# Quantum Reference Frames and Single-Outcome Dynamics — A Compatibility Framework for Generative Presentism Part 03 of 40 (1,772 words) 3.1 The Perspective-Neutral State Space Let G be a finite group with identity e. Consider M + 1 subsystems labelled 0, 1, …, M, each with Hilbert space ℓ²(G), the space of complex functions on G, with orthonormal basis {|g⟩ : g ∈ G}. The basis state |g⟩ represents the subsystem having “orientation” g, for example being at site g of a lattice. This setting is chosen because every statement in it can be proved exactly. Definition 3.1 (QRF system). The kinematical space is H_kin = ℓ²(G)^{⊗(M+1)}. The group acts diagonally by U_h |g_0, g_1, …, g_M⟩ = |h g_0, h g_1, …, h g_M⟩, and the physical, or perspective-neutral, space is the invariant subspace H_phys = { Ψ ∈ H_kin : U_h Ψ = Ψ for all h ∈ G }, with projector Π = (1/|G|) Σ_{h∈G} U_h. A state Ψ in H_phys assigns amplitudes only to relations among subsystems: its wave function satisfies Ψ(h g_0, …, h g_M) = Ψ(g_0, …, g_M) for all h. 3.2 Frame Reductions Definition 3.2 (Frame reduction). Choosing subsystem A as the reference frame, the frame-relative space is H_A = ℓ²(G)^{⊗M}, describing the other M subsystems relative to A. The reduction map R_A : H_phys → H_A conditions the frame on the identity orientation and rescales: (R_A Ψ)(x_1, …, x_M) = √|G| · Ψ(e, x_1, …, x_M) (frame A taken as subsystem 0). Its inverse embeds a frame-relative state into the invariant sector: R_A^{−1} φ = |G|^{−1/2} Σ_{a∈G} Σ_{x} φ(x_1, …, x_M) |a, a x_1, …, a x_M⟩. The same definitions apply with any other subsystem as the frame, with the remaining subsystems listed in a fixed order. Proposition 3.3 (Reductions are unitary). For every subsystem A, R_A is a unitary map from H_phys onto H_A, with inverse given above. Proof. Let Ψ ∈ H_phys. Invariance gives Ψ(g_0, g_1, …, g_M) = Ψ(e, g_0^{−1} g_1, …, g_0^{−1} g_M). Write φ(x) = Ψ(e, x_1, …, x_M). Then ‖Ψ‖² = Σ_{g_0} Σ_{x} |φ(x)|² = |G| ‖φ‖², so ‖R_A Ψ‖ = √|G| ‖φ‖ = ‖Ψ‖. Conversely, for any φ ∈ H_A the vector R_A^{−1} φ defined above is invariant (left multiplication by h permutes the summation index a), and R_A R_A^{−1} φ = φ by direct substitution. Hence R_A is an isometry onto H_A, that is, unitary. □ Definition 3.4 (Valid frame change). For frames A and B, the frame change is V_{A→B} = R_B R_A^{−1} : H_A → H_B. Proposition 3.5 (Groupoid law). V_{A→B} is unitary; V_{A→A} = 1; V_{B→A} = V_{A→B}^{−1}; and V_{B→C} V_{A→B} = V_{A→C}. Proof. Immediate from Proposition 3.3 and Definition 3.4: V_{B→C} V_{A→B} = R_C R_B^{−1} R_B R_A^{−1} = R_C R_A^{−1}. □ Explicit form. If A is subsystem 0 and B is subsystem 1, then frame-A coordinates are x_j = g_0^{−1} g_j and frame-B coordinates are g_1^{−1} g_j. Substituting, g_1^{−1} g_0 = x_1^{−1} and g_1^{−1} g_j = x_1^{−1} x_j, so V_{A→B} |x_1, x_2, …, x_M⟩ = |x_1^{−1}, x_1^{−1} x_2, …, x_1^{−1} x_M⟩, where on the right the first slot is subsystem 0 (frame A) described relative to B. V_{A→B} permutes basis states, so a superposition over x_1 in frame A becomes, relative to B, a correlated state of frame A and all other subsystems. This is the familiar QRF effect: a frame-A product state can be a frame-B entangled state (§6.3). 3.3 Invariant Operators and Their Frame Forms An operator O on H_kin is invariant if U_h O = O U_h for all h. Invariant operators map H_phys to itself. Definition 3.6 (Frame form). The frame-A form of an invariant operator O is O^A = R_A O|_{H_phys} R_A^{−1}, an operator on H_A. Lemma 3.7 (Intertwining). For every invariant O and all frames A, B: V_{A→B} O^A V_{A→B}^{−1} = O^B. Moreover, for every Ψ ∈ H_phys, ‖O Ψ‖ = ‖O^A R_A Ψ‖ and ⟨Ψ, O Ψ⟩ = ⟨R_A Ψ, O^A R_A Ψ⟩. Proof. V_{A→B} O^A V_{A→B}^{−1} = R_B R_A^{−1} R_A O R_A^{−1} R_A R_B^{−1} = R_B O R_B^{−1} = O^B. The second statement holds because R_A is unitary and O^A R_A = R_A O on H_phys. □ Remark 3.8 (Every frame-A operator is a frame form). Because R_A is unitary, every operator X on H_A is the frame-A form of an invariant operator, namely R_A^{−1} X R_A on H_phys extended by zero on the orthogonal complement of H_phys. (This operator commutes with every U_h because U_h acts as the identity on H_phys and preserves its complement.) The invariance requirement therefore does not restrict what a single frame can say. It restricts the relation between what different frames say: if frame A uses X, frame B must use V_{A→B} X V_{A→B}^{−1}, not an operator that merely looks the same in B’s coordinates. Section 5.3 turns this into the validity condition for frame-relative laws, and §8 shows numerically that violating it produces a different world. Remark 3.9 (Covariance is built into the framework, not discovered). By Lemma 3.7 and Remark 3.8, any law that is well defined on H_phys has frame forms related by the frame-change unitaries. Frame covariance of the dynamics below is therefore a consequence of the perspective-neutral framework, not an independent finding. The substantive assumption is that the universe is closed, so that laws act on H_phys rather than on H_kin with an external frame. The contribution of this preprint lies elsewhere: in showing that a CTGP interpretation can be placed on such dynamics consistently, in proving what that interpretation implies, and in identifying what CTGP adds (§11). ---------------------------------------------------------------------------------------------------- 4. Dynamics, Ontology, and the Complete State ---------------------------------------------------------------------------------------------------- 4.1 The Generative Law Stages are indexed by σ in a discrete, linearly ordered set, written σ_0 < σ_1 < σ_2 < …, with σ′ denoting the successor of σ. The discrete index is used in Part II because every statement about it can be proved by finite induction; §9 gives the continuous-time form and states how the two are related. Definition 4.1 (Admissible generative law). An admissible generative law consists of an invariant unitary W on H_kin (the deterministic part of one stage-to-stage transition) and an instrument {F_k : k ∈ 𝒦} of invariant operators on H_kin with Σ_k F_k† F_k = 1. The outcome set 𝒦 is countable, and its elements are called transition events. Definition 4.2 (Generative kernel). For a normalized Ψ_σ ∈ H_phys, the transition to σ′ is the random pair (k, Ψ_{σ′}) with Pr(k | Ψ_σ) = ‖F_k W Ψ_σ‖², Ψ_{σ′} = F_k W Ψ_σ / ‖F_k W Ψ_σ‖. The kernel K(Ψ_{σ′}, k | Ψ_σ) is the joint distribution so defined. Because W and F_k are invariant, Ψ_{σ′} lies in H_phys again. States are rays: Ψ and e^{iθ} Ψ represent the same state, written Ψ ≐ e^{iθ} Ψ. A history of length n is a sequence ω = (k_1, …, k_n) of transition events. Its probability given Ψ_{σ_0} is the product of the successive transition probabilities, equivalently ‖F_{k_n} W ⋯ F_{k_1} W Ψ_{σ_0}‖². 4.2 Primitive Ontology: What Exists at a Stage A stochastic outcome label is not by itself an ontology. The following reading states what the formalism is taken to represent. Ontological Reading 4.3 (Primitive ontology). At stage σ, the present world consists of: (i) the present physical state [Ψ_σ], a ray in H_phys. It is physically real and governs the probabilities of all future events, but it is not itself material stuff located in space; it is a holistic state of the relational degrees of freedom; (ii) the present world’s past-tensed profile L_σ (§4.4). The material ontology consists of transition events. A transition event is a primitive occurrence: something that happens, once, in the generation of a stage from its predecessor. When it occurs it is part of the generating present; at every later stage it is something that was the case, recorded in L and carried forward causally through its effect on Ψ. Macroscopic facts, such as where a pointer points, are patterns of transition events. This is the flash ontology of GRW-type theories, with the wave function read realistically as part of the present physical state. An alternative reading treats the wave function as nomological, so that the events are the only physical ontology. On that reading, Tier 2 of the determinacy criterion (§5.1) would concern the law’s current parameter rather than a physical state, but Tiers 1 and 3 and every theorem below would be unchanged. Reading 4.3 is adopted because CTGP identifies S_σ as the present physical state governing successor generation (C-f). 4.3 Primitive Events Versus Sharp Relational Properties Each transition event k has an associated effect E_k = F_k† F_k, which is a positive operator on H_phys. The effect determines the probability of the event: Pr(k | Ψ_σ) = ⟨W Ψ_σ, E_k W Ψ_σ⟩. The event and the effect are different kinds of thing. The effect is part of the law. The event is an additional fact: that, of all the events the law made possible, this one occurred. That additional fact is what single-outcome realism (C-d) asserts and what Everettian ontology denies. In particular, an event concerning a relational quantity need not be the possession of a sharp value of that quantity. In the model of Part III, an event (i, j, d) is a flash centred on the separation d between particles i and j, with finite width s. Its occurrence does not make the proposition “the separation of i and j is d” true. Proposition 4.4 (Finite-width events do not create sharp values). Let F_k = Σ_r c_k(r) P(r) where {P(r)} is a complete family of orthogonal invariant projectors and c_k(r) ≠ 0 for every r. If, before the event, W Ψ_σ is not an eigenvector of a given P(r), then after the event Ψ_{σ′} is not an eigenvector of P(r) either. Proof. Since F_k is diagonal in the decomposition {P(r)} with nonzero coefficients, P(r′) Ψ_{σ′} ∝ c_k(r′) P(r′) W Ψ_σ is nonzero exactly when P(r′) W Ψ_σ is nonzero. The set of r′ with nonzero component is therefore unchanged by the event, and a vector is an eigenvector of P(r) (with eigenvalue 1 or 0) exactly when that set is {r} or omits r. □ So in the finite-width model the event is a Tier 1 fact, while the sharp-value proposition about separation remains without a determinate value (Tier 3 of §5.1). The sharp limit s → 0, in which c_k(r) vanishes for r ≠ d, is the special case in which an event does produce a sharp value. This distinction answers the objection that a flash ontology merely renames a measurement outcome: the event is not the readout of a pre-existing value, and not the acquisition of a sharp one, but a primitive occurrence whose probability the law fixes.