Question, main result, and scope

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, 15, 16, 17, 18, 10]. The measurement constitutions supply separate model-relative examples [11, 20, 21]; none of their dynamics is used to prove the Bell results.

Theorem 5.1 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.

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 $Z$,

Both bounds are sharp in the no-signalling model class. The quantum guessing bound supplies the additional bound $H(Z\mid T)\ge\fQ(S)$ 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 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 separates illustrative evaluations at the published CHSH central values [23, 26, 27] from a genuine finite-sample translation of the entropy-production and soundness theorems in [28]. 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 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, 5, 22]. 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, $\TV(P,Q)=\sup_A|P(A)-Q(A)|\in[0,1]$; the variation norm of their signed difference is $2\TV(P,Q)$. 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 $E$,

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. 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 is a set.

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.