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).
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4. Dynamics, Ontology, and the Complete State
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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.