpness assertion. □ **Remark 2.6 (Reading the entropies).** *Status: Scope.* The bounds concern source information not determined by the nominated initial classical readout, given the displayed access premise. They are not observer-independent statements of permanent inaccessibility. They refer to the finite response $Z$, not differential entropy of a continuous source state. Under (QT), $H(Z\mid T)\ge\max\{g_{\rm NS}(S),{f_{\mathrm{Q}}}(S)\}$ follows, but the two available bounds do not establish an optimal quantum Shannon curve. A postselected finite run needs the separate analysis of Section [3](/sealed-or-leaky/experimental-anchoring#exp:section). ### 2.3 The four explicit escape models and their exact logical status **Theorem 2.7 (Source tetralemma: exclusion and corrected necessity).** *Status: Proved.* A source-complete model with $S>2$ cannot jointly satisfy (SO), (D), (MI) and (NS). Within the single-outcome model class, each of (D), (MI) and (NS) is separately necessary for this exclusion: models (C2)–(C4) below satisfy the other premises, have identity readout, and reproduce $$ P_Q(a,b\mid x,y)=\tfrac14(1+ab\,c_{xy}),\qquad c_{00}=c_{01}=c_{10}=-c_{11}=1/\sqrt2. $$ Model (C1) is a source-complete branching realization of the same branch-weight table, outside the single-outcome formalism. It is not a counterexample to Lemma [2.2](/sealed-or-leaky/the-source-tetralemma#tet:SOredundant). Thus four descriptions of escape remain, but the claim of four logically independent premises is false. **Proof.** The exclusion is Theorem [2.4](/sealed-or-leaky/the-source-tetralemma#tet:hidden). The following four models all take the readout to be the complete specified initial state. **(C1) Branching: outside (SO) and the stated (D).** Let the source be a unit ray of the two-qubit-plus-apparatus Hilbert space, prepared in $\Phi^+\otimes{\lvert\mathrm{ready}\rangle}$. Independent settings select unitary measurement interactions with final vector $$ \sum_{a,b}(P_a^x\otimes P_b^y)\Phi^+\otimes{\lverta,b\rangle}. $$ Use $A_0^{\rm op}=\sigma_z$, $A_1^{\rm op}=\sigma_x$, $B_0^{\rm op}=(\sigma_z+\sigma_x)/\sqrt2$, $B_1^{\rm op}=(\sigma_z-\sigma_x)/\sqrt2$. Orthogonal pointer branches have squared norms $P_Q(a,b\mid x,y)$, a no-signalling table. The complete vector is a deterministic function of source and settings, but no single actual pair $A,B$ has been selected. This is complete-state determinism, *not* (D). The branch-weight reading is the relative-state alternative [[30](/sealed-or-leaky/references#bib-Everett57), [31](/sealed-or-leaky/references#bib-Deutsch99), [32](/sealed-or-leaky/references#bib-Wallace12)]; derivation of subjective probabilities from branch weights is not a premise or result here. **(C2) Fundamental chance: violates only (D).** Take $\Lambda$ to be pure rays with $\nu=\delta_{\Phi^+}$, $\tau=\mathrm{id}$, and independent settings. Sample one pair from $P_Q$ using fundamental chance not encoded in the initial source state. Then (SO), (MI) and (NS) hold, while (D) fails. Completeness here is completeness of the initial physical state, not a hidden initial encoding of subsequent chance. Pure-ray chance ontologies supply the relevant contrast [[2](/sealed-or-leaky/references#bib-BB95), [33](/sealed-or-leaky/references#bib-GRW86)]. **(C3) Measurement dependence: violates only (MI).** Let $\lambda=(a_0,a_1,b_0,b_1)\in\{\pm1\}^4$, $A=a_x$, $B=b_y$, and take any positive setting law $q(x,y)$ with $$ \nu(\lambda\mid x,y)=\tfrac14 P_Q(a_x,b_y\mid x,y). $$ The two unused entries are uniform. Summing over them reproduces $P_Q$ and normalizes each conditional source law. All its entries are positive, so each setting pair remains in the support at every $\lambda$. Conditional on $T=\lambda$, the responses are local deterministic and satisfy (NS). They satisfy (SO) and (D), but the source law depends on the settings. This is a measurement-dependent construction of the Brans type [[35](/sealed-or-leaky/references#bib-Brans88)]. **(C4) Accessible nonlocality: violates only (NS).** Take $\lambda=(u,v)$ uniform on $[0,1]^2$, independently of the settings, with $T=\lambda$, and define $$ A=\mathrm{sgn}(1/2-u),\qquad B=A\,\mathrm{sgn}((1+c_{xy})/2-v), $$ assigning $\mathrm{sgn}(0)=+1$ on the irrelevant threshold sets. Its averaged joint probabilities are $P_Q$. It satisfies (SO), (D) and (MI). For $y=1$, on the positive-measure interval between $(1-1/\sqrt2)/2$ and $(1+1/\sqrt2)/2$, Bob's definite output changes with $x$. Its box conditional on the readable $\lambda$ therefore violates (NS), although its unconditioned marginals do not signal. No claim that such complete access is physically available is required for the countermodel. □ **Proposition 2.8 (The three-branch form fails).** *Status: Proved.* The assertion that every source-complete realization of quantum Bell statistics is branching, source-indeterministic, or an operational surrogate violating convex-linear realism is false. Neither convex-linearity nor its failure is needed in the exclusion theorem. **Proof.** (C3) and (C4) are single-outcome, source-deterministic and source-complete. Neither is forced into the proposed surrogate branch. With a single preparation, convex-linearity imposes no nontrivial requirement; random mixtures of admitted preparation laws can moreover be represented by their mixtures. A density-matrix-only stochastic surrogate is already in the chance branch for the nondegenerate Tsirelson marginals. None of the arguments of Theorem [2.4](/sealed-or-leaky/the-source-tetralemma#tet:hidden) uses convex-linearity. These observations refute the proposed universal three-way claim; they do not classify every possible surrogate. □ **Proposition 2.9 (Where convex-linearity bites: the chance branch).** *Status: Proved conditional on a convex-linear ontological model reproducing the trine table and tomographically complete qubit statistics.* A pure-state-complete chance model is possible, as in (C2). Nevertheless its preparation-law readout cannot in general be identified with only the density matrix if convex-linearity is imposed. The fibre over $I/2$ contains ontic preparation laws at pairwise TV distance at least $1/4$, and at summed distance at least $1/2$ from the trine mixture. This is a statement about preparation laws, not a proof that each pure ray is an incomplete individual state. **Proof.** Apply Corollary [8.4](/sealed-or-leaky/what-a-quantum-preparation-description-necessarily-omits#ctx:noninjectivity) and Theorem [8.3](/sealed-or-leaky/what-a-quantum-preparation-description-necessarily-omits#ctx:sum). The Beltrametti–Bugajski model distinguishes proper-mixture laws while retaining rays as individual states [[2](/sealed-or-leaky/references#bib-BB95)]. □ --- # Section 3: Experimental anchoring ## 3 Experimental anchoring ### 3.1 A finite-sample likelihood statement, not a central-value entropy claim *Status: Statistical model.* Let $C_i=(A_i,B_i)$ and $W_i=(X_i,Y_i)$, with history $\mathcal H_i=(C_1,W_1,\ldots,C_{i-1},W_{i-1})$. For a fixed classical initial $T$, assume $$ {\mathbb P}(W_i=xy\mid T,\mathcal H_i)=q_{xy}>0 $$ and that the box of $C_i$ conditional on $T,\mathcal H_i$ is no-signalling. These are sequential premises, not consequences of the one-round assumptions. Their chain factorization is $$ P(c^n,w^n\mid t)=\prod_{i=1}^n q_{w_i} P_i(c_i\mid w_i,\mathcal H_i,t). $$ Equation (5). Summation over the outcomes gives $P(w^n\mid t)=\prod_iq_{w_i}$. Thus conditioning on the whole setting word does not introduce a future-input factor into the outcome likelihood. This causal factorization is indispensable for the following likelihood calculation. Put $q=\min_{xy}q_{xy}$, $\sigma_{xy}=(-1)^{xy}$, $$ D_i=\frac{\sigma_{X_iY_i}A_iB_i}{q_{X_iY_i}},\qquad \widehat S=\frac1n\sum_iD_i,\qquad S_i={\mathbb E}[D_i\mid T,\mathcal H_i]. $$ This defines the estimator; it is not automatically the same as a published setting-frequency-normalized central value. **Lemma 3.1 (Conditional-range CHSH concentration).** *Status: Proved conditional on the sequential setting law above.* For $0<\alpha<1$, define $$ r_n=\frac1q\sqrt{\frac{2\ln(1/\alpha)}n}. $$ For almost every $t$, ${\mathbb P}_t\{n^{-1}\sum_i S_i<\widehat S-r_n\}\le\alpha$. The constant uses the conditional range width $2/q$, not the larger bound $2(1/q+4)$ on centered increments. **Proof.** 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 $$ {\mathbb E}_t[e^{\theta(D_i-S_i)}\mid\mathcal H_i] \le e^{\theta^2/(2q^2)}. $$ Iteration and Markov's inequality give ${\mathbb P}_t(\sum_i(D_i-S_i)\ge nr)\le \inf_{\theta>0}e^{-\theta nr+n\theta^2/(2q^2)} =e^{-nq^2r^2/2}$. Substitution proves the assertion. Independence of device outputs between rounds was not assumed. □ **Proposition 3.2 (Finite-sample outcome-likelihood certificate).** *Status: Proved conditional on [(5)](/sealed-or-leaky/experimental-anchoring#exp:causalfactor) and conditional no-signalling; quantum alternative conditional on per-history (QT).* Define $F_{\rm NS}(s)=0$ for $s\le2$, $F_{\rm NS}(s)={f_{\mathrm{NS}}}(s)$ for $2 $$ {\mathbb P}_t\!\left\{-\log_2 P_t(C^n\mid W^n) $$ {\mathbb P}_t\{P_t(C^n\mid W^n)>\delta_*,\ J_*=1\}\le\alpha. $$ Equation (7). Neither display alone is a bound on unsmoothed min-entropy of a postselected distribution. **Proof.** Each joint outcome probability is no larger than an Alice marginal, so Lemma [2.3](/sealed-or-leaky/the-source-tetralemma#tet:guess) bounds it by $2^{-F_{\rm NS}(S_i)}$. By [(5)](/sealed-or-leaky/experimental-anchoring#exp:causalfactor), $$ -\log_2P_t(c^n\mid w^n) \ge\sum_iF_{\rm NS}(S_i) \ge nF_{\rm NS}\!\left(n^{-1}\sum_i S_i\right). $$ 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](/sealed-or-leaky/experimental-anchoring#exp:range) makes the last expression at least the right side of [(6)](/sealed-or-leaky/experimental-anchoring#exp:finitecorrect). 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](/sealed-or-leaky/references#bib-Pironio10)], 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](/sealed-or-leaky/experimental-anchoring#exp:tailtransfer) supplies that conversion explicitly. □ ### 3.2 Published CHSH values: illustrations only *Status: Imported central values; derived illustrative evaluations.* The reported values are $2.42\pm0.20$ in Hensen et al. [[23](/sealed-or-leaky/references#bib-Hensen15)], $2.221\pm0.033$ in Rosenfeld et al. [[26](/sealed-or-leaky/references#bib-Rosenfeld17)], and $2.0747\pm0.0033$ in Storz et al. [[27](/sealed-or-leaky/references#bib-Storz23)]. The corresponding reported trial counts and local-realist significance bounds are $245$, $P\le0.039$; $10^4$ in the cited run, $P<2.57\times10^{-9}$; and more than $10^6$, $P<10^{-108}$, respectively. These significance bounds test the authors' stated nulls; they are not source-entropy estimates. | Experiment | Central $S$ | ${f_{\mathrm{NS}}}(S)$ | ${f_{\mathrm{Q}}}(S)$ | $g_{\rm NS}(S)$ | | --- | --- | --- | --- | --- | | Hensen et al. | 2.42 | 0.1600 | 0.2075 | 0.2100 | | Rosenfeld et al. | 2.221 | 0.0820 | 0.0926 | 0.1105 | | Storz et al. | 2.0747 | 0.0272 | 0.0283 | 0.03735 | Table 1. Evaluations of ideal-box bounds, in bits per round, not finite-sample certifications. ${f_{\mathrm{NS}}}$ and ${f_{\mathrm{Q}}}$ bound the appropriate conditional min-entropy and also Shannon entropy; $g_{\rm NS}$ is the sharper no-signalling Shannon bound. The ${f_{\mathrm{NS}}}$ evaluations at one reported error bar below and above the central values are, respectively, $[0.0816,0.2430]$, $[0.0695,0.0946]$, and $[0.0260,0.0284]$; for ${f_{\mathrm{Q}}}$ they are $[0.0921,0.3838]$, $[0.0769,0.1091]$, and $[0.0270,0.0296]$. These transformed error bars are not confidence-certified entropy intervals. An ideal identical-round interpretation is needed to read the central entries as constant per-round rates. *Status: Not established from the supplied summaries.* No finite-sample source-entropy number for these three data sets is certified here. Applying Proposition [3.2](/sealed-or-leaky/experimental-anchoring#exp:finite) needs the exact score or equivalent sufficient statistics, the actual setting law, applicable sequential assumptions and a specified confidence rule; a central estimate and its standard error do not supply those items. Trial lists have not been reconstructed. The photonic analyses in [[24](/sealed-or-leaky/references#bib-Giustina15), [25](/sealed-or-leaky/references#bib-Shalm15)] use different Bell statistics; the CHSH expressions are not substituted for those statistics. ### 3.3 Bierhorst-admissible information and the actual protocol word **Definition 3.3 (Bierhorst-admissible Tier-1 information).** *Status: Imported model class; notation translated from [[28](/sealed-or-leaky/references#bib-Bierhorst18), Eqs. (2)–(3)].* A classical random variable $T$, fixed before the protocol, with no access to data produced during it, is *B-admissible* when, for each trial and almost every supported history and $t$, $$ \begin{aligned}{\mathbb P}(X_i=x,Y_i=y\mid T=t,\mathcal H_i)&=\tfrac14,\\ {\mathbb P}(A_i=a\mid X_iY_i,T=t,\mathcal H_i) &={\mathbb P}(A_i=a\mid X_i,T=t,\mathcal H_i),\\ {\mathbb P}(B_i=b\mid X_iY_i,T=t,\mathcal H_i) &={\mathbb P}(B_i=b\mid Y_i,T=t,\mathcal H_i). \end{aligned} $$ Equation (8, 9, 10). Here $\mathcal H_i$ includes the settings and outcomes of earlier trials. All public training choices and the fixed Bell function are part of the conditioning context. This is a classical pre-existing isolated-adversary condition; it is not a theorem for quantum side information, later-acquired protocol data, or the internal state of a predictable settings generator. The published bounded-bias variants require their own revised parameters and are not silently invoked in the numerical application below. **Lemma 3.4 (Finite-seed obstruction to exact sequential setting independence).** *Status: Proved.* Suppose the full setting word $W$ is a deterministic function of a finite-valued generator resource $G_{\rm set}$, including any genuinely fresh randomness used during the protocol. The exact setting condition [(8)](/sealed-or-leaky/experimental-anchoring#exp:Bsettings) implies $$ H(W\mid T)=2n\le H(G_{\rm set}\mid T)\le\log_2|\mathcal G_{\rm set}|. $$ In particular, a generator resource with less than $2n$ conditional entropy cannot satisfy that condition, even when $T$ does not reveal the generator state. **Proof.** Average [(8)](/sealed-or-leaky/experimental-anchoring#exp:Bsettings) 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_{ #### Alphabet and symbol translation. *Status: Definitions.* Bierhorst et al. label detection by $+$ and nondetection by $0$. The bijection $+\mapsto+1$, $0\mapsto-1$ puts both into this article's alphabet; it changes no probability or entropy and discards no nondetections. Their Bell function is denoted here by $\mathcal B$, not $T$; their extractor seed by $R$, not CHSH $S$; and their extracted string by $K$. Write $W=X^nY^n$, let $C$ be the protocol's processed outcome word, and let $J$ be its pass indicator. #### Causal padding is part of the record map. *Status: Imported implementation detail; not reconstructed data.* For Data Set 5, SI S.6 of [[28](/sealed-or-leaky/references#bib-Bierhorst18)] reports first crossing the fixed threshold at trial $41{,}243{,}976$, followed by relabelling every later outcome to nondetection. Thus $C$ is the raw word through that crossing, followed by deterministic $(-1,-1)$ pairs. Its remaining Bell factors equal one. This history-dependent change is covered by the paper's adaptive theorem: it is selected from the completed past, not from a current distant outcome. The reported raw final product $2.018\times10^{41}$ is *not* the product of this padded word. We use the reported threshold crossing, not that raw product as a replacement threshold or as a recomputation of $C$. #### Source and numerical quantities. *Status: Imported published quantities.* Data Set 5 of [[28](/sealed-or-leaky/references#bib-Bierhorst18)] ([arXiv:1803.06219v1](https://arxiv.org/abs/1803.06219v1) and SI) uses $n=55{,}110{,}210$ trials after $5\times10^6$ training trials; the reported Bell-function parameter is $m=0.0100425$. The fixed threshold, errors, seed length and output length are $$ \begin{gathered} v=1.5\times10^{32},\quad p=9.025\times10^{-25},\quad \kappa=9.5\times10^{-13},\quad p=\kappa^2,\\ \epsilon_{\rm ext}=5\times10^{-14},\quad d=315{,}844,\quad \ell=1024,\quad \epsilon=10^{-12}. \end{gathered} $$ Equation (11). The probability $\pi={\mathbb P}(J=1)$ is a property of the repeated-protocol law, not of the particular observed word. The hypothesis $\pi\ge\kappa$ remains explicit; observing $J=1$ does not prove it. The training-based i.i.d. success estimate in the paper is not substituted for an adversarial lower bound on $\pi$. **Lemma 3.5 (Rounding-controlled entropy-production parameter).** *Status: Proved from the printed Bell table and its stated downward-rounding rule in [[28](/sealed-or-leaky/references#bib-Bierhorst18)].* The printed coefficients $b_{xy}^{ab}$, in the original $\{+,0\}$ column order, are $$ \begin{array}{c|rrrr} xy & ++ & +0 & 0+ & 00\\\hline 00&1.0243556353&0.9704647804&0.9735507658&1\\ 01&1.0256127409&0.9491951243&0.9960775334&1\\ 10&1.0227274988&0.9962782754&0.9461091383&1\\ 11&0.9273040563&1.0037217225&1.0039224645&1 \end{array} $$ Under uniform settings, their maximum local expectation is $1$ and maximum no-signalling expectation is $1.01004250775$. If the original coefficients are rounded down at the tenth decimal place, their actual no-signalling excess $m_{\rm act}$ satisfies $$ 0.01004250775\le m_{\rm act}\le m_+:=0.01004250785. $$ Equation (12). Put $$ \delta_+=\left[1-\frac{\exp(\ln(pv)/n)-1}{2m_+}\right]^n, \qquad b_+=-\log_2\delta_+. $$ Equation (13). For the parameters [(11)](/sealed-or-leaky/experimental-anchoring#exp:parameters), $$ 1344.914017261073.05>1064. $$ Equation (14). Consequently $\delta_+$ is a conservative entropy-production threshold, and the extraction constraint for $\ell=1024$ remains satisfied. These checks do not reconstruct the experimental product from the rounded table. **Proof.** Enumerate the sixteen local assignments $(a_0,a_1,b_0,b_1)\in\{0,1\}^4$ and their means $\frac14\sum_{xy}b_{xy}^{a_xb_y}$. The maximum is one, attained by all nondetections. The other eight vertices of the binary no-signalling polytope are the PR boxes [[22](/sealed-or-leaky/references#bib-Pironio10), App. A.3]; their means are $$ \frac18\sum_{x,y,a\in\{0,1\}} b_{xy}^{a,\ a\oplus xy\oplus rx\oplus sy\oplus t}, \qquad (r,s,t)\in\{0,1\}^3. $$ Their maximum occurs at $(r,s,t)=(0,0,0)$ and equals $4040170031/4000000000$; each of the other seven means is below one. These finite sums can be checked directly with the displayed decimal rationals. Increasing every coefficient by less than $10^{-10}$ changes the expectation under any normalized distribution by less than $10^{-10}$. Monotonicity of the maximum then proves [(12)](/sealed-or-leaky/experimental-anchoring#exp:mupper). The largest printed local mean other than the all-nondetection strategy is $1-5.25\times10^{-10}$. Raising every uncertain coefficient by $10^{-10}$ still leaves each such mean below one. The all-nondetection coefficients are fixed to exactly one by the published training constraint. Thus the local bound for the original function also follows from this rounding envelope; no unchecked numerical optimization tolerance is needed. For $pv>1$, the entropy-production expression for $\delta$ is increasing in $m$ wherever its bracket is positive. The actual theorem therefore also holds with the larger $\delta_+$. Its admissible range $pv\le(1+3m_{\rm act}/2)^n$ holds already with the lower endpoint in [(12)](/sealed-or-leaky/experimental-anchoring#exp:mupper). Numerically stable evaluation is $$ b_+=-\frac{n}{\ln2}\ln\!\left(1- \frac{\exp(\ln(pv)/n)-1}{2m_+}\right). $$ The intervals in [(14)](/sealed-or-leaky/experimental-anchoring#exp:numericalintervals) follow by bounding logarithms with $\ln x=2\sum_{j\ge0}((x-1)/(x+1))^{2j+1}/(2j+1)$ after binary range reduction, and bounding the positive exponential and $-\ln(1-u)$ series by their geometric tail bounds. Twenty terms in the range-reduced logarithms and four terms in each of the latter small-argument series suffice for the displayed intervals. Finally $\ell+4\log_2\ell=1064$. The conservative value agrees with the fragment's rounded $1344.9$, but the rounded printed $m$ is not treated as an exact extremal value. □ #### The exact imports used. *Status: Imported conditional theorems from [[28](/sealed-or-leaky/references#bib-Bierhorst18), Eq. (4), Eqs. (5)–(6), SI S.2 and S.5].* With $T$ B-admissible, the entropy-production theorem gives, almost everywhere in $t$, $$ {\mathbb P}_t\{P_t(C\mid W)>\delta_+,\ J=1\}\le p. $$ Equation (15). Here $P_t(C\mid W)$ is the probability assigned to the *realized* word and settings, not a maximum taken after observing the data. For the soundness theorem, additionally take $R$ uniform on $\{0,1\}^d$ and independent of $(C,W,T,J)$, use the specified extractor $K=\mathrm{Ext}(C,R)$, and suppose $\pi\ge\kappa$. Then $$ {\mathrm{TV}}\!\left(P_{KWRT\mid J=1},\, U_\ell\otimes P_{WRT\mid J=1}\right) \le p/\pi+\epsilon_{\rm ext}\le10^{-12}. $$ Equation (16). Here $U_\ell$ and $U_d$ are uniform laws and $P_{WRT\mid J=1}$ is the product of $P_{WT\mid J=1}$ and $U_d$, with coordinates put in the displayed order. The independence of $R$ makes the real and ideal $(W,R,T)$ marginals identical after passing. Equations [(15)](/sealed-or-leaky/experimental-anchoring#exp:EPT)–[(16)](/sealed-or-leaky/experimental-anchoring#exp:soundness) are not assumed for a larger class of side information than Definition [3.3](/sealed-or-leaky/experimental-anchoring#exp:admissible). ### 3.4 Extracted randomness translated into source-response entropy **Lemma 3.6 (Uniformity-to-entropy continuity with the correct dimension).** *Status: Proved; imports the entropy-continuity inequality of [[29](/sealed-or-leaky/references#bib-Audenaert07)].* Let $K$ take $M=2^\ell$ values and let $E$ be classical. If ${\mathrm{TV}}(P_{KE},U_\ell\otimes P_E)\le\epsilon\le1-1/M$, then $$ H(K\mid E)\ge\ell-h_2(\epsilon)-\epsilon\log_2(M-1) \ge\ell(1-\epsilon)-h_2(\epsilon), \quad p_{\rm guess}(K\mid E)\le M^{-1}+\epsilon. $$ Equation (17). **Proof.** Write $\delta_e={\mathrm{TV}}(P_{K\mid e},U_\ell)$. Equality of the $E$ marginals gives ${\mathbb E}\delta_e\le\epsilon$. Always $0\le\delta_e\le1-1/M$, since a point mass is farthest from the uniform law on $M$ points. Apply Audenaert's bound to the diagonal $M$-dimensional density matrices: $$ H(K\mid E=e)\ge\log_2M-h_2(\delta_e)-\delta_e\log_2(M-1). $$ The loss function is concave and nondecreasing on $[0,1-1/M]$, since its derivative is $\log_2((M-1)(1-u)/u)\ge0$ there. Jensen and monotonicity give the first inequality of [(17)](/sealed-or-leaky/experimental-anchoring#exp:continuity); $\log_2(M-1)\le\ell$ gives the second. No uncontrolled $\delta_e>1/2$ exception or auxiliary majorant is needed. Finally, under the ideal law any guess measurable in $E$ succeeds with probability $1/M$. Total variation changes that success probability by at most $\epsilon$, including for the optimal real-law guess. □ **Lemma 3.7 (Translation of the certified extracted string).** *Status: Proved conditional on B-admissibility, the published protocol and seed hypotheses, $\pi\ge\kappa$, single outcomes and runwise outcome determinism.* Suppose the processed word is a deterministic function $C=F(\lambda,W)$ of an initial source state and the full setting word; let $T=\tau(\lambda)$ be B-admissible. Let $Z_{\rm run}=F(\lambda,\cdot)$ be the finite response function on the finite set of setting words. With the extraction hypotheses of [(16)](/sealed-or-leaky/experimental-anchoring#exp:soundness), $$ \begin{aligned}H(Z_{\rm run}\mid T,J=1) &\ge H(Z_{\rm run}\mid T,W,R,J=1)\\ &\ge H(K\mid T,W,R,J=1)\\ &\ge1024(1-10^{-12})-h_2(10^{-12}) >1024-1.1\times10^{-9},\\ p_{\rm guess}(Z_{\rm run}\mid T,W,R,J=1) &\le2^{-1024}+10^{-12}. \end{aligned} $$ Equation (18, 19). Thus the unsmoothed conditional min-entropy is at least $-\log_2(2^{-1024}+10^{-12})>39.86$ bits, also after dropping $W,R$ from the conditioning. No independence of $\lambda$ from $W$ is needed beyond the stated B-admissibility of $T$ for these implications. **Proof.** Apply Lemma [3.6](/sealed-or-leaky/experimental-anchoring#exp:Audenaert) 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)](/sealed-or-leaky/experimental-anchoring#exp:extractedShannon). 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)](/sealed-or-leaky/experimental-anchoring#exp:soundness) gives [(19)](/sealed-or-leaky/experimental-anchoring#exp:extractedguess). Dropping conditioning cannot improve guessing. All distributions in this argument are conditional passing-ensemble laws, not entropies assigned to one realized string. □ **Corollary 3.8 (The correctly scoped 1024-bit Shannon consequence).** *Status: Proved conditional on all hypotheses of Lemma [3.7](/sealed-or-leaky/experimental-anchoring#exp:translation).* The Data Set 5 soundness certificate implies the lower bound in [(18)](/sealed-or-leaky/experimental-anchoring#exp:extractedShannon) for every separately specified B-admissible classical readout $T$. It is a Shannon bound on the finite response function conditional on passing. The number $1024$ is the extracted output length, not its certified unsmoothed min-entropy and not unsmoothed min-entropy of an unrestricted $\lambda$. **Proof.** This is Lemma [3.7](/sealed-or-leaky/experimental-anchoring#exp:translation) with the published parameters. Keeping its probability space, conditioning and side-information class is essential to the statement. □ ### 3.5 A stronger finite-sample consequence without an extractor or source–setting independence **Lemma 3.9 (Atom-tail transfer through determinism and postselection).** *Status: Proved.* Let $Z,C$ be finite, $E$ classical standard Borel, $C=f(Z,E)$, and $J$ a pass event of probability $\pi>0$. Suppose $0 $$ {\mathbb P}\{J=1,\ P(C\mid E)>\delta\}\le p. $$ Equation (20). Then $$ \begin{aligned}p_{\rm guess}(Z\mid E,J=1)&\le\min\{1,(\delta+p)/\pi\},\\ H(Z\mid E,J=1)&\ge (1-p/\pi)\left[\log_2\frac{\pi-p}{\delta}\right]_+. \end{aligned} $$ Equation (21, 22). For clarity, define *deletion-smoothed conditional min-entropy* of a normalized law $P$ by optimizing over subprobability measures $Q\le P$ of mass at least $1-\eta$: $$ H_{\min}^{\mathrm{del},\eta}(Z\mid E)_P =\sup_{\substack{0\le Q\le P\\Q(\mathrm{all})\ge1-\eta}} \left[-\log_2\int\max_z\frac{dQ(z,\cdot)}{d\mu}(e)\,d\mu(e)\right], $$ where $\mu$ dominates the $E$ measures. The value is independent of that dominating choice. For the conditional passing law, $$ H_{\min}^{\mathrm{del},p/\pi}(Z\mid E,J=1) \ge-\log_2\delta+\log_2\pi. $$ Equation (23). The smoothing parameter is discarded probability mass; no claim that it equals a purified-distance or normalized-TV smoothing convention is made. **Proof.** Let $B$ indicate the good event $P(C\mid E)\le\delta$, and restrict the conditional passing law to $B=1$: $$ Q(z,de)={\mathbb P}(Z=z,E\in de,B=1\mid J=1). $$ This deletes mass at most $p/\pi$. For every $(z,e)$ that survives, determinism and the good-event definition give $P(Z=z\mid E=e)\le P(C=f(z,e)\mid E=e)\le\delta$. Taking $\mu=P_E$ therefore gives $$ \frac{dQ(z,\cdot)}{dP_E}(e)\le\frac{\delta}{\pi},\qquad p_{\rm guess}(Q;Z\mid E)\le\frac{\delta}{\pi}. $$ This proves [(23)](/sealed-or-leaky/experimental-anchoring#exp:smoothraw). The deleted submeasure has guessing probability at most its mass, proving [(21)](/sealed-or-leaky/experimental-anchoring#exp:rawguess). Let $g={\mathbb P}(B=1\mid J=1)\ge1-p/\pi$. The normalized good law has guessing probability at most $\delta/(\pi g)$ and hence Shannon entropy at least $[\log_2(\pi g/\delta)]_+$. Conditioning on $B$ can only reduce average Shannon entropy, and the bad-law entropy is nonnegative. Therefore $$ H(Z\mid E,J=1)\ge g[\log_2(\pi g/\delta)]_+. $$ The right side is nondecreasing in $g\ge0$, giving [(22)](/sealed-or-leaky/experimental-anchoring#exp:rawShannon). No independence between $Z$ and $E$ was used. In particular the lemma is stronger, for postselected posterior-response entropy, than a transfer argument requiring an independent prior response law. □ **Corollary 3.10 (Published-parameter finite-sample source-response certificate).** *Status: Proved conditional on B-admissibility, the published entropy-production certificate, single outcomes, runwise outcome determinism and $\pi\ge\kappa$.* For Data Set 5, take $E=(T,W)$, $Z=Z_{\rm run}$ and $\delta=\delta_+$ of [(13)](/sealed-or-leaky/experimental-anchoring#exp:deltaplus). Without using the extractor or assuming source–setting independence, the published quantities imply $$ \begin{aligned}H(Z_{\rm run}\mid T,W,J=1)&>1304.97\ \text{bits},\\ H_{\min}^{\mathrm{del},\,9.5\times10^{-13}} (Z_{\rm run}\mid T,W,J=1)&>1304.97\ \text{bits},\\ H_{\min}(Z_{\rm run}\mid T,W,J=1)&>39.93\ \text{bits}. \end{aligned} $$ Equation (24, 25, 26). The Shannon and ordinary guessing conclusions remain true when $W$ is dropped from the conditioning. These are not obtained by substituting a central CHSH value into ${f_{\mathrm{NS}}}$. **Proof.** Average [(15)](/sealed-or-leaky/experimental-anchoring#exp:EPT) over $t$ and apply Lemma [3.9](/sealed-or-leaky/experimental-anchoring#exp:tailtransfer) to $C=Z_{\rm run}(W)$. Since $\pi\ge\kappa$, deletion mass is at most $p/\kappa=\kappa$, and the smooth bound is at least $b_++\log_2\kappa$. The Shannon bound is at least $$ (1-p/\kappa)[b_++\log_2(\kappa-p)] >1304.9768>1304.97. $$ Here monotonicity in $\pi$ follows directly because both nonnegative factors in [(22)](/sealed-or-leaky/experimental-anchoring#exp:rawShannon) increase with $\pi$. The unsmoothed bound is $$ -\log_2\bigl((\delta_++p)/\kappa\bigr)>39.93. $$ Numerically $b_++\log_2\kappa$ lies between $1304.97687$ and $1304.97689$; the Shannon correction to it is less than $1.3\times10^{-9}$. These conservative intervals follow from Lemma [3.5](/sealed-or-leaky/experimental-anchoring#exp:rounding); they do not identify $1344.9$ with an unconditional or unsmoothed entropy. Dropping public settings can only reduce guessing and increase Shannon entropy. □ **Lemma 3.11 (The fragment's prior-response tail and its additional premise).** *Status: Proved conditional on B-admissibility, the entropy-production certificate, runwise outcome determinism and $Z_{\rm run}\perp W\mid T$.* With the additional conditional independence just stated, for almost every $t$, $$ {\mathbb P}_t\{P_t(Z_{\rm run}=z_{\rm obs})>\delta_+,\ J=1\}\le p, $$ Equation (27). where $z_{\rm obs}$ is the realized response function. Independence of the full $\lambda$ and $W$ conditional on $T$ is sufficient but not necessary: response-level conditional independence suffices. In particular, runwise (MI) and $T=\tau(\lambda)$ imply that sufficient condition. This does not redefine the (MI) premise of Section [2.1](/sealed-or-leaky/the-source-tetralemma#tet:setting). **Proof.** At the realized setting word $w$, $$ P_t(C=z_{\rm obs}(w)\mid W=w) \ge P_t(Z_{\rm run}=z_{\rm obs}\mid W=w) =P_t(Z_{\rm run}=z_{\rm obs}). $$ The event in [(27)](/sealed-or-leaky/experimental-anchoring#exp:prioratomtail) is therefore contained in that of [(15)](/sealed-or-leaky/experimental-anchoring#exp:EPT). This is an exceptional-event bound on a *prior* atom, not itself a conditional entropy after passing. Corollary [3.10](/sealed-or-leaky/experimental-anchoring#exp:rawcor) instead proves a posterior entropy statement without this independence premise. □ **Lemma 3.12 (Threshold test of a complete deterministic admissible readout).** *Status: Proved conditional on B-admissibility and the published entropy-production theorem; reported threshold crossing imported.* For a predetermined threshold $v>1$ and the same Bell-function protocol, any source-complete model with single outcomes and $C=F(\lambda,W)$ satisfies $$ {\mathbb P}(J=1)\le v^{-1}. $$ At the published $v=1.5\times10^{32}$ this is $2/(3\times10^{32})$. Unlike Corollaries [3.8](/sealed-or-leaky/experimental-anchoring#exp:bierhorst) and [3.10](/sealed-or-leaky/experimental-anchoring#exp:rawcor), this test bound does not require $\pi\ge\kappa$. Its rejection event is the reported crossing of the fixed threshold, not a threshold fitted to the final raw product. **Proof.** 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. □ ### 3.6 Who certifies what, and what remains a gap **Remark 3.13 (Certification and source interpretation).** *Status: Scope; imported certificate and proved translation distinguished.* **Published certificate.** Bierhorst et al.'s theorem concerns pre-existing classical information isolated from the protocol. Their passing-ensemble soundness inequality, seed requirement and statistical assumptions are [(16)](/sealed-or-leaky/experimental-anchoring#exp:soundness) and Definition [3.3](/sealed-or-leaky/experimental-anchoring#exp:admissible), not security against every physically imaginable no-signalling observer. The Methods section reports pseudorandom settings generated by the Mersenne Twister and explains the additional device-trust basis for effective independence [[28](/sealed-or-leaky/references#bib-Bierhorst18)]. Possession of its predictive internal state is outside the present admissible class. Neither the article nor this translation certifies independence from such a state. Merely hiding that state is also insufficient to prove the exact sequential law: Lemma [3.4](/sealed-or-leaky/experimental-anchoring#exp:seednecessity) requires $2n=110{,}220{,}420$ bits of conditional settings-resource entropy. Counting this many pseudorandom output bits does not certify that entropy. The source does not supply an independent verification of this exact generator-entropy condition, and no quantified replacement error for it is imported here. Thus the numerical implications remain conditional model statements, not unconditional information-theoretic certification of the pseudorandom implementation. **Proved here under determinism.** Determinism turns recorded-word unpredictability into lower bounds on an initial finite response function, which is a function of $\lambda$. The extracted-string route gives nearly $1024$ conditional Shannon bits but only $39.86$ guaranteed unsmoothed min-entropy bits. The entropy-production route gives over $1304.97$ Shannon and deletion-smoothed min-entropy bits and over $39.93$ unsmoothed bits. The smooth and unsmoothed quantities are not interchangeable. No cardinality, differential entropy, or universal observer-access assertion about an arbitrary $\lambda$ is inferred merely from the output length. Without deterministic source responses, the same statistics can instead represent fresh chance. **Passing matters.** All near-$1024$ and $1304.97$ entropy bounds refer to the passing-ensemble law and require the displayed lower bound on its pass probability. Conditioning on passing need not preserve no-signalling; none of the proofs assumes it does. One observed pass is not a measurement of $\pi$, and an ensemble conditional entropy is not a property of one known output string. If $L$ denotes the conditional Shannon lower bound, the unconditioned conclusion is only $H(Z_{\rm run}\mid T)\ge\pi L\ge\kappa L$, by conditioning additionally on $J$ and nonnegativity. A large unconditional entropy would need an independently justified much larger lower bound on $\pi$. #### Explicit unresolved empirical scope. *Status: Not established.* The supplied summaries do not establish the actual pass probability, exact sequential setting independence for the pseudorandom implementation, independence from every proposed additional variable, security with quantum side information, or an unconditional high-entropy bound for $\lambda$. Nor do they provide the exact scores and setting analyses needed to certify the three central-value CHSH examples. Those gaps are not filled by inventing trial lists. For the Data Set 5 theorem-level translation, the reported threshold crossing and published aggregate parameters do suffice under the stated hypotheses; independent reproduction of the experiment or extractor execution is not claimed. A claim about another error level, a different data set, or biased settings requires its own published or recomputed parameters. --- # Section 4: The escape routes and their boundaries ## 4 The escape routes and their boundaries *Status: Scope of the explicit countermodels.* Theorem [2.7](/sealed-or-leaky/the-source-tetralemma#tet:tetralemma) is an implication in a specified model class, not a census of all physical theories. Its examples establish three separate premise countermodels and one branching alternative outside its single-outcome domain. Multiple routes may be used together. #### Branching. The explicit state in (C1) has a complete initial ray and deterministic vector evolution. Its branch-weight table is $P_Q$, but assigning a unique actual outcome requires additional structure. The relative-state literature discusses the interpretation of these weights [[30](/sealed-or-leaky/references#bib-Everett57), [31](/sealed-or-leaky/references#bib-Deutsch99), [32](/sealed-or-leaky/references#bib-Wallace12)]. No such interpretation can be smuggled into the scalar outcome functions of (D). Accordingly (C1) refutes an unrestricted single-outcome ontology claim, not Lemma [2.2](/sealed-or-leaky/the-source-tetralemma#tet:SOredundant). #### Fundamental chance. Model (C2) keeps the initial state complete while outcomes are intrinsically stochastic. This suffices as a logical countermodel; a particular collapse dynamics is not required. The preparation-contextuality obstruction in Proposition [2.9](/sealed-or-leaky/the-source-tetralemma#tet:chance) concerns the density matrix as a readout of preparation laws, not incompleteness of each pure-state ontic value [[2](/sealed-or-leaky/references#bib-BB95), [12](/sealed-or-leaky/references#bib-S05)]. #### Measurement dependence. Model (C3) preserves local deterministic responses while correlating source and settings. Specific independence hypotheses can have empirical consequences, but independence against an unrestricted omitted variable is not certified merely by observing random-looking settings. Brans-type constructions show the logical role of the premise [[35](/sealed-or-leaky/references#bib-Brans88)]. No stronger impossibility of testing every specified setting mechanism is asserted. #### Accessible nonlocality. Model (C4) has no-signalling unconditioned marginals and signalling responses conditional on the readable source. This is an explicit counterexample to inferring (NS) for arbitrary side information from ordinary Bell marginals. The transition-set literature and Theorem [12.1](/sealed-or-leaky/setting-dependence-controlled-nonequilibrium-and-access#sig:budget) quantify mathematical nonequilibrium reweightings [[13](/sealed-or-leaky/references#bib-V02)]; physically preparing them is a separate premise. The retained finite-pilot setting dependence in Theorem [11.7](/sealed-or-leaky/finite-pilot-resources-exact-inventories-and-retained-records#pil:signal) is an already specified nonrelativistic constitutive prediction, not a demonstrated spacelike experiment. | Model | Formal status | What it establishes | | --- | --- | --- | | (C1) Branching | Outside (SO) and stated (D) | Complete-state determinism is different from single-outcome determinism. | | (C2) Chance | (SO), (MI), (NS); not (D) | (D) is indispensable within the single-outcome class. | | (C3) Measurement dependence | (SO), (D), (NS); not (MI) | (MI) is indispensable in that class. | | (C4) Accessible nonlocality | (SO), (D), (MI); not (NS) | Conditional-access (NS) is indispensable in that class. | --- # Section 5: Operational equivalence and its restricted alternatives ## 5 Operational equivalence and its restricted alternatives The objects compared must be fixed before a completeness claim is made. An individual ontic state, a preparation law on ontic states, its density matrix, a present detector configuration, and a complete experimental record are different objects. A failure of injectivity for one of their maps does not establish failure of injectivity for the others. Let $S$ be a set of physically reduced source states or preparation laws, and let $p:S\to T$ be a nominated readout. For each admissible complete experiment $e$, write $\mu_{e,s}$ for the probability law of its complete data on a measurable space $(D_e,\Sigma_e)$. The admissible family includes exactly the preparations, interventions, repetitions, stopping rules, and retained records that the model permits. These permissions are physical premises, not consequences of the following definitions. Put $$ s\sim_{\rm op}s'\quad\Longleftrightarrow\quad \mu_{e,s}=\mu_{e,s'}\quad\hbox{for every admissible }e, \qquad q:S\longrightarrow S/{\sim_{\rm op}}. $$ This quotient is a predictive equivalence class, not necessarily an observable at one instant or a finite-dimensional physical state. **Theorem 5.1 (Operational surrogate and testing ceiling).** *Status: Proved.* The family $\mu^\circ_{e,[s]}:=\mu_{e,s}$ is well defined on $S^\circ=S/{\sim_{\rm op}}$ and reproduces every stipulated record law. Taking the readout of this statistical model to be $\operatorname{id}_{S^\circ}$ gives an injective-readout surrogate. If an alternative class contains this surrogate for every source model under consideration, a level-$\alpha$ test of that class cannot have power exceeding $\alpha$ against any such source model. More generally, if for a complete experiment its source law $P$ has $\inf_{Q\in\mathcal Q}{\mathrm{TV}}(P,Q)\le\varepsilon$, every test $\varphi:D\to[0,1]$ satisfying $\sup_{Q\in\mathcal Q}\int\varphi\,dQ\le\alpha$ obeys $\int\varphi\,dP\le\alpha+\varepsilon$. **Proof.** Equality of all the laws is the definition of $\sim_{\rm op}$, so the surrogate family is well defined and preserves them. Identity readout is injective. The same test has the same rejection probability under equal laws. In the approximate statement, for every $Q\in\mathcal Q$, $\int\varphi\,dP\le\int\varphi\,dQ+{\mathrm{TV}}(P,Q)$; take the infimum. □ This is empirical non-identifiability in a specified model class, not logical undecidability. The surrogate is a statistical family. The construction does not give it a local action, a finite-dimensional Markov state, deterministic dynamics, a specified resource bound, or a particular ontology. If these structures are required of alternatives, membership must be established separately. If $S$ is measurable and the maps $s\mapsto\mu_{e,s}(A)$ are measurable, equipping $S^\circ$ with the quotient sigma-algebra makes the induced kernels measurable: the inverse image under $q$ of each kernel level set is measurable in $S$. The quotient need not be standard Borel. No regular conditional distribution on an arbitrary quotient is asserted here. #### Temporal and intervention consistency. Equal one-time marginals do not suffice: a single fair bit repeated at every time and independently drawn fair bits have identical one-time laws and different two-time laws. Complete experiment laws avoid this error. When the source supplies a coherent causal experiment system, copying all its complete laws preserves all equalities expressing prefix consistency, randomization, and admissible adaptive composition. For finite action and outcome alphabets, its history representation is explicit. At a positive-probability history $h_j$, with the next action $a_j$ externally selected under the stipulated intervention semantics, define $$ K_{j+1}(y\mid h_j,a_j) =\frac{{\mathbb P}(h_j,y\mid a_0,\ldots,a_j)} {{\mathbb P}(h_j\mid a_0,\ldots,a_{j-1})}. $$ Causal consistency makes the denominator independent of $a_j$; at zero-probability histories choose any probability vector. Multiplying these kernels, and the action-policy probabilities when actions are randomized, reconstructs the source record law by the chain rule. Thus the same causal law has a representation using observed histories alone. It need not be Markovian on the present readout. An arbitrary collection of unrelated protocol laws would not supply this consistency, and the present theorem does not invent it. In general standard Borel settings one additionally needs suitable measurable conditional kernels. **Theorem 5.2 (Quantitative failure of readout-only prediction).** *Status: Proved.* For a fixed experiment, suppose $p(s_0)=p(s_1)$ and $\delta={\mathrm{TV}}(\mu_{e,s_0},\mu_{e,s_1})$. Every proposed law $M_{e,p(s)}$ based only on the readout satisfies $$ \max_{i=0,1}{\mathrm{TV}}\bigl(\mu_{e,s_i},M_{e,p(s_i)}\bigr) \ge\frac{\delta}{2}. $$ The constant is sharp for this pair, attained by the common law $M=\tfrac12(\mu_{e,s_0}+\mu_{e,s_1})$. More generally, suppose candidate laws obey the independently specified continuity bound ${\mathrm{TV}}(M_{e,t},M_{e,t'})\le Ld(t,t')$. Then $$ \max_i{\mathrm{TV}}\bigl(\mu_{e,s_i},M_{e,p(s_i)}\bigr) \ge\frac{[\delta-Ld(p(s_0),p(s_1))]_+}{2}. $$ If certified comparison laws $\widehat\mu_i$ satisfy ${\mathrm{TV}}(\widehat\mu_i,\mu_{e,s_i})\le\eta_i$, replace $\delta$ in the last expression by $[{\mathrm{TV}}(\widehat\mu_0,\widehat\mu_1)-\eta_0-\eta_1]_+$. **Proof.** Apply the triangle inequality through the candidate law, or through the two candidate laws in the continuity version. For the midpoint, $\mu_{e,s_0}-M=(\mu_{e,s_0}-\mu_{e,s_1})/2$, giving equality. The comparison-law bound follows by a second triangle inequality. □ Theorem [5.2](/sealed-or-leaky/operational-equivalence-and-its-restricted-alternatives#op:radius) turns a verified leak into a lower bound on the error of *every* model using only the nominated readout, subject to its stated continuity restriction. It proves inadequacy of that description. It neither identifies a unique completion nor excludes an unrestricted theory whose state is the complete operational history. For general spaces the theorem asserts a pairwise lower bound; it does not assert that a globally measurable family of minimax centers exists. --- # Section 6: Deterministic completions require unresolved information ## 6 Deterministic completions require unresolved information **Theorem 6.1 (Conditional randomness and completion size).** *Status: Proved.* Let $S,T$ be standard Borel spaces, let ${\mathbb P}$ be a probability law on $S$, let $T_0=p(S_0)$ for a measurable $p$, and let a future finite-valued record be a measurable deterministic function $R=R(S_0)$. 1. If $R$ is binary, write $q(T_0)={\mathbb P}(R=1\mid T_0)$. The minimum probability of error of a measurable deterministic predictor of $R$ from $T_0$ is $$ \inf_h{\mathbb P}\{R\ne h(T_0)\} ={\mathbb E}\min\{q(T_0),1-q(T_0)\}. $$ If this is positive, $R$ does not descend almost surely through $p$; on a positive-measure set of readout fibers there are source states with different records. 2. If a finite-valued completion variable $Z$ makes $R=f(T_0,Z)$ almost surely, then, with entropy in bits, $$ H(Z\mid T_0)\ge H(R\mid T_0). $$ In particular, a conditionally uniform $n$-bit record requires at least $n$ bits of conditional completion entropy and $|\mathcal Z|\ge2^n$. 3. At a fixed readout $t$, suppose a deterministic simulator uses at most $M$ values of $Z$. If its record law is within $\varepsilon$ in total variation of the uniform law on $\{0,1\}^n$, then $$ M\ge(1-\varepsilon)2^n. $$ The bound is sharp when $M\le2^n$: a uniform distribution on any $M$ record strings has distance $1-M/2^n$ from the uniform target. **Proof.** Conditionally on $T_0=t$, predicting $0$ incurs error $q(t)$ and predicting $1$ incurs $1-q(t)$; the measurable threshold predictor at $q=1/2$ attains their minimum. Standard Borel regular conditional laws of $S_0$ given $T_0=t$ are concentrated on $p^{-1}(t)$ for almost every $t$. Where $0