Two ways to exclude a classical operational model
Hardy and Chiribella–D’Ariano–Perinotti (CDP) identify different extra premises: reversible connectivity of pure states, and purification of mixed states. Their classical alternatives are operational probability models. Applying either reconstruction to mechanics first requires specifying which preparations, transformations and measurements constitute a closed system description. A dimensional action parameter then needs a dynamical bridge.
Q01 first milestone, 2026-09-13. This is a dependency audit of selected primary passages, with source coverage in B67. The full reconstruction proofs remain outside this milestone.
Premise-to-use map
| Dependency | Hardy, version 4 | CDP, version 3 |
|---|---|---|
| Operational class used here | Finite distinguishability capacity N and finite probability-coordinate count K; preparations, measurements, reversible transformations and composition | Finite-dimensional state/effect spaces defined through operational statistics; tests, transformations and composites |
| Additional premise | Any two pure states admit a continuous reversible connection within the system | Every state has a pure extension; fixed purifying systems give uniqueness up to their reversible transformations |
| Classical failure | A finite simplex has isolated pure states and reversible permutations | Pure states of the standard classical composite are products, whose marginals remain pure |
| Other assumptions retained | Probability, simplicity, subspace and composition axioms | Causality, perfect distinguishability, ideal compression, local distinguishability and pure conditioning |
| Action step | Match transformations to physical time, energy and mechanical action | The same dimensional identification is needed after state/composite reconstruction |
Hardy §1 states the axiom package; §§4 and 7 explain the classical failure. CDP §II specifies finite statistical dimension, and §§III.1–III.2 locate the classical-compatible principles and purification. These are conditional reconstructions, not assumptions already supplied by the receiver model. Hardy, CDP.
What the classical alternatives actually fail
For the finite simplex, a reversible affine map permutes its vertices. A continuous path starting at one vertex cannot reach a different vertex while remaining in that finite set. This is the elementary content of Hardy’s classical exclusion; the rest of his axiom package matters for selecting the quantum alternative rather than merely rejecting the simplex.
For a classical joint probability table, a pure normalized state is concentrated at one pair of labels. Summing over either label still gives a concentrated state. A mixed marginal therefore has no pure classical extension. Already the existence clause of purification fails; uniqueness adds an independent requirement to the reconstruction. A correlated classical ensemble can encode ignorance, but remains mixed globally. These are the standard elementary classical comparisons identified in the source passages, not new no-go theorems.
The Newtonian connection is closure during a transformation
Hardy §7 explicitly motivates continuity with a ball moving between the two boxes representing a bit. Its intermediate positions exceed the two-state description. This supports a more precise question than whether classical motion is continuous: can the operational description be finite-capacity, exact and closed under the entire reversible transformation?
Project inference: ordinary phase-space motion and a finite bit description operate at different descriptive levels. A proposed classical derivation must justify its measurement restrictions and transformation closure together. Suppressing intermediate distinctions by coarse-graining alone does not prove that the resulting transformations are reversible on the reduced states. Likewise, calling a global mechanical microstate definite does not establish CDP’s operational pure extension of a locally mixed preparation. Its allowed effects, composition and operational equivalence must be specified.
This preserves the useful physical motivation in Hardy while keeping the ambient-model dependency visible. The paper’s countably infinite extension is not audited here; this finite-simplex comparison is not a rejection of every continuous classical phase-space theory.
What would supply the action unit?
The selected axioms concern probabilities, distinguishability, composition and transformations. The outstanding project step is to relate their transformation parameters to measured mechanical time and energy, and then to a phase accumulated per unit action. The checkerboard model supplies this relation through its chosen K, amplitudes and measurement rule; see the maintained comparison. This audit identifies that input rather than deriving a positive value or universal lower bound.
Strategic consequence
Retain both reconstruction packages as candidate premises. The next quantum decision is whether an independently motivated operational closure/composition principle forces one of them for the admitted mechanical class. K01 is useful only if its context/measurement assumptions decide that question. Further receiver calibration coefficients do not currently resolve it.
The bounded passage milestone is complete. Subsequent C125–C128 tests now locate a concrete failure of the finite operational premise: the reversible spin interaction cannot close on any finite enlargement of the retained observables for all preparations; see the finite-closure proof. C129/B77 accepts the initial generator restriction; STATE selects the next bounded technical dependency. Full reconstruction proofs remain separately unaudited.