A Structural Proof of the No-Signaling Theorem

The no-signaling principle in quantum mechanics—that local operations on one
subsystem of an entangled pair cannot transmit information to a distant subsystem—
has never been structurally proven from independent physical principles. Every
existing derivation is circular: von Neumann constructed the tensor product for-
malism specifically to enforce no-signaling [1], and all subsequent “proofs” derive
no-signaling from a structure that was designed to contain it [2, 3]. We present a
non-circular structural proof. Starting from two premises independent of quantum
formalism—the relativistic propagation limit c and the experimental record of Bell
tests, quantum erasers, and which-path experiments—we derive the no-signaling re-
sult as a mathematical consequence of the tensor contraction structure. The proof
proceeds in four steps: (i) the inscription tensor formalism, which encodes correla-
tions in a rank-r tensor whose contraction rules are fixed by the mathematics, not
by postulate; (ii) an exhaustive enumeration of all nine measurement configurations
for an entangled pair, showing that partial contraction yields 1
2 in every case, in-
dependent of the distant party’s choice; (iii) the cross-term kill theorem, proving
that one environmental inscription produces an exact, immediate, and idempotent
transition from quantum to classical probability structure; and (iv) the proof by
contrapositive via which-path and quantum eraser experiments, confirming both
directions of the theorem against the complete experimental record. Four inde-
pendent confirmations—information-theoretic, conservation-law, operational, and
symmetry-based—close every remaining escape route. The proof requires no new
physics, no new axioms, and no ontological commitment beyond what the existing
formalism and experimental record already contain. We note that the Entropic Lat-
tice Ontology’s fourth spatial degree provides a geometric explanation for why the
proof works—the correlation occupies a spatial degree with no three-dimensional
address—but this theoretical framework is not required for the proof’s validity