# What Must a Model Preserve? Synthetic Tests Toward Evaluating Artificial Knowledge

**Wolfram Horstmann**  
[Humboldt-Universität zu Berlin](https://www.hu-berlin.de/)

**Journal of Artificial Knowledge, volume 1, issue 1**  
**Manuscript accepted:** 2 October 2026  
**Publication date:** 3 October 2026

## Abstract

Scientific knowledge produced through models depends not merely on obtaining
an output, but on what a representation preserves and which inferences that
preservation warrants. This problem becomes especially consequential when
artificial systems generate predictions, transformations, or scholarly claims
that may appear knowledgeable before their relation to a target has been
tested. Starting from Rosen's modelling relation, this article distinguishes
predictive, structural, and purpose-relative adequacy. Bounded synthetic
studies provide background evidence for the distinction: a declared predictive
rule accepted a same-family case, rejected an obvious misspecification, and
preserved an invalid evaluation as undefined; coarse states concealed different
successors and paths; and richer representations recovered different
information at different descriptive costs. These studies test representational
predicates, not an AI knowledge-production system. Rather than treating the
local results as general validation, the article uses them to formulate three
prospective experiments with explicit failure conditions. It then extends the
question tentatively to scholarly production itself: whether transformations
among research records, AI-produced articles, reviews, and revisions preserve
claims, evidential relations, uncertainty, and declared purpose. The proposals
position the synthetic results as methodological steps toward evaluating
artificial knowledge and, later, iterative and potentially scaled AI-supported
science. Correspondence is not presented as sufficient for knowledge,
purpose-relative adequacy remains untested, and increased throughput is not
equated with epistemic progress.

## 1. Introduction: the problem of modelling change

Scientific models are epistemic instruments: investigators operate on them to
draw inferences about targets that may be difficult to observe, manipulate, or
reason about directly (Hughes, 1997; Suárez, 2004). Scientific representation
therefore concerns more than associating one object with another. It concerns
what inferences the association permits, how those inferences are interpreted,
and under which questions they are justified. A fluent conclusion, accurate
prediction, or structurally detailed simulation is not knowledge merely by
virtue of having been produced.

This matters for **artificial knowledge**. The term is used here cautiously for
epistemic products—predictions, inferences, and structured claims—generated or
materially transformed by artificial systems and offered for scientific use.
It does not presuppose that the producing system is an epistemic agent or that
every successful output constitutes knowledge. It instead names an evaluation
problem: when artificial operations produce a claim about a target, which
features of the target–representation relation make that claim scientifically
usable and open to correction? Questions of truth, justification, explanation,
provenance, and social acceptance extend beyond representational correspondence;
but correspondence is one place where such claims can be made explicit and
tested rather than inferred from an output's plausibility.

Three labels must therefore be kept apart. **Automated scientific output** is
any prediction, text, dataset, or other result produced through an automated
operation; it is a production category and carries no knowledge claim by
itself. **AI-supported science** denotes a research process in which artificial
systems contribute materially within evidential, critical, and organizational
controls; it likewise does not make every output knowledge. **Artificial
knowledge**, as used here, names the narrower evaluative question of whether an
artificially generated or transformed epistemic product warrants scientific
use. For a product even to enter the evaluation proposed here, it must expose
at least a declared target, a representation and transformation, a claim or
observation, an intended use with an evaluative criterion, and enough evidence
and provenance for the relation to be challenged and corrected. These are
minimum conditions of assessability, not jointly sufficient conditions for
knowledge.

Picture a toy model intended to predict where a fly on a tabletop will move
next. The model records only whether the fly is near or far from a window, but
not the direction in which it is moving. Two situations can therefore look
identical to the model even though the fly is moving toward the window in one
and away from it in the other. The model may describe the fly's present
category correctly while remaining unable to predict its next category.
Adding direction may improve that prediction, but the extra detail may be
unnecessary if the task is only to count how often the fly enters the area near
the window. The example exposes three different questions: whether a prediction
is accurate, whether the representation preserves consequential change, and
whether the retained information is adequate for the intended use. An
artificial system that reasons from this representation inherits the same
limitations; the plausibility of its answer cannot compensate for information
that the representation discarded.

The same design problem becomes practically consequential when a model guides
a drone. A representation that retains distance from a target but discards
heading or motion may be adequate for displaying a present location category
and inadequate for predicting the next position or avoiding a collision. The
fly and drone cases are illustrations of the representational problem, not
validated biological or engineering models.

An initial match does not guarantee that a model remains informative after the
represented system changes. If two detailed states receive the same simplified
label but have different possible successors, the label may be adequate for
reporting their present category and inadequate for predicting their next
movement. If two paths finish in the same category, an endpoint comparison may
miss a difference in when a threshold was crossed. Neither result alone says
whether the representation is useful for a particular scientific purpose.

Rosen's modelling relation is a natural point of departure. It brings causal
entailment in a natural system and inferential entailment in a formal system
into a relationship through encoding and decoding or interpretation. His
earlier treatment expresses modelling through mappings and commutative diagrams
in predictive contexts (Rosen, 1978); his subsequent work also treats dynamics,
time, error, and the stability of modelling relations (Rosen, 1985, 1991).
The present article therefore does **not** propose to add dynamics to a
supposedly static Rosen theory. It takes a restricted local alignment condition
as a reference and asks which different claims must be checked when actual
representations are coarse, partial, or evaluated for different uses.

An empirically testable approach is needed because neither formal coherence nor
a persuasive output establishes that a model preserves the features on which a
knowledge claim depends. This is particularly important for AI-based knowledge
production. Artificial systems can generate and transform claims rapidly, while
the resulting text or prediction may leave unclear whether an observation, a
transition, or a task-relevant distinction was retained. Operational tests make
those dependencies answerable to evidence: the target and representation can be
declared, predictions can be committed before held-out observations are opened,
transitions can be evaluated as preserved, failed, or undefined, and purposes
can be fixed before competing representations are compared. Synthetic tests do
not by themselves validate claims about natural or engineered systems, but they
allow the proposed distinctions to fail under controlled conditions before they
are applied in more complex settings.

The central question is therefore a minimal one for evaluating model-mediated
and artificially produced knowledge claims:

> What must a model preserve as the represented system changes, and how can
> distinct claims of predictive, structural, and purpose-relative adequacy
> be tested without assuming exact correspondence everywhere?

The contribution sought here is a selective *operational decomposition* of
this question. A mapping between states is distinguished from an adequate
observation prediction; a prediction is distinguished from preservation of a
transition or path; and preservation is distinguished from usefulness for a
declared purpose. Each distinction creates a different place where a knowledge
claim can succeed, fail, or remain undefined. Together they provide a small,
challengeable vocabulary for subsequent experiments without requiring one
all-purpose measure of correspondence. The paper does not claim that these
distinctions or their formal ingredients were previously unknown. It reports
what a small sequence of synthetic tests made visible, where that evidence
ends, and how the proposal could be challenged by established accounts and
further experiments.

## 2. Theory: three claims that should not be collapsed

Representation connects modelling to knowledge because it licenses reasoning
with one system in place of another. Hughes's denotation–demonstration–
interpretation account makes the movement explicit: a model denotes a target,
is used to demonstrate a result, and requires interpretation back toward that
target (Hughes, 1997). Inferential approaches likewise make surrogate reasoning
central, while emphasizing that the relevant consequences must be scientifically
justified answers to questions about the target (Khalifa et al., 2022; Suárez,
2004). The present decomposition does not replace these accounts. It asks which
more specific adequacy claim is being made when an inference concerns an
observation, a transition or path, or a declared use.

This distinction is useful for artificial knowledge because an artificial
system can transform a representation and produce a conclusion without thereby
settling what the conclusion preserves. The three dimensions below locate
separate burdens of evaluation. They are necessary questions within the scope
of this article, not jointly sufficient conditions for knowledge.

### 2.1 A restricted alignment reference

For a selected represented state \(w_1\), a local, encoding-side
Rosen-inspired condition can be written

\[
R(E(w_1))=E(L(w_1)).
\]

Here, \(w_1\in W\) is a state of the represented system; \(L:W\to W\) is
its declared operation; \(E:W\to M\) encodes or represents that state in a
modelling system \(M\); and \(R:M\to M\) is a model operation or inference.
The equation asks whether operating after encoding reaches the same model
state as encoding after the represented-system operation. It is a local
condition for specified states and operations, not an assumed identity for
every state, and it is **not** Rosen's complete modelling relation: decoding,
interpretation, and the distinction between causal and inferential entailment
also belong to that fuller account. It supplies a stringent reference case
against which less exact or differently evaluated relations can be examined.

The synthetic tests below do not instantiate a natural system in Rosen's full
sense. Their targets are generated or constructed systems. They test whether
selected distinctions can be specified and evaluated, not whether a formal
model has been empirically validated against nature.

### 2.2 A small carrier for transition questions

For explicit change, let

\[
S_A=(X_A,T_A),\qquad S_B=(X_B,T_B),\qquad
e:X_A\rightharpoonup X_B.
\]

\(X_A\) and \(X_B\) are typed state spaces of two distinct systems;
\(T_A\subseteq X_A\times X_A\) and \(T_B\subseteq X_B\times X_B\) are
their permitted directed transitions; and \(e\) is a partial directed
correspondence. A transition \(x_k\to x_{k+1}\) uses a discrete index \(k\),
not necessarily clock or physical time. Partiality permits an assessment to
be *undefined* when a necessary state correspondence is absent. The tuple

\[
\mathcal C=(X_A,T_A,X_B,T_B,e)
\]

is a candidate comparison carrier: it says which states, transitions, and
correspondence have been declared, but it does not pronounce them adequate.
Observations, purposes, explicit boundaries, mechanisms, and relation changes
can be added only where a test needs them. The carrier's minimality is a
working design choice, not a demonstrated theorem; simulation relations or
other established structures may express the same content more economically.

The carrier can be read as a relational relaxation of the restricted Rosen
condition rather than as a new system theory:

| Restricted Rosen reference | Transition carrier | Relation between them |
| --- | --- | --- |
| represented states \(W\) | \(X_A\) | \(X_A\) supplies the states being represented |
| represented operation \(L\) | \(T_A\) | the graph of a deterministic \(L\) is a special case of \(T_A\) |
| model states \(M\) | \(X_B\) | \(X_B\) supplies the representing states |
| encoding \(E\) | partial correspondence \(e\) | \(e\) permits missing state correspondences |
| model operation \(R\) | \(T_B\) | the graph of a deterministic \(R\) is a special case of \(T_B\) |

When \(L\), \(R\), and \(E=e\) are total functions, one-step preservation in
the carrier reduces to the restricted commutation condition. Relations,
partiality, branching, path comparisons, observations, and declared purposes
extend the kinds of question that can be asked; they are not derived from the
restricted equality alone.

### 2.3 Predictive, structural, and purpose-relative adequacy

**Predictive adequacy** concerns a declared observation. In the first synthetic
test, an input condition \(s\) has a model prediction \(\hat y_s\) and a
target response \(y_s\). When both are defined, the local rule is

\[
A_{\epsilon}(s)
=\mathbf 1\!\left[\left|y_s-\hat y_s\right|\le\epsilon\right],
\]

where \(\epsilon\) is a tolerance fixed before the held-out responses are
revealed. If prediction, observation, or required alignment is missing, the
comparison is undefined rather than automatically successful or failed.
Passing this rule is an observation-level result, not proof of a correct
mechanism or preserved trajectory.

**Structural adequacy** concerns change itself. For a declared transition
\((x,x')\in T_A\), define the local verdict

\[
V_1(x,x')=
\begin{cases}
\mathrm{undefined}, & e(x)\text{ or }e(x')\text{ is undefined},\\
\mathrm{preserved}, & (e(x),e(x'))\in T_B,\\
\mathrm{failed}, & (e(x),e(x'))\notin T_B.
\end{cases}
\]

For deterministic operations \(F_A\) and \(F_B\), the preserved case reduces
to \(e(F_A(x))=F_B(e(x))\). For a coarse state \(b\in X_B\), let
\(D_b\subseteq T_A\) be the set of detailed transitions declared before
evaluation as originating in \(b\). Complete mapping coverage requires

\[
\operatorname{Cov}_{e}(D_b,b)
\iff
D_b\neq\varnothing\ \land\
\forall (x,x')\in D_b:\ e(x)=b\ \land\ e(x')\text{ is defined}.
\]

Only when this condition holds is the mapped successor set defined:

\[
\operatorname{Succ}_{A/e}(b;D_b)
=\{e(x')\mid(x,x')\in D_b\}.
\]

If the declared set is empty or any required source or successor mapping is
missing, the successor-set verdict is undefined; unmapped transitions may not
be silently removed. Comparing a defined set with
\(\operatorname{Succ}_B(b)=\{b'\mid(b,b')\in T_B\}\) requires the claim to be
fixed in advance. Equality asks for exact correspondence, inclusion
\(\operatorname{Succ}_{A/e}(b;D_b)\subseteq\operatorname{Succ}_B(b)\) asks whether
the model covers generated alternatives, and the reverse inclusion asks
whether every modelled alternative is generated within the declared domain.
A unique deterministic coarse successor additionally requires

\[
\left|\operatorname{Succ}_{A/e}(b;D_b)\right|=1.
\]

For a finite path \(\pi=(x_0,\ldots,x_n)\), its mapped path
\(e(\pi)=(e(x_0),\ldots,e(x_n))\) is defined only when every required mapping
is defined. Equality of final mapped states does not imply equality of mapped
paths. These one-step, branching, and path predicates are related but not
interchangeable, and they do not form one universal structural score.

**Purpose-relative adequacy** asks whether a representation \(r\) retains the
information required for a declared task \(q\) at an acceptable descriptive
cost. A minimal prospective template fixes a task-performance measure
\(P_q(r)\), a performance threshold \(\tau_q\), a descriptive-cost measure
\(C(r)\), and an admissible cost \(\kappa_q\) before comparison:

\[
V_q(r)=
\begin{cases}
\mathrm{undefined}, & q,P_q,\tau_q,C,\text{ or }\kappa_q\text{ is not fixed},\\
\mathrm{adequate}, & P_q(r)\geq\tau_q\text{ and }C(r)\leq\kappa_q,\\
\mathrm{inadequate}, & \text{otherwise}.
\end{cases}
\]

Predicting a next category, identifying a reachable category, and
distinguishing a steering mechanism can require different information. No
universal metric, threshold, or cost unit is proposed. A purpose and its
decision rule must be fixed *before* comparing representations; otherwise they
can be used retrospectively to excuse whichever representation happened to
perform well. This template is a research proposal in the present article,
not an experimentally established result.

The three dimensions may coincide in a particular case but do not follow from
one another. State alignment does not entail predictive adequacy; predictive
adequacy does not entail transition or mechanism preservation; and structural
detail is not unconditionally valuable without a task (Bokulich & Parker,
2021; Suárez, 2004).

The contribution is therefore an operational taxonomy for separating test
claims, not a claim to a novel formal object. Its present evidential status is:

| Dimension | Operational object | Present status | Principal limit |
| --- | --- | --- | --- |
| predictive | \(A_\epsilon(s)\) | locally tested with same-family and constant synthetic models | no near-miss discrimination test |
| one-step structural | \(V_1(x,x')\) | tested in constructed and mechanism-generated cases | local declared domains only |
| branching and path structural | successor sets and mapped paths | explored using reused synthetic trajectories | no completed same-case comparison with an established formalism |
| purpose-relative | \(V_q(r)\) template | prospective and untested | task metrics, thresholds, and costs remain to be fixed experimentally |

### 2.4 Position among adjacent accounts

The proposal is positioned here through a bounded, claim-led selection rather
than a systematic literature review. The selected accounts are the closest
formal or philosophical comparators for the operations used in this article:
commutation, preservation of transitions, aggregation of states, retention of
paths, and adequacy for a declared use. The comparison asks the same questions
of each account—what relation is evaluated, what kind of failure it can expose,
and what remains outside its immediate scope:

| Account | Principal object or relation | Question made precise | Relation to the present taxonomy |
| --- | --- | --- | --- |
| Rosen's modelling relation | causal and inferential entailments connected by encoding and decoding | whether inference in a model is appropriately related to operation in a represented system | supplies the broader modelling frame; the local equality in Section 2.1 is only a restricted reference |
| Simulation and approximate simulation | state relations that preserve possible transitions exactly or within a metric | whether one transition system can match the behaviour of another | may absorb much of the structural predicate; it does not by itself select the scientifically relevant observation map, task, or admissible descriptive cost (Girard & Pappas, 2007) |
| Abstraction and lumpability | grouped states whose aggregate dynamics remain well defined | when coarse states support closed deterministic, nondeterministic, or probabilistic dynamics | directly bears on branching under coarse mappings; the present tests do not evaluate probabilistic lumpability (Kemeny & Snell, 1960) |
| Symbolic dynamics | symbolic sequences induced by partitions of a dynamical system | which path distinctions survive symbolic coding | provides established machinery for trajectory-sensitive comparison; it may express the path component more economically (Lind & Marcus, 1995) |
| Adequacy for purpose | representational performance relative to a specified scientific use | which properties of a representation matter for a question or decision | motivates the prospective task-and-cost predicate rather than one unconditional ranking (Bokulich & Parker, 2021) |

What remains after this comparison is not a claimed new formalism. It is a
small operational synthesis that places observation tests, transition and path
tests, and purpose-relative tests in one inspectable sequence while preserving
failure and undefined outcomes. A completed same-case translation could show
that an established account expresses any individual component better; that
would refine the synthesis rather than count against its central separation of
evaluative burdens. The selection excludes broader literatures on model-based
explanation, measurement, and social epistemology except where they bear
directly on these declared comparisons. Its limits should therefore not be
mistaken for evidence that no closer or more economical formal treatment
exists.

## 3. Methods: synthetic tests of distinct correspondence claims

The tests move from observation comparison to local transitions,
mechanism-generated dynamics, multi-step paths, and alternative
representations. Each test posed a distinct correspondence question and used
its own specified system, comparison rule, and failure conditions. The
sections that follow report those tests together to show how the questions
accumulate; they are not independent replications or one prospectively
pre-registered multi-part experiment.

For observation-level prediction, a synthetic stimulus–response
characteristic supplied seven training conditions and four held-out
conditions. A one-parameter model was fitted before the held-out responses
were opened. Predictions were committed first, and an inclusive absolute-error
tolerance of \(0.06\) was fixed. A negative control then used a deliberately
constant predictor against a stimulus-dependent target. Its first attempted
evaluation became invalid after a post-reveal evaluator failure; that attempt
was preserved as undefined. A governed repeat used a checked executable
environment and new held-out responses before evaluating the same
kind of negative control. The word “characteristic” refers to the shape of
the synthetic input–output relation; no neuron or biological response was
observed.

The first structural test used a deliberately small, constructed pair of
transition systems. Seven detailed positions and a partial coarse mapping
allowed declared cases of preservation, endpoint mismatch, and missing
correspondence. This test checked operational differentiation, not uncertain
behavior discovered from an independent mechanism.

The later structural tests used a two-sensor, two-wheel, Braitenberg-style
synthetic vehicle (Braitenberg, 1984) as a mechanism-generated example. Its
detailed state contained position and heading. A radial representation
reduced that state to **goal**, **near**, or **far** relative to a source.
The fly, drone, and vehicle presentations instantiate the same abstract
pattern: a coarse location category retains distance-related information while
discarding heading or motion that may determine what happens next.
For sensor distance \(d\), the signal was \(I(d)=1/(1+d^2)\). With left and
right wheel speeds \(v_L,v_R\), wheel base \(b\), and Euler step \(\Delta\),
the implementation used

\[
v=\frac{v_L+v_R}{2},
\qquad
\omega=\frac{v_R-v_L}{b}.
\]

Sensor offsets were \(0.20\) forward and \(0.10\) lateral; wheel base was
\(0.40\), gain \(1.0\), and step size \(0.50\). The radial thresholds were
\(G:r\leq0.75\), \(N:0.75<r\leq2.00\), and \(F:r>2.00\). The state grid
intentionally stressed those thresholds: eight radii
\(0.70,0.74,0.76,0.80,1.95,1.99,2.01,2.05\) were crossed with four headings
\(0,\pi/2,\pi,-\pi/2\). One Euler step generated 32 detailed transitions.
Paired crossed and uncrossed wheel mechanisms were subsequently run for eight
steps from 32 common starting states. A final comparison constructed
branching, path-sensitive, and orientation-enriched representations and
examined a held-out subset of 16 cases. That split reused existing synthetic
trajectories and was not a blind replication.

**Study-to-evidence map.** Paths in the last column are relative to the
extracted standalone supplement directory.

| Test | Target or representation | Frozen design and evaluation status | Primary reported outcome | Failure or undefined condition | Supplement entry |
| --- | --- | --- | --- | --- | --- |
| Same-family prediction | synthetic logistic stimulus–response characteristic | 7 training and 4 prospectively held-out conditions; \(A_{0.06}\) | 4/4 passed; maximum error \(0.02336968\) | missing required value is undefined; error above \(0.06\) fails | `source/prototype/generate_synthetic_neuron_characteristic_v02.py`; `source/artifacts/synthetic-neuron-characteristic-v0.2/`; matching `reference/` entries |
| Constant misspecification | constant predictor against a fresh logistic target | 7 training and 4 sealed holdout conditions; governed repeat after an invalid predecessor | 4/4 failed; errors \(0.09567552\)–\(0.38962421\) | predecessor evaluator failure remains traceable as undefined; governed errors above \(0.06\) fail | three programs under `source/prototype/`: `generate_autonomous_misspecification_v02.py`, `fit_constant_misspecification_v02.py`, and `evaluate_autonomous_misspecification_v02.py`; matching `reference/` entries |
| Constructed transitions | 7 integer positions, 3 coarse states, partial map covering 5 positions | 4 prospectively declared transition pairs | 2 preserved, 1 failed, 1 undefined | endpoint mismatch fails; missing correspondence is undefined | `source/artifacts/s-v0.4-braitenberg/input.json`; `source/prototype/evaluate_transition_correspondence.py`; matching `reference/` entries |
| One-step vehicle | 32 radius–heading microstates under crossed coupling | frozen mechanism, grid, thresholds, and persistence prediction; no holdout | 20 preserved and 12 failed; every coarse state branched | mapped successor differing from predicted persistence fails | `source/artifacts/s-v0.5-braitenberg/config.json`; `source/prototype/run_braitenberg_dynamic_consistency.py`; matching `reference/` entries |
| Eight-step vehicle | 32 common starts under crossed and uncrossed mechanisms | 8 Euler steps; no blind holdout | 0 final-category differences, 16 final-radius differences, 4 path differences | endpoint agreement is retained as a negative result; mapped paths may still differ | `source/artifacts/s-v0.6-multistep/config.json`; `source/prototype/run_braitenberg_multistep.py`; matching `reference/` entries |
| Representation comparison | branching, mapped paths, and orientation-enriched branching | 16 construction and 16 evaluation cases split from existing trajectories; not blind replication | branching variants covered 16/16 endpoints; paths distinguished 2/16 pairs; enriched mean width \(1.3333\) | no common task metric or aggregate ranking was declared | `source/artifacts/s-v0.7-comparative/config.json`; `source/prototype/run_comparative_representation.py`; matching `reference/` entries |

These settings are entirely computational. They contain no biological data,
physical robot, empirical measurement, or evidence that the selected rules
describe naturally occurring systems.

## 4. Results

### 4.1 Prediction can pass, fail, or be undefined

The same-family stimulus–response model passed all four prospectively
held-out comparisons. The largest absolute error was \(0.02336968\), below
the fixed \(0.06\) tolerance. The result showed that state/input alignment
alone did not determine the verdict: a typed observation and predeclared
comparison rule were required.

The first constant-model negative-control attempt supplied **no valid
scholarly test result** because the evaluator failed after reveal. It was not
counted as a weak pass or numerical failure. In the governed repeat, all
four constant-model comparisons were defined and all four failed: errors
ranged from \(0.09567552\) to \(0.38962421\), each above \(0.06\). The
protocol therefore rejected this declared mismatch rather than producing
success automatically.

This contrast is deliberately easy. The positive response was generated from
the fitted function family, while the negative was an extreme constant
misspecification. The evidence establishes executable local discrimination
between these two synthetic cases; it does not establish calibrated model
selection, robustness to plausible alternatives, statistical power, or a
general prediction theory.

### 4.2 A mapped state need not preserve its next state

The constructed local-transition evaluation returned two preserved cases,
one defined failure, and one undefined case:

\[
(1,1,0,\bot_{\mathrm{map}}).
\]

Only five of seven detailed positions were mapped. Two cases agreed in their
starting coarse-state signature but differed after their endpoint was
evaluated. The static starting match therefore did not decide whether the
declared transition was preserved. Because the examples were selected to
exercise the predicate, this result demonstrates inspectable operational
differentiation, not independent empirical confirmation of a new law.

The mechanism-generated one-step vehicle study produced a stronger stress
case. Of 32 frozen detailed transitions, 20 preserved the proposed coarse
persistence outcome and 12 failed it. Every starting category had more than
one observed coarse successor. At each of eight sampled radii, changing
heading while keeping radius fixed yielded different successor categories.
The radial category consequently did **not** define a unique coarse
successor for this mechanism, grid, and step size. Heading was information
discarded by the coarse map but consequential for the generated transition.
This does not make the goal/near/far representation useless: it makes unique
next-category prediction an unsupported use of that representation in the
tested setting.

### 4.3 Equal endpoints can conceal paths; responses have different costs

The eight-step paired-mechanism test found no crossed/uncrossed pair with a
different final coarse category among 32 starts. That negative endpoint
result was retained. Sixteen pairs nevertheless had different final radii,
and four followed different intermediate coarse-category paths. The paths
made the timing of threshold crossing visible where the endpoint category
did not. This supports a bounded reason for path retention when route or
timing matters, not a claim that paths are always superior.

The final representational comparison produced the following outcomes on its
held-out evaluation subset:

| Response | Information retained | Held-out observation | Descriptive cost |
|---|---|---|---|
| Coarse branching | possible endpoint categories | 16/16 endpoints covered; mean branching width 2.0 | 3 source keys, 6 relation pairs |
| Ordered paths | intermediate category order | 2/16 paired mechanisms distinguished despite equal endpoint labels | longer represented object |
| Orientation-enriched branching | coarse category plus heading-related distinction | 16/16 endpoints covered; mean branching width 1.3333 | 12 source keys, 16 relation pairs |

Enrichment reduced local ambiguity but did not eliminate branching. Its lower
mean branching width partly follows from dividing examples among more keys;
it is not a complexity-adjusted gain. The path comparison used a different
performance question from endpoint coverage, so the three rows do not supply
one common ranking. The results identify trade-offs. They do **not** decide
which response is adequate for a scientific purpose that was never declared
in the comparison.

## 5. Discussion

### 5.1 What the studies distinguish

The completed tests support a narrow but important distinction: a claim that
two states correspond cannot stand in for a claim that the model predicts an
observation, preserves a transition, retains a path difference, or serves a
purpose. The positive and negative characteristic cases show observation
adequacy can be evaluated separately from input alignment. The local
transition cases distinguish preservation, contradiction, and missing
comparison. The generated vehicle cases show why one coarse state label may
hide different successors. The multi-step and alternative-representation
cases show why more information is not one uniformly better answer.

The transition carrier kept *construction of a comparison* separate from
*evaluation of its success*. It provided a stable place to specify states,
allowed changes, and a directed mapping while leaving observation functions,
mechanisms, boundaries, path memory, and tasks optional. That modularity is
methodologically useful but not proof that the carrier is an irreducible new
formal object. Existing simulation and approximate-simulation relations may
already express transition preservation under specified conditions (Girard &
Pappas, 2007). Deterministic quotients and nondeterministic abstractions offer
ways to understand grouped-state dynamics; probabilistic lumpability poses a
different condition that these deterministic tests did not evaluate (Kemeny &
Snell, 1960). Symbolic dynamics treats path sequences through partitions
(Lind & Marcus, 1995). An honest comparison may show that the carrier should
be translated into, or absorbed by, an established formalism.

The same caution applies to Rosen. His modelling relation is not one state
equation, and the restricted equality in Section 2.1 is not a claim that his
larger account is silent about change or error. A future comparison must
express the *same* synthetic case in the fuller Rosen vocabulary and in the
present one, then ask whether any distinction here remains unavailable there.
Formal redundancy is a legitimate outcome, not a failed publication strategy.

### 5.2 From representational correspondence to artificial knowledge

The experiments suggest a disciplined way to approach claims of artificial
knowledge without deciding in advance whether an artificial system itself
knows. They test components of a representational evaluation, not AI-supported
science or an AI-produced scholarly claim as such. The primary object of
evaluation is the epistemic product and its
traceable relations: what target was represented, what operation produced the
result, which observation or structural feature was preserved, and what use
the result is supposed to support. On this view, provenance and reproducibility
can make an artificial contribution inspectable, but they do not replace tests
of what its representation preserves.

Correspondence is therefore neither a complete theory of knowledge nor a
dispensable technical detail. Predictive adequacy asks whether a declared
observation is recovered under a fixed rule. Structural adequacy asks whether
the transformation preserves the changes on which an inference depends.
Purpose-relative adequacy asks whether the retained information is appropriate
for the question being answered. Because these burdens can be tested
separately, later studies can strengthen, revise, or reject one dimension
without treating every artificial knowledge claim as one indivisible success
or failure.

This makes the framework productive beyond the examples reported here. It
provides explicit objects for criticism: the encoding, operation, observation,
transition, task, and cost. New experiments can vary one of these while holding
the others fixed; formal comparisons can ask whether established theories
already capture the same distinction; and negative results can identify which
kind of epistemic claim failed. The value of the decomposition lies in enabling
those tests, not in conferring knowledge status on AI output by definition.

The same logic can later be applied reflexively to scholarly production. A
research record can be treated as a represented system and an AI-produced
article as a representation generated from it. The question is then which
claims, evidential relations, uncertainties, and declared purposes survive the
transformation into a manuscript and through subsequent review and revision.
This possibility follows from the framework but has not been tested by the
experiments reported here.

### 5.3 Evidential limits

The studies provide local synthetic evidence, not natural-system validation.
The positive prediction example was easy by construction; the negative model
was conspicuously wrong. The local-transition example was also
construction-driven. The vehicle grid concentrated threshold cases and used
one sensor law, parameter set, Euler integrator, and eight-step horizon. Its
counts cannot be generalized to all detailed states, parameter choices, or
physical vehicles. The final comparison reused existing trajectories and
mixed endpoint coverage with mechanism discrimination; it was not a common
task-based contest or a blind replication.

No result establishes biological accuracy, causal explanation, robust model
selection, universal structural adequacy, or the explanatory necessity of a
new carrier. The focused literature positioning is not a systematic review;
the formal comparisons above remain prospective. Most importantly, the
purpose-relative criterion has been argued for but **not tested**. It cannot
be counted as a third successful adequacy result simply because the preceding
experiments made its need intelligible.

## 6. Outlook: three prospective experimental tests

### 6.1 Three immediate tests

The next studies should be selected by experimental readiness, not by an
arbitrary desired paper count. Each requires a fixed synthetic system,
intervention, predeclared outcome and failure condition, held-out evaluation,
and independent review. The following three are distinct candidates aligned
with the three adequacy dimensions; none has been executed for this article.

**Predictive adequacy beyond trivial mismatch.** Use a synthetic moving-agent
generator, expressed through the accessible fly or drone illustration, with
held-out positions and plausible near-miss models rather than only an extreme
constant predictor. Candidate models could differ in whether they retain
direction, motion, or only distance from a target; controlled noise and
outliers could be added prospectively. Freeze the model families, tolerance or
scoring rule, training/holdout split, and criteria for false acceptance and
false rejection before reveal. The adequacy rule would be challenged if it
accepts the declared wrong model too easily, rejects a model adequate for the
declared use, or depends strongly on post hoc treatment of outliers.

**Structural adequacy under generated change.** Reuse the synthetic
moving-agent system with detailed states, coarse mappings, and unseen
transitions or short paths. Systematically hide or retain direction, motion,
or path information, and predeclare cases intended to exhibit preservation,
failure, and undefined comparison. Compare the proposed structural predicates
with an established transition or abstraction formalism on the same cases
(e.g., Girard & Pappas, 2007; Kemeny & Snell, 1960).
The claim of added discrimination would fail if the predicates do not separate
the cases, or if the established account expresses all distinctions more
economically.

**Purpose-relative adequacy under declared tasks.** On the same synthetic
system, fix distinct uses before evaluation—for example, displaying the
present location category, predicting the next region, distinguishing a route,
and identifying potential collision cases. Compare branching, path retention,
and state enrichment on held-out cases with predeclared measures of task
performance and descriptive cost. This operationalizes an established concern
with adequacy for purpose (e.g., Bokulich & Parker, 2021). The proposed
purpose-relative dimension would be challenged if the declared use does not
predict which retained information matters, if the ranking changes only
through arbitrary cost weights, or if simpler representations perform as well
as more elaborate ones.

The following protocol skeletons make explicit what must be frozen before any
of the three studies begins. They do not constitute registrations: numerical
thresholds, sample sizes, random seeds, and final program hashes remain to be
set in separate prospective protocols.

**Predictive protocol skeleton.** The unit of analysis is one held-out initial
state and prediction horizon. A deterministic moving-agent generator and a
registered noise-and-outlier schedule provide target trajectories. The
intervention removes or perturbs heading, motion, or distance information in
predeclared model families. Comparators include a full-state model, a
distance-only model, and plausible near-miss models with lagged or biased
heading—not merely a constant predictor. The primary metrics are false
acceptance and false rejection rates under a frozen prediction tolerance;
success requires both to remain below their preregistered bounds. A case is
undefined when a required target, prediction, or alignment is missing. Initial
states and noise draws are divided into construction and sealed holdout sets
before fitting. Sensitivity analysis varies the tolerance and registered
outlier rate across a fixed grid. The proposal is falsified as a useful
discriminator if it accepts the known near-miss families too readily, rejects
the full-state family too readily, or reverses its conclusion across the
admissible sensitivity grid.

**Structural protocol skeleton.** The unit is one unseen transition or
fixed-length path from a held-out initial state. The same generator is frozen,
while the intervention changes the state partition or removes heading, motion,
or path memory. The present one-step, successor-set, and mapped-path predicates
are compared on the same cases with the approximate-simulation metric of
Girard and Pappas (2007), specialized to the frozen discrete transition
systems. The primary outcome is correct classification of prospectively
constructed preserved, failed, and undefined cases, followed by a comparison
of which distinctions each formalism expresses. Missing state mappings or
incomplete successor-set coverage or required paths are undefined, not
failures. Holdout cases use
initial states and, where applicable, mechanism variants excluded from
construction. Sensitivity analysis varies partition thresholds and path
horizons on a registered grid. The added structural vocabulary is unsupported
if it fails to separate the declared cases or if the established comparator
expresses every distinction at least as clearly and economically.

**Purpose-relative protocol skeleton.** The unit is a representation–task pair
evaluated over held-out trajectories. The generator is fixed, and the
intervention changes retained information among distance-only,
distance-plus-heading, and path-memory representations. Tasks are declared
before evaluation—for example, displaying current region, predicting next
region, distinguishing route, and identifying synthetic collision cases.
Primary outcomes are task-specific loss and descriptive cost. Cost is recorded
as a vector—number of retained scalar state variables, number of retained path
steps, and encoded storage—rather than hidden immediately in one weighted
score. A representation is adequate only if it meets the preregistered task
threshold and cost constraint; missing task, threshold, cost definition, or
required output makes the verdict undefined. Trajectories are stratified by
task-relevant conditions before a sealed holdout is drawn. Sensitivity analysis
varies thresholds and, if scalarization is used, a preregistered grid of cost
weights; conclusions must also be reported as Pareto relations without weights.
The purpose-relative claim is undermined if tasks do not predict which
information matters, if rankings depend only on arbitrary weights, or if the
simplest representation performs equivalently across all declared tasks.

For all three studies, independent review means that a reviewer instance not
used for authoring, program construction, or execution receives the frozen
protocol, locked results, and reproduction package after the outcome is fixed.
The review report is preserved before the Human Sponsor decides whether the
study supports, revises, or rejects its claim.

The three tests could share the moving-agent generator, but a common test bed
must not collapse their different criteria. Each experiment must remain
independently interpretable. Taken together, they ask whether the three
proposed burdens can organize progressively harder claims about artificial
knowledge: that a generated answer fits observations, that its representational
process preserves consequential change, and that what it preserves is adequate
for a declared inquiry. Failure of any test would refine the vocabulary
available to the others rather than merely reduce a single aggregate score.
The moving-agent setting remains synthetic and cannot establish the safety or
adequacy of a real drone system.

### 6.2 Reflexive successors: scholarly artifacts as representations

A later experiment can treat a primary research record as the represented
system and an AI-produced article as its representation. A fixed comparison
would ask whether the article preserves accepted claims, links between claims
and evidence, relevant uncertainty, negative or undefined results, and the
purpose for which the study was conducted. This reformulates an earlier
tentative reconstruction question without requiring a special ontological
category for scholarly texts. Reproducibility can provide a minimum evidential
standard for computational claims in such a comparison (e.g., Peng, 2011),
although reproducibility alone would not establish representational adequacy.

An inverse experiment could then ask whether a materially different article
can produce the same result under a fixed, source-blind review procedure.
Equivalent review outcomes would not establish identity of meaning, evidence,
or quality. They could instead reveal an equivalence class under the rubric,
an insensitive review instrument, or optimization against the reviewer. These
alternatives require discriminators fixed before authoring begins.

Repeated authoring, review, and revision would add a dynamic question: does
each transformation preserve or improve the relevant evidential relations, or
does it propagate and legitimize an earlier loss? Context-isolated and
context-exposed AI contributions could be compared within such a design, but
role separation alone would not guarantee epistemic independence.

### 6.3 Iterative and scaled knowledge production

At a more tentative level, scholarly work may be studied as a changing
meta-system of representation systems:

\[
\mathcal{M}_k=(S,R_k,x_k),
\qquad
(x_k,R_k)\leadsto(x_{k+1},R_{k+1}).
\]

Here, \(S\) denotes participating representation systems, \(x_k\) their
relevant states, and \(R_k\) the relations active at transition index \(k\).
For a journal, changes in \(R_k\) could include which system authors or reviews,
which evidence is available, and which evaluative rule is in force. The formal
question is whether this factorization adds explanatory or predictive value
beyond an ordinary enlarged state \(z_k=(x_k,R_k)\). The framework should be
retained only if the changing-relation description supports a testable
distinction that the enlarged fixed-state formulation does not express as
economically.

Scaling this organization to many AI Authors, Reviewers, experiments, and
computational resources would make the process itself an empirical object.
Automated cycles of hypothesis generation, experiment selection, execution,
and interpretation provide an important precedent (e.g., King et al., 2004),
but do not by themselves establish the epistemic adequacy of scaled scholarly
production. More recent systems occupy distinct points in that space. A-Lab
joined computational screening, literature-derived synthesis proposals,
robotics, characterization, and active learning in inorganic-materials
synthesis (Szymanski et al., 2023). Coscientist used a large-language-model
system with search, code, and laboratory tools to design and execute chemical
tasks (Boiko et al., 2023). The AI Scientist extended automation across idea
generation, code, experiments, manuscript production, and automated review in
machine-learning research (Lu et al., 2026). These systems demonstrate
substantial automation of scientific operations; they do not remove the need
to ask which target relations, uncertainties, failures, and purposes their
outputs preserve. The present proposal is therefore an evaluative complement
to workflow automation, not an alternative automation architecture.

Relevant outcomes include error propagation, convergence on one vocabulary,
loss of alternative explanations, circular validation, provenance loss,
reproducibility, correction rates, and resource consumption. Increased
throughput would not by itself demonstrate increased knowledge production.
The stronger claim of self-organization would require explicit rules for how
new questions and relations arise, how they are constrained, and how recursive
uptake differs from repeated automated output.

These reflexive studies could eventually test whether contemporary AI requires
a qualitatively new representational relation or can be described through
existing functional roles such as modeller, operator, interpreter, reviewer,
and experimental object. That preserved null-hypothesis question cannot be
decided merely because AI systems generate articles. Comparisons with formal
systems theory should distinguish broad general-systems concepts (e.g., von
Bertalanffy, 1968) from precise formal state-transition accounts (e.g., Girard
& Pappas, 2007). A bounded future test inspired by sociological accounts of
communicative continuation (e.g., Luhmann, 1992, 1995) could operationalize
uptake as an identifiable claim from artifact \(a_k\) becoming a cited premise,
constraint, objection, or revision target in artifact \(a_{k+1}\), with its
provenance retained. The comparison condition would be repeated automated
output without identifiable uptake. Failure to distinguish these conditions
would undercut the proposed explanatory value of communicative continuation;
success would test only this bounded implication, not systems theory as a
whole. These comparisons are follow-ups to the immediate experiments, not
conclusions of the present one.

## 7. Conclusion

The synthetic record does not yield one verdict that a model “corresponds,”
and it does not turn an artificial output into knowledge by declaration. It
yields several testable questions whose answers can differ. Prediction was
locally accepted and rejected under an observation rule; transitions could
preserve, fail, or be undefined; coarse states concealed different successors;
paths recovered some differences hidden by equal endpoints; and alternative
representations traded information against descriptive cost.

For artificial knowledge, the central implication is methodological: a claim
should be connected to the target, transformation, criterion, and purpose that
make it answerable to evidence. The three adequacy dimensions provide distinct
ways to build and challenge that connection. The open task is to specify
*which* achievement a scientific use requires and to test that requirement
prospectively. The proposal is best treated as a disciplined operationalization
to compare with Rosen and neighboring formal theories, not as a completed or
uniquely original theory of representation or knowledge.

The same questions can later be applied reflexively to transformations among
research records, AI-produced articles, reviews, and revisions. That extension
remains prospective: scaled production would provide evidence of artificial
knowledge only where preservation, correction, and purpose are tested rather
than inferred from the volume or fluency of outputs.

## AI-assistance disclosure

This article was developed with substantive assistance from multiple OpenAI
ChatGPT and Codex instances coordinated through AIMan, a system initiated by
the author. The recorded development period covered 31 August through 2 October
2026. These were hosted service environments, not locally executed language
models. Exact backend model identifiers and version changes were not
consistently exposed or preserved for every historical instance; no
model-specific replay claim is therefore made.

| Activity | AI contribution | Verification and human authority |
| --- | --- | --- |
| Formalization and study design | candidate relations, examples, protocols, and counterexamples | iterative Sponsor decisions and explicit protocol gates |
| Implementation and execution | code generation, deterministic execution, and result assembly in Codex workspaces | preserved programs, configurations, outputs, validation checks, and rerunnable package |
| Interpretation and writing | candidate analyses, objections, revisions, and manuscript text | claim-to-evidence checks and Sponsor acceptance of interpretations |
| Critical review | context-separated AI review of formal, methodological, and manuscript claims | preserved review reports; the Sponsor retained final acceptance authority |

Wolfram Horstmann determined the research direction, approved the protocols
and interpretations, and accepts responsibility for the published work. The AI
systems are not listed as authors. The supplementary package supports
reproduction of the reported computations but does not reproduce
model-internal reasoning or make unavailable hidden reasoning inspectable.

## Research provenance

The reported experiments were conducted in separate, versioned research
cycles and are assembled here into one scientific argument; this manuscript
does not claim they formed a single prospectively registered study. A human
researcher set direction, made material methodological decisions, and retained
authority to accept or reject interpretations. Deterministic programs generated
and evaluated the reported synthetic cases. Protocols, outputs, decisions, and
reviews remain available in the research archive. The v0.15 manuscript
underwent AI-based scholarly review and was accepted by the Human Sponsor for
Issue 1. The Human Sponsor accepted the v0.16 publication wording and curated review
history as final Issue 1 content on 3 October 2026.

The complete AIMan archive remains local and private. The public
record comprises the accepted article, the licensed standalone computational
supplement, its claim-to-evidence map and checksums, and the approved canonical
article URL, https://www.perscientiam.org/joak/issue-1/what-must-a-model-preserve/. Raw conversational material, internal
coordination records, credentials, and non-public workspace paths are excluded
unless separately reviewed and released.

## Data and code availability

The computational supplement is `supplement.zip`, SHA-256
`5c895bfa0c44fd7c0b87cf83cfddd88fd02630a9b878c47d069d5511baf1d0c8`. Its approved public URL is
https://www.perscientiam.org/joak/issue-1/what-must-a-model-preserve/supplement.zip.
After extraction, a reader or AI using Python 3.10 or later can run
`python3 run_and_verify.py` without network access or third-party Python
packages. A fresh-extraction test verified all 27 computational inputs and
passed 20/20 article-level checks. On that runtime, the exact JSON-value
audit also passed. A predecessor test on another runtime found six
floating-point endpoint-radius differences of at most approximately
\(4.44\times10^{-16}\). The scientific reproduction verdict is therefore
reported at the article's declared precision; exact numeric agreement may
vary by runtime.

The package computationally reproduces the six completed result groups and
links them to frozen configurations, source programs, and reference outputs.
The internal study reports remain in the private AIMan archive and are not in
the supplement ZIP. The package does not independently replicate the
science or reproduce
historical blindness, human or AI reasoning, review, or editorial decisions;
those aspects are traceable only through provenance records.

The supplement omits the older manuscript and internal provenance
documents. It supplies a claim-to-evidence map, CC BY 4.0 notices for its
documentation and data, and an MIT notice for original code. The article is
not bundled in the ZIP. The checksum identifies the exact supplement file.

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