Background

The von Neumann measurement chain — the regress of entanglement that arises when system, apparatus, environment, and observers interact unitarily — has lacked a formal, non-ad-hoc termination criterion for nearly a century. Copenhagen imports an undefined cut; objective-collapse models add stochastic parameters that modify linear dynamics; decoherence displaces, but does not terminate, the chain; relational and Everettian accounts deny that termination is needed but supply no criterion for when a definite relational fact obtains. We are not aware of a prior account that states a structural criterion for which system terminates the chain and attaches a falsifiable laboratory signature to it; the nearest neighbour, Rovelli’s relational quantum mechanics, relationalises facts but provides no termination criterion, no algebraic structure for the terminating system, and no empirical discriminator (Table 1.2).

Gap

Missing: (a) a definition of the terminating system that does not import classical primitives; (b) an algebraic account of why the terminating system’s internal frame is inaccessible from outside; (c) a dynamical (not anthropocentric) derivation of the preferred pointer basis; (d) a laboratory discriminator separating the proposed termination from ordinary environmental decoherence.

Approach

The chain is analysed under Algebraic Quantum Mechanics: a Quantum Reference Frame (QRF) transformation into the internal frame of a candidate observer [D1, conditional]; a Type III₁ modular characterisation of the internal reference algebra with a CPTP instrument replacing prohibited projections [D2]; the Absolute Self-Origination (ASO) condition, stated for the first time as a consolidated three-part definition (§2.1); and driven-dissipative dynamics whose active feedback binds the Liouvillian to an Exceptional Point at g_C = γ/2 [D3]. The informational privacy of the internal frame rests on the IRM trace-norm bound, whose status — a programme-internal theorem awaiting independent re-derivation — is stated explicitly throughout [L1].

Results

(1) The QRF transformation yields a separable internal relata with zero local von Neumann entropy, preserving global linearity [D1]. (2) The modular CPTP instrument gives the coherence decay ρ₊₋(t) = ρ₊₋(0)·e^{−2g²γt} (dimensional bookkeeping corrected in this revision) [D2]. (3) The Jordan block at g_C = γ/2 produces the secular envelope t·e^{−γt} — the cluster-shared Class A signature [D3, E1]. (4) ASO-distinctive flagship prediction [E2, PREDICTION]: for any system satisfying the ASO condition, the environmentally einselected pointer basis must coincide with the eigenbasis of the internal feedback Hamiltonian Ĥ_ctrl = gσ̂ₓ, independently of the environmental coupling — a coincidence ordinary decoherence theory gives no reason to expect; Revision 3.0 derives the coincidence at leading perturbative order via Zeno screening [D4, Appendix A], making the fidelity curve F(λ) a registered functional-form test. (5) Wigner’s Friend, the preferred-basis problem, and the PBR boundary are addressed with explicit identification of which assumption is denied in each [§7].

Implications

If the two-branch laboratory programme of §9 confirms — the sigmoidal EP transition (shared Class A) and the pointer-basis coincidence (ASO-distinctive) — the chain terminates at thermodynamically self-stabilising systems as a matter of algebraic structure, not interpretation [D1–D3, E1–E2]. The Lawvere/terminal-coalgebra reading of the privacy bound is classified a structural conjecture [C2], and the framework makes no claim about consciousness: the SOO criterion is structural and thermodynamic [§11, L7]. Falsification is symmetric and immediate: both branches are pre-registered with decision rules, and a pointer-basis mismatch alone falsifies the ASO selection mechanism.