This note checks, one axiom at a time, whether the change-substrate of Graphic Equalisation supplies the operational axioms from which the informational reconstructions of quantum theory derive the complex Hilbert space and the Born rule. The papers named above carry the surrounding framework. The honest verb throughout is recover, not derive: the substrate reproduces the known rule within its own structures, given its admitted primitives and a contingent, self-located selection, rather than compelling it from necessity. Each step is rated solid, earned, plausible or contingent, and the residuals are named rather than hidden. Two findings carry the note. First, the substrate instantiates the operational-probabilistic core — convex states, affine effects, a transformation monoid with reversibles, composition, normalised probabilities — from the classical total-probability backbone and not from any quantum assumption. Second, two independent reconstructions have two different most-quantum axioms, and the substrate supplies each natively from a different one of its own principles: Hardy’s continuity of reversible transformations from free continuation, and the purification of Chiribella, D’Ariano and Perinotti from entanglement as a superframe dimension. Complex over real is then not a gap to be closed by a forcing argument but the signature of a contingent selection, self-located exactly as the number of spatial dimensions is (the physics companion’s Proposition 14.2). The only claim made here is the arrangement — that the two crux axioms above are already-present substrate structure; each component is individually known, and the uniqueness is supplied by the cited theorems.
Seven notions carry the mapping below. Each is defined in one of the three papers named above; this section fixes the references and adds nothing. A frame is a subgraph that runs (Graphic Equalisation’s Definition 6.2), and frames compose by nesting, a graph being a subset of its container (its Proposition 6.6). A frame’s state is its Narrative, the standing structure it carries forward (the momentum paper’s Definition 2.2), and where a frame instantiates a direction — optional in general, present in this setting — that Narrative has a magnitude and a direction (the momentum paper’s Definition 4.1); the state proper is the internal Narrative, as against what another frame reads off it (its Definition 5.3). A closure cycle is the loop a realisation must close, the cycle being the unit of inferred time (Graphic Equalisation’s Proposition 13.7), and phase is position on that cycle rather than a signed time (its Remark 13.8). A realisation is a collapse, a local write that picks one outcome from the admissible set and carries a weight (Graphic Equalisation’s Definition 4.1), and realisations are strictly ordered, the order carried by the rules rather than by a clock (its Proposition 11.2). Free continuation is what a frame does when no rule fires, the information-preserving default (the momentum paper’s Definition 5.2); free means information-preserving and not absent, since evaluating a rule is itself a write (Graphic Equalisation’s Proposition 7.16), and all the mapping below uses is that free continuation is injective, hence reversible. Nothing is privileged: there is no god’s-eye frame, and therefore no privileged phase origin and no privileged measurement basis (its Axiom -1). Entanglement is a superframe dimension: a subframe’s local state is mixed because the purity is a dimension of the larger frame (the physics companion’s Proposition 6.1 and its Corollary 6.2).
The substrate recovers the Born rule — reproduces it within its own structures, given its admitted primitives and a contingent, self-located selection — rather than deriving it as a forced necessity.
Remark 2.1 (Recovery is the honest verb, and the right one). Derivation per se would compel Born from primitives. The framework forces almost nothing, and that is the point of a graph substrate: what survives every attempted removal is a container relation across which rules run (Graphic Equalisation’s Remark 3.2), and essentially everything else is contingent rule-content, as with the number of spatial dimensions (the physics companion’s Section 15) or the gauge group (its Remark 11.3). A contingent law can only be recovered given its rule-content, never derived as compelled. Some steps below are forced given the premises — the squared modulus, from no-privilege together with the additivity of decohered alternatives — while the premises themselves are admitted and contingent: complex, linear, self-similar.
Every distinctively-quantum ingredient is supplied by a substrate principle; the cited reconstruction theorems supply the uniqueness (Gleason, 1957; Hardy, 2001; Masanes and Müller, 2011; Chiribella et al., 2011). The only real claim is the arrangement — each component is individually known. That sizing is not modesty about the result but a statement of what kind of result it is, and it governs every rating below.
Before any reconstruction axiom applies, the substrate must be an operational-probabilistic theory (OPT): states in a convex set, effects, transformations, and a composition rule. That framework is a precondition for either axiom set, so the mapping has two levels, and this is the lower one.
The key that makes it tractable is seeing what the OPT convex structure is. The convexity of the state space and the affinity of effects are the classical total-probability backbone — total probability over ordered branchings — distinct from, and prior to, the quantum superposition linearity that Section 4 worries about. Every probabilistic theory has this backbone. The substrate has it by construction, because realisations carry weights, compose in order, and depend only on the present.
Proposition 3.1 (The substrate instantiates the operational-probabilistic core). Convex states, affine effects, a transformation monoid with reversibles, composition, and normalised probabilities are each supplied by the substrate’s weighted, ordered realisations together with the history-optional, present-structure-acts first law (the momentum paper’s Section 6; Graphic Equalisation’s Corollary 12.5). The instantiation rests on the classical total-probability backbone and on no quantum assumption.
Ingredient by ingredient, with the substrate structure that supplies each. Four ratings are used here and in Section 4: solid, the substrate structure is the requirement’s statement; earned, derived here from a substrate principle rather than read off one; plausible, a commitment of the right kind, with statement-match not shown; contingent, rule-content, admitted and self-located rather than supplied.
OPT requirement | Supplied by | Rating |
Systems | Frame types: a subgraph together with an outcome set. | solid |
States are well-defined equivalence classes | Statistics depend only on the current Narrative — history is optional and present structure acts, so there is no preparation-memory and operational equivalence holds. | solid |
Convex state space | Mixing two preparations by a weighted coin-realisation: realise a λ-weighted coin, prepare P on one branch and Q on the other. | solid |
Affine effects | Total probability over the coin–prepare–measure branching, e(λP + (1 −λ)Q) = λe(P) + (1 −λ)e(Q). | solid |
Transformations (monoid, with reversibles) | Rules compose, in order; free continuation is the identity, information-preserving and hence reversible. | solid |
Composition | Subframe nesting; a joint preparation lives over the product outcome set. Existence is Level 0; the complex-specific tomography is the contingent layer of Section 5. | solid |
Probabilities | Realisation weights in [0,1], normalised over a complete measurement. | solid |
Remark 3.2 (The affinity is not the superposition linearity). The affinity of effects is the law of total probability applied to ordered realisations, and nothing about it is quantum. It is worth marking because the two linearities are easy to run together, and the argument would be circular if they were: Level 0 earns the classical backbone, and Section 4 must earn the superposition linearity separately, on other grounds.
What remains at this level is hygiene rather than core: a finite tomographic dimension, in the sense that a state is fixed by finitely many fiducial effects — plausible for finite frames by the aperture, unproven in general; topological closure and compactness of the convex state space; the full composition axioms, associativity of sequential and parallel composition and the interchange law, which should follow from the ordering and nesting of realisations but each need checking; and the no-restriction and effect-realisability niceties. None of these is the question whether the state space convex-linearises at all, which Level 0 answers; they are carried forward in Section 7.
Two independent reconstructions are used. Hardy (2001) gives five axioms, of which classical probability theory satisfies the first four and fails the fifth. Chiribella et al. (2011) — Chiribella, D’Ariano and Perinotti, CDP below — give six principles that single out quantum theory over the real, the quaternionic and the classical. They have different most-quantum axioms, which is what makes the pair worth running rather than either alone.
Proposition 4.1 (Each reconstruction’s crux axiom is supplied natively, from a different substrate principle). Hardy’s quantum-selecting axiom is the continuity of reversible transformations between pure states; the substrate supplies it from free continuation, which is information-preserving and hence reversible, and whose phase is a continuous U(1), so it moves continuously between pure states — this note reads an amplitude as rei𝜃, modulus and cycle-phase. CDP’s characteristically quantum principle is purification; the substrate supplies it from entanglement as a larger-frame dimension, a mixed subframe state being the restriction of a pure superframe state. Neither is borrowed from the other, and neither was introduced for this purpose.
That two separate reconstructions’ crux axioms are both already-present substrate structure, from two different principles, is the substantive positive finding of this note. The remaining correspondences are weaker and are marked as such:
Axiom | Substrate structure | Rating |
Continuity of reversible transformations (Hardy; classical theory fails exactly this) | Free continuation: information-preserving, hence reversible, with a continuous cycle-phase. | solid |
Purification (CDP; fails classically) | Entanglement as a superframe dimension: a mixed subframe state is the restriction of a pure superframe state. | solid |
Phase- and context-independence | No privileged phase origin and no privileged frame: probability is invariant under global phase and the same across measurement bases. | earned |
Probability | Weighted realisations; frequencies over an ensemble of collapses from one possibility structure. | plausible |
Simplicity and ideal compression | Minimum description length as a native commitment (Graphic Equalisation’s Remark 2.4) (Rissanen, 1978). | plausible |
Subspaces | Subframe-as-frame: restriction to a subset of distinctions behaves as a smaller frame. | plausible |
Perfect distinguishability | Distinctions are first-class; a state that is not completely mixed carries one. | plausible |
Pure conditioning | Collapse of a pure Narrative yields a definite realised outcome. | plausible |
Local tomography (Hardy’s composite axiom; CDP’s local distinguishability) | Subframe composition over the product outcome set — contingent, and the subject of Section 5. | contingent |
The plausible ratings are genre-matches rather than statement-matches: the substrate has a commitment of the right kind, and that the specific axiom is the one the substrate enforces is not shown. Marking them so is the point of the table; a reader who upgrades one of them is disagreeing with a rating, which is the intended form of the disagreement. The table is a correspondence map, not a proof that the operational axioms follow from the substrate: “solid” marks a direct structural match, while “earned”, “plausible”, and “contingent” mark respectively an argument still carrying a premise, a right-shaped identification, and admitted rule-content. Only the cited reconstruction theorems establish their own axiomatic consequences once their premises are granted.
The squared modulus is selected among the natural candidates by phase-averaged additivity, and made unique by Gleason, in the physics companion (its Proposition 7.1 and the argument following it); this note uses the result. What the framework supplies to Gleason (Gleason, 1957) are his premises — a vector-valued state, from the linearity step, and a context-independent measure, from the absence of a privileged rule set — and Gleason supplies what the framework does not.
Remark 4.2 (Two supporting identifications). The inner product is the cross-frame projection: one frame’s Narrative read in another’s basis, so it is not imported structure but the framework’s own cross-frame reading. Unitarity is that a free continuation destroys no distinction — an information-preserving write rather than the absence of one, which is the right shape, since unitary evolution plainly does change the state and is reversible for exactly that reason. Both are identifications rather than derivations, and the rigorous step from “uncoupled” to full additivity and norm-preservation is the standing residual of Section 7.
Local tomography is the axiom that selects complex Hilbert space over real, and it is where a forcing argument would have to sit. Writing K for Hardy’s fiducial degrees of freedom including normalisation, for two two-state systems:
| Theory | K single | K composite | KA⋅KB | Local tomography |
| Classical | 2 | 4 | 4 | holds (but fails continuity) |
| Real (rebit) | 3 | 10 | 9 | fails (10 > 9) |
| Complex (qubit) | 4 | 16 | 16 | holds |
A closure-phase frame is consistent with complex amplitudes at one system — the Bloch sphere, K = 4, rather than the real disk’s K = 3 — though a phase-less frame is equally constructible and the substrate requires neither. Read that way, a superframe, being a frame, is a complex closure-phase Narrative over the product outcome set, giving KAB = 16: local-tomographic, complex.
Proposition 5.1 (Complex over real is a self-located selection, not a forced consequence). Complex-over-real is contingent rule-content, as with the number of spatial dimensions. Local tomography is therefore not a gap to be closed by a forcing argument; it is the signature of a contingent selection. A real-amplitude composition, KAB = 10, is another admissible configuration. Our universe is the complex one, self-located (the physics companion’s Proposition 14.2).
Remark 5.2 (Why forcing is the wrong question). Whether the substrate forces the complex composite is not the open question, and asking it would be an error of the framework’s own kind rather than merely an unproven step. Seeking to force complex over real would contradict the framework’s stance that it never forces contingent physics. The reading that survives is narrower and does real work: the admissible complex configuration yields KAB = 16 and the admissible real one yields 10; a real composite would be a superframe that is not a closure-phase frame like its subframes, so real amplitudes cost self-similarity. That gives a direction and a reason without giving a compulsion, and it is the same status every other contingent constant in this programme has.
Given the contingent choice, the Born rule and the Hilbert structure follow via the mapping and the cited reconstructions — which is how the framework treats all physics: shapes follow from contingent rule-content, the rule-content itself never forced. Of the post-2011 reconstructions, Masanes and Müller (2011) obtain finite-dimensional quantum theory from five physical requirements on states and transformations, and theirs is the formulation this note would most like tested against, since those requirements are stated on single systems plus a composition rule — exactly the seam where the substrate’s answer turns contingent rather than forced. Independent support for reading the selection as contingent comes from Höhn (2017) and Höhn and Wever (2017), whose rules on an observer’s acquisition of information yield the correlation structure of qubits and rebits alike and so do not by themselves select complex over real. The rest of the reconstruction landscape is the physics companion’s (its Section 17).
Special and general relativity reconcile on this substrate as one change-budget spent two ways, with the invariance of c a definitional consequence rather than a postulate, and quantum collapse and relativity are joined at c as the in-frame event and the cross-frame map of one change-substrate (the physics companion’s Proposition 4.1, its Corollary 4.2, and its Section 4.1). With Born recovered on the same primitives, Born, SR and GR stand reconciled on one substrate — the reconciliation is that companion’s, argued in its Section 7.2, and it is a reconciliation of the conflict at the level of shape, not a quantum theory of gravity: no field is quantised, no metric derived, no dimensionless constant fixed.
What is open, unaddressed, and named:
Each commitment below is marked [shape] or [detail]. A [detail] failing is a revision: the account survives with that mechanism replaced. A [shape] failing costs range rather than a mechanism, because what fails is something the substrate is claimed to require or to permit — so the account does not hold where it claimed to, which is the more serious of the two and is still a boundary rather than an annihilation.
Scope note (not a falsifier). This note is shape-level and not quantitative: it fixes no dimensionless constant, derives no field equation, and offers no theory of quantum gravity. Nor is it a derivation: the claim is the arrangement, and no step above is offered as compelling the Born rule from primitives.
Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. Informational derivation of quantum theory. Physical Review A, 84(1):012311, 2011.
Andrew M. Gleason. Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics, 6:885–893, 1957.
Lucien Hardy. Quantum theory from five reasonable axioms. arXiv:quant-ph/0101012, 2001.
Philipp A. Höhn. Toolbox for reconstructing quantum theory from rules on information acquisition. Quantum, 1:38, 2017.
Philipp A. Höhn and Christopher S. P. Wever. Quantum theory from questions. Physical Review A, 95(1):012102, 2017.
Lluís Masanes and Markus P. Müller. A derivation of quantum theory from physical requirements. New Journal of Physics, 13(6):063001, 2011.
Jorma Rissanen. Modeling by shortest data description. Automatica, 14(5):465–471, 1978.