4.4 The Complete Ontological State and the Truth Profile
Definition 4.5 (Complete ontological state). The complete ontological state at stage σ is
Ω_σ = (S_σ, L_σ),
where S_σ = [Ψ_σ] is the ray of the perspective-neutral state, and L_σ, the Lucretian truth profile,
is a set of past-tensed propositions of two kinds:
Had(σ″, [Ψ_{σ″}]), read “at stage σ″ the state was [Ψ_{σ″}]”, for σ″ < σ;
Had(σ″ → σ‴, k), read “in the generation of σ‴ from σ″, transition event k occurred”, for σ‴ ≤
σ.
The second kind extends the stage-indexed properties Had(σ′, φ) of CTGP – explained, Definition
17.20, to transitions. This adds no commitment: a transition from σ″ to σ‴ is something that
occurred, and the present world can have been such that it occurred.
Definition 4.6 (Truth-profile update). When the transition σ → σ′ realizes event k,
L_{σ′} = L_σ ∪ ΔL_{σ→σ′}, ΔL_{σ→σ′} = { Had(σ, [Ψ_σ]), Had(σ → σ′, k) }.
The truths acquired in a transition are exactly two: the state that was present has become past, and
the event that occurred in the transition is something that was the case. The initial condition is
L_{σ_0} = ∅ and S_{σ_0} = [Ψ_{σ_0}].
4.5 Admissible Histories
The truth profile is not an arbitrary set of past-tensed propositions. It ranges over a space fixed
by the law.
Histories, actuality, and futurity. Three uses of the word history must be kept apart, because the
mathematics speaks of histories in a sense that carries no ontological commitment. The actual past
history at a stage consists of the states and events that have occurred and are now past. Past
stages do not continue to exist; truths about them are grounded in the present world’s Lucretian
profile. The actual history through the present, where that phrase is needed, adds the presently
existing state to its past; it does not mean that the present has already become past. An admissible
history, by contrast, is a mathematical object: a possible sequence of events permitted by the law,
an element of the probability space defined below. The set of admissible histories lists the
alternatives the law permits. It does not represent alternative histories that exist side by side,
and a probability measure on future trajectories describes possible continuations of the present,
not an existing future.
Within CTGP no state or event exists as future. Each state first exists as the present, and each
realized transition becomes part of the actual past only after it has occurred. The actual past is
therefore not a completed structure containing future events: it grows as successive presents are
generated, without revision of what has occurred. What has occurred cannot later become something
else (C-b), although records, memories, and evidence about it can change, which is the distinction
between truth and recoverability (C-i). Only one history is ever realized. At each stage its actual
past consists exclusively of events that have already occurred, and the other admissible histories
are possibilities represented by the law, not additional existing histories. When the text speaks of
a complete history, completeness is relative to a specified finite stage, or to a trajectory used in
probabilistic analysis; it never means that the future has occurred or exists tenselessly.
Definition 4.7 (Admissible histories and complete states). Fix the law and the initial state
Ψ_{σ_0}. Let
ℋ_n = { ω ∈ 𝒦^n : Pr(ω | Ψ_{σ_0}) > 0 }
be the set of admissible histories of length n, the possible event sequences of nonzero probability.
For ω ∈ ℋ_n write Ψ_m(ω) for the state after the first m events (Ψ_0(ω) = Ψ_{σ_0}), and define its
truth profile
L(ω) = { Had(σ_m, [Ψ_m(ω)]) : 0 ≤ m < n } ∪ { Had(σ_{m−1} → σ_m, k_m) : 1 ≤ m ≤ n }.
The admissible truth profiles and admissible complete states at stage σ_n are
ℒ_n = { L(ω) : ω ∈ ℋ_n }, 𝔒_n = { ([Ψ_n(ω)], L(ω)) : ω ∈ ℋ_n }.
Theorem 4.8 (Characterization of admissible profiles). A set L of past-tensed propositions belongs
to ℒ_n if and only if all of the following hold:
(o) Restricted language. Every element of L is a past-tensed proposition of one of the two forms
of Definition 4.5: a state proposition Had(σ_m, [Ψ]) with m ≥ 0 and Ψ ∈ H_phys, or a transition
proposition Had(σ_{m−1} → σ_m, k) with m ≥ 1 and k ∈ 𝒦.
(a) One event per transition. For each m = 1, …, n, L contains exactly one proposition of the
form Had(σ_{m−1} → σ_m, k).
(b) One state per past stage. For each m = 0, …, n−1, L contains exactly one proposition of the
form Had(σ_m, [Ψ]).
(c) Correct initial state. The state proposition for σ_0 is Had(σ_0, [Ψ_{σ_0}]).
(d) Lawful consistency. For each m = 1, …, n−1, if L contains Had(σ_{m−1}, [Ψ]) and Had(σ_{m−1}
→ σ_m, k), then F_k W Ψ ≠ 0 and the state proposition for σ_m is Had(σ_m, [F_k W Ψ]); and, for
the last event k_n, F_{k_n} W Ψ ≠ 0 where [Ψ] is the state recorded for σ_{n−1}.
(e) Past-stage restriction. Every state proposition in L is indexed to a past stage σ_m with 0 ≤
m ≤ n−1, and every transition proposition is indexed to a transition σ_{m−1} → σ_m with 1 ≤ m ≤
n. Thus L contains no state proposition about the present stage σ_n and no proposition of either
kind about any later stage.
When these hold, the history ω with L = L(ω) is unique. Moreover, admissible profiles are nested: if
ω ∈ ℋ_n and ω|m is its initial segment of length m ≤ n, then L(ω|m) ⊆ L(ω).
Proof. (⇒) Each condition holds by construction of L(ω) and the definition of ℋ_n. (⇐) By (a),
reading off the event proposition for each transition gives a unique sequence ω = (k_1, …, k_n). By
(c) and (d), induction on m shows that the state proposition for σ_m is Had(σ_m, [Ψ_m(ω)]) and that
each factor of Pr(ω | Ψ_{σ_0}) is nonzero, so ω ∈ ℋ_n. By (o), every element of L is a state or
transition proposition. By (e), its state propositions concern only the stages σ_0, …, σ_{n−1} and
its transition propositions only the n transitions up to σ_n; by (a) and (b), each of these carries
exactly one proposition, namely the one just identified. Hence L = L(ω). Uniqueness follows because
ω is read off from L. Nesting holds because the propositions defining L(ω|m) are among those
defining L(ω). □
Theorem 4.8 makes precise the requirements that a truth profile must meet: propositions of the two
permitted forms only, exactly one realized event per transition, state propositions only about past
stages and consistent with the law, no incompatible state assignments, preservation of acquired
truths, and no claim that an unrealized future event has occurred. The restriction of state
propositions to past stages matters: the present state is part of S, not of the past-tensed profile
L, and a set that added a state proposition about σ_n to an admissible profile would satisfy every
other condition without belonging to ℒ_n. Theorem 4.8 establishes that the Lucretian profile is a
consistent interpretation of the model’s realized histories. It does not establish that Lucretian
properties exist in nature; that is a philosophical thesis of CTGP, which the model shows to be
coherent.