Predictive sufficiency and the descent of transport to records

State the future tests before declaring a record complete

Let a family of generative histories H act on microscopic states X. A supplied observation map Q:X→R produces records, and o is a permitted test of those records. A future experiment has the form x↦o(Q(hx)) for an admissible future history h. This notation can describe a deterministic output or a probability distribution over outcomes.

Two microscopic states are behaviorally equivalent when every declared future experiment agrees:

In a probabilistic experiment, equality means equality of the complete predicted outcome law, not that two individual trials happened to have the same result. The definition also makes clear that available preparations and reference resources belong to the test specification. Including a coherent reference path can distinguish histories that an intensity-only detector cannot. §2.1, p.3.

The inner/outer terminology supplies an interpretation of microscopic generation and accessible completion. It does not determine H, Q, or o. A model that changes the available tests has changed the operational meaning of a complete record, even if it keeps the same underlying microscopic state space.

Why this is the coarsest test-preserving equivalence

Each test f:X→Y defines an equivalence relation f(x)=f(y). The intersection of all such test kernels is again an equivalence relation. Any relation that preserves every test must be contained in that intersection. Behavioral equivalence therefore identifies as many states as possible while preserving all declared predictions.

Now propose another record map q:X→Y. Sufficiency requires every permitted test to factor through q, so

The direction matters. A sufficient record may preserve distinctions that no permitted test uses. For example, a record containing the full microscopic state can be sufficient but unnecessarily detailed. The minimal behavioral quotient X/∼ removes exactly the distinctions that the tests cannot resolve. A coarser map that identifies a distinguishable pair is insufficient. Proposition 2.1, p.3.

This is an exclusion principle rather than a selection mechanism. It can reject an endpoint-only completion after a separating experiment is exhibited, but it does not automatically construct a finite-dimensional Hilbert carrier or a physically realizable storage channel. Those require additional structure and a rule that implements the sufficient record.

An update descends only when it respects the equivalence

An update k acts unambiguously on the quotient if x∼y implies kx∼ky. This is the descent condition: equivalent representatives must have equivalent updates. It follows from behavioral equivalence when the future-history class is closed under right composition by k. A test of kx has the form o(Q(hkx)); closure says that hk is already among the future tests used to compare x with y.

Without that closure, two states may agree for every currently admitted future but separate after the proposed new update. The claimed quotient dynamics would then depend on which hidden representative had been chosen. A completion program must specify the admissible operations as carefully as the output map. Proposition 2.1, p.3.

For a group action, there is a further constraint. If every element and every inverse descends through a surjective record map, the descended transformations remain mutual inverses. A many-to-one observation map alone does not create irreversible quotient evolution. Irreversible completion must restrict the operation family, prevent inverse descent, discard access to an environment, or apply a specified operational projection. Proposition 2.2, p.4.

Construct a future experiment that separates two histories

Let γ₁,γ₂:x→y have identical endpoint structural descriptions but unitary transports U₁,U₂:V_x→V_y. Prepare a normalized ψ, place the two histories in separate coherent control paths, and recombine them with calibrated relative phase φ. The plus-port probability is

If U₁≠U₂, the unitary W=U₂†U₁ has an eigenvector ψ with eigenvalue e^iθ≠1. For φ=0, that input gives p₊=(1+cosθ)/2<1, whereas comparing γ₂ with itself gives one. A declared future test has separated the histories. A sufficient record cannot therefore identify them solely from their common endpoint structures. Theorem 2.3, p.4.

This proof is frame independent. Both transports acquire the same endpoint factors, W transforms by conjugation at x, and ψ changes with the base frame. The probability is unchanged. The coherent control and calibrated reference are physical parts of the test, not consequences of passive frame freedom. Restricting them can weaken the conclusion.

What it means for transport to descend to a completed record

Suppose a completion map π identifies some comparison histories, with their boundary carriers already specified. A transport assignment descends to these completed comparison classes exactly when π(γ₁)=π(γ₂) implies U_γ₁=U_γ₂, interpreted as intrinsic maps between the identified boundary carriers. One may then define Ū([γ])=U_γ independently of representative. Composition descends as well when π respects composition and the transport assignment is compositional.

If the retained description stores only endpoints, this condition demands that all admitted histories between those endpoints have the same retained transport. A nontrivial relative holonomy is an obstruction. The interference theorem supplies an operational witness: when its coherent tests are admitted, identifying such histories would change a future probability. Theorem 2.3 and Corollary 2.4, pp.4–5.

The repair is to enrich the record with enough relational information, or to declare a more restricted test family and justify its restriction. The retained information need not be a literal microscopic trajectory. It can be a transport class, a sufficient comparison register, or a prepared reference that reconstructs all the admitted statistics. Calling that enlarged object an “endpoint record” does not remove its relational content.

The test family determines whether central phases survive

Consider the operation ρ↦UρU† on carrier density matrices. Replacing U by e^iθU leaves this channel unchanged. If all future tests use only this conjugation action and no coherent control of alternatives, the scalar phase is operationally invisible. An adjoint-only record can therefore have a different quotient from a defining-carrier transport record with phase-sensitive tests.

The distinction is measurable in the supplied examples. A forced carrier–dual singlet circuit has success |TrW/m|², so W=e^iθI always returns successfully. A controlled comparison against identity instead gives p₋=(1−ReTrW/m)/2=sin²(θ/2), resolving that same phase. §5.2, pp.10–11; Proposition 17.3, p.53.

One must therefore avoid claiming that every physical implementation can distinguish every U(m) element. The theorem says this under its calibrated coherent test assumptions. Conversely, phase blindness of one detector does not prove that the phase may be erased when the full experiment family contains a more sensitive detector. The family, not an isolated measurement, defines sufficiency.

A shared reference must be accounted for explicitly

An endpoint comparison might be supplied by an already prepared joint state or a common reference register. That possibility does not invalidate the no-natural-comparator theorem, which concerns separate structural data alone. It changes the record data available to the experiment.

The exact preservation theorem makes this unavoidable. A unital completely positive channel that fixes both full local carrier algebras also fixes their joint algebra. Independent selector forgetting on separate left factors therefore retains pre-existing right-carrier correlations. An argument that uses such channels to claim that all common frames have been erased would contradict its own operator-preservation assumptions. Proposition 4.7, p.8.

A reference-free preparation class is one legitimate choice; a class with prepared shared references is another. The chosen model must state which is used. A common temporal origin alone implies neither choice. Likewise, transferring a product-volume reference from a coherent reservoir can implement a desired record tensor, but invariant covariant processing cannot create it from symmetric inputs. The preparation cost and its later retention are separate parts of the completion mechanism. §6.2, pp.14–15.

Use sufficiency as a practical model diagnostic

A proposed completion can be assessed by a short sequence of precise tasks: list its admissible future operations and probes; identify its proposed equivalence classes; test whether each update respects those classes; and search for a coherent or correlated experiment that separates any pair the map identifies. When a witness exists, either retain more relational information or explain why the witness is physically unavailable.

Approximate completion needs an additional quantitative criterion. A finite detector tolerance can motivate bounded differences of outcome distributions rather than exact equality, but an approximate relation need not be transitive and error can accumulate under future updates. The source’s exact equivalence theorem does not provide those stability estimates automatically. Its selector semigroup supplies one exact attenuation bound for a particular channel, not a general theory of approximate behavioral compression. Proposition 2.1 and Theorem 4.3, pp.3 and 7.

Once sufficient comparison histories compose smoothly, they can define a connection. That conclusion still leaves carrier ranks, adjacency, curvature selection, field dynamics, and physical matching open. The operational theorem sharpens the source problem by stating what a successful completed record cannot forget; it does not derive the rest of the physical theory from the word “completion.”

Continue reading

• Exact completion channels
• From history maps to a connection
• Compare the available experiment families
• Independently presented endpoint carriers
• Shared-reference protocols: what label erasure leaves observable
• Wilson-loop tomography: which measurements determine the transport?