Latent Generativity and Counterfactual Capability - An Identification Framework for Finite Traces
Abstract
A diagnostic for generativity exploitation compares a participant’s effective
generative capability across arrangements. Its object is a set of viable
trajectories reachable under declared conditions. Nothing observable is that
set: what can be recorded is a finite sample of realized trajectories, produced
under conditions that actually obtained and through a channel that may project,
censor, aggregate, or add noise. Three constraints sharpen the difficulty. A
trajectory is an observation of generativity rather than generativity itself,
and two distinct dynamical systems can share one locally. Future repertoires
cannot be predicted and past generativity is hard to evaluate, because
trajectories only sample it. And chained counterfactual reasoning about
unrealized generation is a form legal reasoning generally declines. Together
these threaten the surrounding programme with unfalsifiability. This paper
responds by decomposing the problem into six inferential layers with
independent failure modes, and by establishing where evidence does and does not
constrain the capability object. The negative half is transferred with
attribution from two companion projects, which established that exact equality
of accessible history laws fixes binary discrimination at chance, that
behaviorally silent structure is invisible, and that enlarging a probe family
refines an equivalence partition. The paper’s own contribution concerns a
target those results do not reach: a reachable repertoire is set-valued and is
indexed to conditions that were never realized. Writing the repertoire as a
superlevel set of a reachability functional converts a question about set
equality into a question about how far that functional moves across an
observational equivalence class. Five results follow. The counterfactual
repertoire is sharply bounded by the core and hull of the class. A model class
permitting on-regime behavior to be spliced to arbitrary off-regime behavior
collapses those bounds to whatever the class alone permits, so the lift fails
maximally. Otherwise uncertainty about the organization confines uncertainty
about the repertoire to a band around the reachability threshold, whose width
is the class’s off-regime deviation. A capability comparison is identified
across the entire class whenever the gap between assessed and baseline
functionals exceeds the combined ambiguity. And under uniformly
condition-Lipschitz kernels that ambiguity is at most twice the Lipschitz
constant times the distance from the counterfactual condition to the observed
regime. The surgery construction and the Lipschitz bound bracket the problem:
without a structural tie between off-regime and on-regime behavior no
counterfactual information is available, and with a uniform tie the bounds
degrade gracefully with counterfactual distance. Four theses argue that
non-identifiability is a property of a model class paired with a design rather
than evidence against the object, that the framework is falsifiable exactly
because it returns determinate negatives, that identification failure allocates
an evidential benefit to the party controlling the records, and that existence
claims reach conclusions magnitude claims cannot. The paper establishes no
estimator, no general non-identifiability theorem, and no valuation method, and
classifies no observed relation.
Keywords: identification; partial identification; observational
Document Status, Preparation, and Reuse
This unnumbered section records the manuscript’s status, research genealogy,
preparation process, and reuse conditions. These statements govern the document
and carry no evidential weight for its substantive position.
Status and correspondence.
This paper is a revisable discussion document. Its definitions are proposals,
its mathematical results are conditional on declared assumptions, and its
empirical bridges remain open. Objections, counterexamples, corrections,
alternative formulations, and relevant literature are welcome at
mailto:huangwanhong@serendip.ngohuangwanhong@serendip.ngo.
Research genealogy and transfer.
The author-led exploratory discussion supplied three constraints: that a
trajectory is an observation of generativity rather than generativity itself
and that two dynamical systems can share one locally; that future repertoires
cannot be predicted and past generativity is hard to evaluate; and that
reasoning from a hypothetical chain of unrealized generation to an unrealized
sum is a form legal reasoning declines. These constraints generate the paper
and are not premises of any result.
Section 8 transfers results established for two companion
projects in the same research programme rather than proving them anew. The
exact-equivalence discrimination bound and its total-variation refinement were
proved for a brain-in-a-vat experiment; the silent-factor and arcsine
constructions and the probe-family monotonicity result were proved for a study
of topological warrant in developmental language. The transfer is stated
explicitly at each point of use. Section 9 contains the results
original to this paper.
Generative AI use.
Generative AI systems were used in the preparation of this work. Anthropic’s
Claude and OpenAI systems including ChatGPT and Codex supported exploratory
discussion, formal reconstruction, source discovery followed by verification,
and drafting in . The author determined the research questions,
theoretical commitments, and epistemic status of the claims and bears sole
responsibility for the manuscript, including its errors. None of these systems
is an author or holds authorship credit.
License.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0
International License (CC BY-NC 4.0). The license text is available at
https://creativecommons.org/licenses/by-nc/4.0/.
1. Introduction
A diagnostic that compares a participant’s effective generative capability
across arrangements has an awkward object. Capability, on the construction it
uses, is a set of viable trajectories reachable from the participant’s actual
position under a declared arrangement. Nothing observable is that set. What can
be recorded is a finite sample of realized trajectories, produced under
conditions that actually obtained, through a channel that may project, censor,
aggregate, or add noise. The object is a set of possibilities under conditions
that did not obtain; the evidence is a sample of actualities under conditions
that did.
Three constraints sharpen the mismatch. A trajectory is an observation of
generativity rather than generativity itself, and two distinct dynamical systems
can produce the same trajectory over a limited region, so resemblance of outputs
supports no claim about shared organization. Future repertoires cannot be
predicted and past generativity is hard to evaluate, because trajectories only
sample it. And reasoning of the form had this participant not been
affected, it would have generated one thing, then another, then earned a given
sum is a form legal reasoning generally declines, since such chains compound
unverifiable steps.
Together these threaten the surrounding research programme with
unfalsifiability. A diagnostic whose central object is never accessible can
return only positive or unresolved verdicts, and a test that cannot fail is not
a test. That threat, rather than any application, is what this paper addresses.
The response is neither to solve the identification problem nor to concede it,
but to decompose it and then establish, layer by layer, where evidence does and
does not constrain the object. Six inferential layers are separated: trace
evidence, latent-class identification, reachable-repertoire identification,
feasible-baseline comparison, benefit-through identification, and magnitude.
Each fails independently, and a reporting rule prohibits three specific
promotions across them, since provenance does not establish identity, a
repertoire contraction does not establish a benefit path, and an existence
finding does not establish a valuation.
Two of the paper’s results are transferred rather than new, and the boundary is
stated explicitly wherever they are used. Companion projects in the same
programme have established that exact equality of accessible history laws under
every admissible policy fixes binary discrimination at chance, with a
total-variation ceiling for approximately matched cases; that behaviorally
silent structure is invisible, through an explicit augmentation construction and
a one-variable instance; and that enlarging a probe family refines an
equivalence partition. Adapted here with a design family in place of a policy
class, these establish both directions of the first constraint: trace similarity
licenses no identity claim, and trace dissimilarity licenses no independence
claim, since dissimilarity confounds a difference in organization with a
difference in conditions.
The paper’s own contribution concerns a target those results do not reach. They
answer questions about which model generated observed data. A reachable
repertoire is a set of trajectories under conditions that were
never realized, and non-identification of the generating model is neither
necessary nor sufficient for non-identification of that set. Writing the
repertoire as a superlevel set of a reachability functional converts a question
about equality of sets into a question about how far a scalar functional can
move across an observational equivalence class, and five results follow. The
counterfactual repertoire is sharply bounded by the core and hull of the class.
A model class permitting on-regime behavior to be spliced to arbitrary
off-regime behavior collapses those bounds to whatever the class alone permits,
so on-regime evidence constrains the counterfactual exactly as much as no
evidence. Otherwise uncertainty about the organization confines uncertainty
about the repertoire to a band around the reachability threshold, so membership
is entailed outside that band despite the organization being unidentified. A
capability comparison is identified across the entire class whenever the gap
between assessed and baseline reachability exceeds the combined ambiguity. And
under uniformly condition-Lipschitz kernels the ambiguity is at most twice the
Lipschitz constant times the distance from the counterfactual condition to the
observed regime. The surgery construction and the Lipschitz bound bracket the
problem: without a structural tie between off-regime and on-regime behavior no
counterfactual information is available, and with a uniform tie the bounds
degrade gracefully with counterfactual distance.
Four further insights are argued rather than derived. Non-identifiability is a
property of a model class paired with a design, so it is evidence about
evidence rather than about the object. The framework is falsifiable exactly
because it returns determinate negatives, and a design’s discriminating power is
itself a reportable quantity, so an underpowered design can be shown to be
underpowered. Identification failure is not evidentially neutral between the
parties, since the most measurable layer is the one whose records are held by
the party setting the terms, and the affected participant’s own evidence is
precisely the evidence that licenses no inference. And a supported existence
claim reaches conclusions an unsupported magnitude claim cannot, which locates
what survives the refusal of chained counterfactual valuation.
The paper establishes no estimator, no general non-identifiability theorem, and
no valuation method, and it classifies no observed relation. Its results are
identification results rather than inference procedures, and they rest on
idealizations that realistic model classes may not satisfy.
The argument proceeds in ten further sections. Section 3 states
the scope and the three constraints. Section 4 performs the
problem analysis and states the layer decomposition.
Section 5 reviews traditions that confront underdetermined
targets. Section 6 argues the four theses.
Section 7 develops the political economy of evidence.
Section 8 transfers the non-identification results.
Section 9 proves the paper’s own results.
Section 10 supplies a complementary bound that bypasses latent
identification. Section 11 specifies a worked design,
Section 12 collects open questions, and Section 13
states limits and defeat conditions.
The philosophical and formal parts are formally independent, since every
proposition follows from declared assumptions and no derivation cites a thesis,
and justificatorily dependent, since the formal results are a schema
whose claim to be worth constructing rests on the theses.
Section 6.5 records what fails if each is rejected.
2. Inferential Scope and Target Separation
This section states the problem, the constraints that generate it, the threat
they pose to the surrounding programme, and the paper’s stopping point. It
closes by fixing the relation between the paper’s philosophical and formal
parts.
2.1 The Object and What Can Be Observed
This subsection states the mismatch the paper addresses.
A companion diagnostic evaluates whether a participant’s effective generative
capability is materially worse under an assessed arrangement than under an
independently admissible comparison. Its primary object is a reachable viable
repertoire: the set of trajectories the participant could realize, from its
actual position, under a declared arrangement, horizon, and viability standard.
Nothing observable is that set. Observation supplies a finite sample of
realized trajectories, produced under conditions that actually obtained, and
recorded through a channel that may project onto a lower-dimensional summary,
censor whole regions, aggregate across episodes, or add noise. The diagnostic’s
object is a set of possibilities under conditions that did not obtain. The
evidence is a sample of actualities under conditions that did.
2.2 Three Constraints
This subsection states the constraints that generate the paper. Each was raised
in the source discussion as an objection to a formalization then under
construction, and each rules something out.
A trajectory is not a generativity.
Two distinct dynamical systems can produce the same trajectory over a limited
region. A rule inferring shared generative organization from resemblance of
outputs, reasoning steps, or workflows would therefore be unsafe. This rules out
any construction treating an observed trace as the capability object. It was
raised as a representational problem: an instrument that fixes on the trajectory
has not fixed on the object.
Evaluation fails in both temporal directions.
Future generative repertoires cannot be predicted, and past generativity is hard
to evaluate, because trajectories only sample or approximate it. This separates
two problems frequently merged: prospective uncertainty about what would have
been produced, and retrospective non-identifiability of what was there. It rules
out treating the difficulty as solvable by better record-keeping alone.
Counterfactual chains are not accepted forms of reasoning.
An argument of the form had the participant not been affected, it would
have generated this, then that, then earned a given sum is a form legal
reasoning generally declines, and the reluctance is not arbitrary: such chains
compound unverifiable steps. This rules out founding the framework’s practical
relevance on chained counterfactual valuation.
2.3 The Threat and the Response
This subsection states what the constraints endanger and how the paper responds.
Together the constraints threaten the surrounding programme in a specific way.
If the diagnostic’s central object is never accessible, the diagnostic can
return only positive or unresolved verdicts, and a test that cannot fail is not
a test. The programme would then rest on a concept that no evidence could
discipline.
The response is neither to solve the identification problem nor to concede it.
It is to decompose it. Section 4 separates six inferential
layers with independent failure modes and states a reporting rule prohibiting
promotion across them. Sections 8 through 11
then establish, layer by layer, where evidence constrains the object and where
it does not. The central positive finding is that latent non-identification does
not entail capability non-identification, and that the conditions separating the
two cases are statable.
Relative to a declared model class, observation-design family, horizon, and
condition metric, a capability comparison across arrangements can be identified
without identifying the generative organization, exactly when the difference
between assessed and baseline reachability exceeds the residual ambiguity of the
organization’s observational equivalence class.
Claim ? is a conditional statement about a model, not a claim
that any actual capability has been identified. Its ingredients carry modeling
commitments that Section 13 states as defeat conditions.
2.4 Source Use and the Transfer Boundary
This subsection states two constraints governing the material below.
Where a source is used, the text states the specific constraint it supplies and
the inference it withholds. The verified corpus establishes formal analogues,
methodological constraints, and one legal analogue; it establishes neither the
novelty nor the empirical adequacy of the constructions proposed here, and no
source is cited in support of a proposition.
Where a result is transferred from a companion project, the transfer is stated
at the point of use and the assumptions it carries are named. This matters
because the negative half of the paper is largely inherited, and a presentation
that blurred the boundary would claim as new what the surrounding programme
already established.
3. The Structure of the Identification Problem
This section performs the problem analysis. Its objective is to separate
inferential targets that fail independently, and to state the reporting rule
that keeps them separate.
3.1 Two Temporal Directions
This subsection separates the problems the second constraint merges.
Ex ante unpredictability concerns what a generative organization would
produce going forward. It is a problem about the future, and it persists even
for an organization that is perfectly known: a fully specified stochastic
process still has an unrealized future.
Ex post non-identifiability concerns what organization was there. It is a
problem about inference from a record to its source, and it persists even where
the future is irrelevant, as when assessing a past foreclosure.
The two have different remedies and different failure modes. Richer observation
addresses the second and leaves the first untouched. A longer horizon addresses
neither. Conflating them produces the impression that the identification problem
is a single obstacle, when in fact only one of the two is an inference problem
at all.
3.2 Six Inferential Layers
This subsection states the paper’s central methodological proposal.
Table 1 records six layers, each with its own evidence
requirement and its own failure mode. The proposal is that a report must supply
a status for each layer, and that success at one layer promotes nothing at
another.
| >p0.21
X
p0.30
| Layer | Evidence requirement | Characteristic failure |
|---|---|---|
| Trace evidence | Records of realized trajectories, with conditions, channel, resolution, and horizon declared | Channel unrecorded, so the projection cannot be inverted or bounded |
| Latent-class identification | A design family rich enough to separate candidate organizations | Equivalence class is not a singleton; distinct organizations remain indistinguishable |
| Reachable-repertoire identification | Behavior of the equivalence class under conditions not realized | Class members agree on-regime and diverge off it |
| Feasible-baseline comparison | A technically feasible and independently admissible alternative arrangement | No admissible comparison, or comparison selected by its own conclusion |
| Benefit-through identification | A structural model with intervention, mediator, and identification assumptions | Coincident loss and gain with no path between them |
| Magnitude and valuation | A valuation functional over foreclosed trajectories | Existence supported while worth remains unidentified |
Table. Inferential Layers, Their Evidence Requirements, and Their Failure
Modes. The table organizes obligations and supplies no evidence. Layer coupling
is evaluative rather than logical: later layers take earlier outputs as inputs,
while success at one layer entails success at none.
3.3 Three Prohibited Promotions
This subsection states the reporting rule by naming the errors it blocks.
Provenance to identity.
Evidence that one participant had access to another’s work supports a historical
claim about access. It supports no claim that the two share a generative
organization. Section 8 makes this precise.
Repertoire contraction to benefit-through.
Evidence that a participant’s reachable repertoire contracted supports no claim
that another party’s advantage runs through that contraction. Causal mediation
methodology separates the definition of such an effect from its identification,
estimation, and sensitivity analysis (Imai et al., 2010), and the
separation applies here: the contraction is one layer and the path is another.
Existence to valuation.
Evidence that a contraction occurred supports no claim about what the foreclosed
trajectories were worth. Section 6 argues that this
separation has normative content rather than being a mere caution.
The reporting rule requires that an unresolved layer be recorded as unresolved
rather than as a negative finding. This is the operational form of the
three-valued discipline the companion diagnostic uses.
3.4 Requirements on a Replacement Account
This subsection converts the analysis into obligations discharged below.
The account needs a precise statement of what observation leaves open at the
second layer; a statement of when that openness does and does not propagate to
the third; a route to capability claims that bypasses the second layer entirely,
for use when it fails; and a design specification stating what would have to be
measured. Sections 8 through 11 supply these
in order.
4. Identification Traditions and Rival Approaches
This section locates the paper among traditions that confront underdetermined
targets. Its objective is bounded positioning: each supplies a formal analogue
or a methodological constraint, none supplies a result about generative
capability, and the organizing finding concerns how they respond to
underdetermination.
4.1 Partial Identification
This subsection identifies the tradition whose posture the paper adopts.
Manski derives worst-case nonparametric bounds on treatment effects,
establishing that informative statements remain available where point
identification fails, and that the resulting bounds are sharp given the stated
assumptions (Manski, 1990). Tamer surveys the resulting research
programme, including the systematic trade-off between assumption strength and
interval width and the practice of reporting identified sets in place of point
estimates (Tamer, 2010).
Molchanov and Molinari develop random-set theory as a framework for
identification analysis when data and assumptions reveal a set containing the
parameter rather than its value (Molchanov & Molinari, 2018). This is the
closest available vocabulary for the present target, since a reachable
repertoire is set-valued by construction. The paper uses the vocabulary of core
and hull and asserts no random-set theorem.
Two limits apply. The econometric setting has a well-defined parameter and a
sampling process; the present setting has a set-valued object under a condition
regime that was never realized, which is a harder target. And a standard
criticism of worst-case bounds is that they are frequently too wide to be
useful. Section 13 records that this paper inherits the risk.
4.2 Structural Identifiability and Observational Equivalence
This subsection identifies the formal analogues for the equivalence class.
Bellman and ström analyze structural identifiability: whether the
parameters of a model structure can in principle be recovered from ideal
input–output data, independently of noise or sample size
(Bellman & str, 1970). The distinction between structural
identifiability, a property of the model structure paired with an experiment,
and practical identifiability, a property of finite noisy data, is the
distinction this paper requires when it indexes its equivalence class to a
design family.
Givan, Dean, and Greig develop equivalence notions and model minimization for
Markov decision processes, characterizing when distinct models are
indistinguishable with respect to a class of behaviors
(Givan et al., 2003). This supplies the standard a well-posed equivalence
notion should meet. Kaelbling, Littman, and Cassandra formalize acting when the
state is observed only through a partial and noisy channel
(Kaelbling et al., 1998), which is the structure of the observation channel
used below.
None of these addresses generative organizations, capability, or counterfactual
repertoires, and no result from them transfers. What transfers is the framing.
4.3 Underdetermination Proved Rather Than Assumed
This subsection identifies the methodological posture the paper adopts toward
its own negative results.
Shalizi and Thomas prove that homophily and contagion are generically confounded
in observational network studies (Shalizi & Thomas, 2011). The paper takes
this as a model of the correct response to an underdetermined target: prove the
non-identification rather than weakening an estimate and proceeding.
4.4 Capability and the Limits of Observed Choice
This subsection identifies why observed behavior underdetermines the object.
Sen distinguishes achieved functionings from the capability set of alternatives
a person could feasibly achieve and choose (Sen, 1993). The
identification problem follows directly: what is observed is achievement, and
what the diagnostic requires is the set. Robeyns records the choices required to
operationalize any capability analysis, including dimension selection,
weighting, and the admissibility of incompleteness (Robeyns, 2005),
which is why the specification below is a declared input whose variation must be
reported.
Elster analyzes the adjustment of preferences to perceived feasibility, whereby
an agent ceases to want what it has come to regard as unavailable
(Elster, 1983). This matters acutely here. Where feasibility is
inferred from behavior, preference adaptation corrupts the measurement, and it
does so worst in exactly the cases the diagnostic targets, since sustained
constraint is what produces adaptation. Section 10 accordingly
restricts its result to directly documented condition changes.
4.5 Comparative Summary
Table 2 records each tradition’s underdetermined target, its
response, and the constraint it imposes here.
| >p0.20
p0.24
X
| Tradition | Underdetermined target | Constraint imposed here |
|---|---|---|
| Partial identification (Manski, 1990; Tamer, 2010) | Treatment effect under weak assumptions | Report identified sets; accept that weak assumptions may yield uninformative width |
| Random sets (Molchanov & Molinari, 2018) | A set containing the parameter | Supplies core and hull vocabulary; no theorem is inherited |
| Structural identifiability (Bellman & str, 1970) | Model parameters from ideal data | Non-identifiability is a property of structure paired with experiment, not of the world |
| Equivalence notions (Givan et al., 2003) | Models indistinguishable for a behavior class | A well-posed equivalence notion must be relative to a declared class |
| Partial observability (Kaelbling et al., 1998) | State behind a channel | The channel must be declared before the projection is inverted or bounded |
| Confounding results (Shalizi & Thomas, 2011) | Influence against homophily | Prove the non-identification; do not weaken an estimate and proceed |
| Capability measurement (Sen, 1993; Robeyns, 2005) | Opportunity behind achievement | Declare dimensions, weights, and incompleteness; report their variation |
| Adaptive preferences (Elster, 1983) | Feasibility behind behavior | Behavioral inference of feasibility is inadmissible under sustained constraint |
| Mediation discipline (Imai et al., 2010) | Effect through an intermediate | Definition, identification, estimation, and sensitivity remain separate |
Table. Traditions Confronting Underdetermined Targets. Every tradition
reviewed responds to underdetermination by weakening the claim rather than by
strengthening the inference. This paper adopts the same posture.
5. Epistemic and Philosophical Foundations
This section argues the four theses on which the paper’s significance rests.
Each subsection states a thesis, argues for it, states the strongest objection
known to the author, and replies. The theses answer questions the formalism
cannot answer for itself: what a non-identification result shows about the
object, whether the surrounding framework is falsifiable, whether identification
failure is evidentially neutral between parties, and what an existence claim
without a magnitude claim can support.
5.1 Non-Identifiability as an Epistemic Relation
This subsection blocks the strongest misreading of the paper’s negative results.
Non-identifiability is a property of the relation between a model class and an
observation design. It is not evidence that the latent organization is unreal,
that capability talk is empty, or that the diagnostic’s object was badly chosen.
The argument is that every non-identification statement below is doubly
indexed. The equivalence class of Definition ? is a function
of the model class and the design family. Enriching the design family refines
the class, by a monotonicity result proved for a companion project and stated as
Proposition ?; restricting the model class refines it
equally. Neither operation changes anything about the participant. An inference
from “indistinguishable under this design” to “not there” would license the
same conclusion about any quantity whose current measurement is coarse.
The strongest objection is that a disposition no design could ever distinguish
is idle, and that postulating it violates parsimony. The reply concedes the
principle and denies its application. Section 9 exhibits a class of
claims about the object that observation does constrain: outside an explicit
band, membership in the counterfactual repertoire is entailed by the evidence.
An idle posit would admit no such claims. The object earns its place by
supporting bounded inference, not by assertion.
5.2 Falsifiability Through Determinate Negatives
This subsection answers the objection that would otherwise defeat the
surrounding programme.
A framework that frequently returns unresolved verdicts is falsifiable exactly
when it can also return determinate negatives. The results below supply that
capacity in a specific and quantitative form.
The objection to be answered is that a diagnostic yielding only positive and
unresolved verdicts is unfalsifiable and therefore empty. The reply is that the
framework returns determinate negatives in stated circumstances. The
false-negative direction of Section 8 establishes that an
apparent contraction can be shown to reflect a condition difference rather than
a capability difference. Corollary ? has a negative counterpart:
when the assessed reachability functional dominates the baseline functional by
more than the combined ambiguity, no member of the equivalence class exhibits a
disadvantage, and the framework returns a supported negative.
The transferred bound of Proposition ? strengthens this. Residual
distinguishability between candidates sets an explicit ceiling on any
discrimination procedure, so a design’s discriminating power is a reportable
quantity. A framework that reports its own ceiling can be shown to be
underpowered, which is a way of being wrong.
The strongest objection is that determinate negatives may be practically rare,
so the capacity is formal rather than real. The reply concedes this and converts
it into a commitment: Section 13 records that if no negative case
is ever produced in a domain where the framework is applied, that outcome
counts against the framework rather than being absorbed as caution.
5.3 The Evidential Asymmetry
This subsection argues the paper’s most contestable claim and supplies its
political-economic content.
Identification failure is not evidentially neutral between the parties. The
actor controlling realization conditions typically also controls the records
that would identify a contraction, so the default consequence of
non-identifiability systematically favors that actor.
The argument proceeds from the structure of the results below.
Section 10 establishes that the most tractable route to a
capability claim runs through directly documented contraction of the feasible
action set. Section 5 establishes that the alternative,
inferring feasibility from behavior, is corrupted by preference adaptation
precisely under sustained constraint. So the usable route requires documentation
of access terms, permission changes, allocation decisions, and pricing. Those
records are held by whoever sets them.
The affected participant’s own evidence is largely its realized trajectory,
which Section 8 shows licenses no inference about the
organization. The layer that is most identifiable is therefore the layer whose
records are least accessible to the party with the complaint. A regime resolving
non-identifiability against the complaining party allocates the benefit of an
evidential gap to the party that produced it.
The strongest objection is that this is a general argument for discovery
obligations with nothing specific to generativity, and that it proves too much.
The reply concedes the general point and identifies what survives it. What is
distinctive is that the object of the claim is constitutively unobservable to
the complainant. Reachable repertoires under alternative conditions are not the
kind of thing a participant observes about itself: the participant observes what
it did, not what it could have done under conditions that did not obtain. The
evidence is not merely held by the other side; it was never in anyone’s
possession, and the nearest available substitute is documentation the other side
holds.
A second objection is that a burden-shifting rule would penalize a party for the
intrinsic difficulty of a claim, converting non-identifiability into a
presumption of liability. The reply restricts the thesis. It concerns
justificatory burden in the sense the companion diagnostic uses, not legal proof.
Doctrinal implications are outside this paper. The thesis must also be paired
with the constraint against inferring a classification from structural position
(Zwolinski, 2012): an evidential asymmetry never substitutes for
the mechanism and benefit conditions of the diagnostic, which a domination-based
account would locate in the relation itself (Vrousalis, 2013).
5.4 What an Existence Claim Supports
This subsection determines which conclusions the paper’s evidence can reach.
A supported existence claim can ground conclusions that an unsupported magnitude
claim cannot. Conflating the two produces overreach in one direction and
unwarranted dismissal in the other.
The formal argument is that the two come apart in the results below.
Corollary ? supports an inclusion between repertoires without
identifying either, and Proposition ? supports a comparison
without valuing any foreclosed trajectory. Neither yields a magnitude.
The normative argument is that they support different conclusions. An existence
finding can support a demand for disclosure, a prospective obligation to alter a
mechanism, or a shift in justificatory burden, none of which requires knowing
what the foreclosed trajectories were worth. Compensation is the conclusion that
genuinely requires magnitude. The third constraint of Section 3
observed that chained counterfactual valuation is a form legal reasoning
declines; this thesis locates what survives that refusal.
The legal analogue is instructive and must be handled carefully. King argues that
where conduct destroys a chance of a favorable outcome, the appropriate object of
the claim is the lost chance itself rather than the outcome, so that a claim can
proceed where proof of the outcome on the balance of probabilities is
unavailable (King, 1981). The structural point is that legal reasoning
has confronted an interest in unrealized futures and has at least once
reconceived the object of the claim to preserve provability. That is the analogy,
and it is the whole of it. The doctrine concerns injury with a defined adverse
outcome and an estimable probability of avoiding it; generative capability has
neither. This paper characterizes no jurisdiction’s law, proposes no doctrine,
and does not assert that lost-chance reasoning extends to generativity.
The strongest objection is that without magnitude, trivial and serious
contractions are indistinguishable, so existence alone supports no obligation.
The reply is that materiality operates on the existence claim directly, through
declared resolution on the capability coordinates, and does not require valuing
foreclosed trajectories. The distinction between trivial and serious is
available before the distinction between worth-$x$ and worth-$y$.
5.5 Dependence of the Framework on the Four Theses
This subsection records what fails if each thesis is rejected.
Rejecting Thesis ? permits the negative results to be read as
eliminating the diagnostic’s object, which would end the surrounding programme
rather than discipline it. Rejecting Thesis ? leaves the
unfalsifiability objection unanswered, which is the strongest available
objection to the programme as a whole. Rejecting Thesis ?
removes the paper’s normative and political-economic content, leaving a purely
methodological contribution that remains defensible at reduced scope. Rejecting
Thesis ? collapses existence into magnitude, which either
inflates what the evidence supports or discards evidence that supports
something.
Theses ? and ? are load-bearing.
Thesis ? is the most vulnerable, and the author would welcome
direct challenge to it. Every proposition below survives the rejection of all
four.
6. Opacity, Records, and the Political Economy of Evidence
This section develops the consequences of Thesis ?. Its
objective is to describe the structural position of the records the framework
requires, and to state the practical problem that position creates.
6.1 Where the Usable Records Sit
This subsection locates the evidence the framework needs.
Birch analyzes contemporary technoscientific capitalism as organized around
rentiership: the appropriation of value through ownership and control rights,
monopoly conditions, and market or regulatory devices
(Birch, 2020). Control rights are exactly the devices whose
exercise produces the records Section 10 requires. An access
term, a permission change, an allocation decision, and a price are all records of
a control right being exercised, and they are generated and retained by the
party holding it.
This is not incidental to the domain. The framework’s most tractable design
requires documentation of condition changes; condition changes are what control
rights effect; and the controller is the party that documents them. The
correlation between analytic tractability and inaccessibility is structural.
6.2 Two Kinds of Opacity
This subsection distinguishes cases that call for different responses.
Incidental opacity arises when records exist but are not shared, or when
measurement was never undertaken. It is remediable by disclosure requirements,
retention rules, or independent measurement.
Constitutive opacity arises when the object was never recorded by anyone,
because it is not the kind of thing a participant observes about itself. A
participant’s reachable repertoire under conditions that did not obtain has no
record anywhere. No disclosure regime produces it.
The distinction matters for what disclosure achieves. Documented condition
changes address the third layer of Table 1 and leave
latent-class identification, benefit-through, and magnitude untouched. A
disclosure regime would therefore be genuinely useful and sharply limited, and a
proposal presenting it as a solution to the identification problem would
overstate its reach.
6.3 The Practical Problem
This subsection states the difficulty the formal results do not resolve.
The framework’s most usable route to a capability claim is least accessible to
the party that would invoke it. A complainant possesses its own realized
trajectory, which licenses no inference about the organization, and lacks the
documentation that would support the bypass route. This is a practical problem
about the framework’s applicability, and no result below addresses it.
The paper records it rather than solving it. Burden allocation in law, the
design of disclosure obligations, and the institutional question of who should
be required to retain what all belong to a separate project with its own
verified sources. What this paper establishes is that the problem is structural
rather than a contingent feature of current record-keeping practice.
7. Observational Equivalence and the Failure of Both Inferential Directions
This section establishes what finite traces cannot show. Its results are
transferred with adaptation from two companion projects in the same research
programme, and the transfer is stated at each point. Its objective is to make
precise the first constraint of Section 3.
7.1 Objects and Declarations
This subsection fixes the objects used through Section 10.
Fix a domain $d$, a model class $\Gfam_d$ of candidate generative
organizations, a carrier, a history $h$, a horizon $H$, a reachability
threshold $\epsilon\in(0,1]$, and a resolution $\delta>0$. Let
$(\T,\F)$ be a measurable trajectory space whose $\sigma$-algebra contains
the closed neighborhoods $\Nn_\delta(\tau)$. Let $\Cc$ be a space of
realization conditions with metric $\rho_\Cc$, and let
$\Cc_{\mathrm{obs}}\subseteq\Cc$ be the conditions under which observation
occurred.
For each $G\in\Gfam_d$, condition $C$, and action $a$, let
$K_G(\cdot\mid h,a,C)$ be a probability kernel on $\T$. Let
$\A(C,h)$ be a nonempty feasible action set and $\V(C,h)\in\F$ a viability
event; both depend on conditions and history alone, not on $G$. Let
$\Qq_0$ be a declared observation-design family, with channel $O_q$
producing finite observations $Z$ from a trajectory for each $q\in\Qq_0$.
$$[G]{\Qq_0,\Cc{\mathrm{obs}}}
\left{G’\in\Gfam_d:
\mathcal L(Z\mid G’,C,q)=\mathcal L(Z\mid G,C,q)
\ \ \forall q\in\Qq_0,\ \forall C\in\Cc_{\mathrm{obs}}\right}.$$
The class is doubly indexed. Nothing in it is a property of the participant
alone, which is the formal content of Thesis ?.
7.2 Trace Similarity Licenses No Identity Claim
This subsection states the first direction, transferred.
Let $\theta,\theta’$ induce identical accessible history laws under every
admissible policy in a declared class $\Pi$ through horizon $T$. Then under
equal binary priors, every decision rule measurable in the accessible history
succeeds with probability exactly $\tfrac12$. Under approximate matching, the
optimal success probability is
$$P^*_{\mathrm{succ}}
\tfrac12\left(1+\sup_{\pi\in\Pi}
\TV!\left(\mathbb P^T_{\theta,\pi},\mathbb P^T_{\theta’,\pi}\right)\right).$$
This result was proved for a companion brain-in-a-vat project and is used here
without reproof. Its adaptation replaces the policy class $\Pi$ with the
design family $\Qq_0$: a design is a way of interrogating the system, exactly
as a policy is. The adaptation carries the original assumptions, and two of them
bind. The bound is stated for binary hypotheses with equal priors, so its
extension to a model class of more than two candidates requires a separate
argument. And it presumes a fixed interface, so a design family that changes
what is recordable requires re-derivation.
Two constructions instantiate the failure concretely, both proved for a
companion project on topological warrant.
Let a latent process have state $x_t$, kernel $K$, and output
$y_t=h(x_t)$. Adjoin a coordinate $z_t$ evolving under an arbitrary kernel
independently of its own past, and define the output map to ignore it. Then
every observable finite-dimensional distribution of the augmented process
agrees with that of the original, while the latent state space differs.
Let $X$ have the arcsine density on $[-1,1]$ and let $Y=\cos\Theta$ with
$\Theta$ uniform on $[0,2\pi)$. Then $X$ and $Y$ are identically
distributed while their supports differ topologically.
Proposition ? is a precise formalization of the first constraint
of Section 3: two different dynamical systems share every
observable trajectory law. Proposition ? gives a one-variable
instance requiring no process notation. Together they establish that trace
similarity, however extensive, supports no claim that two participants share a
generative organization. A rule inferring copying from resemblance of outputs
would be unsafe in exactly the way the source discussion anticipated.
7.3 Trace Dissimilarity Licenses No Independence Claim
This subsection states the converse direction.
For a single $G$, there exist conditions and designs with
$$\mathcal L(Z\mid G,C_1,q_1)\neq\mathcal L(Z\mid G,C_2,q_2).$$
Immediate from the dependence of $K_G$ on $C$ and of the channel on $q$.
Any $G$ whose kernel is non-constant in $C$, together with any two
conditions at which it differs, supplies an instance.
The proposition is elementary and its consequence is not. Observed dissimilarity
of expressions confounds a difference in generative organization with a
difference in history, tools, permissions, collaborators, or institutional
position. Two participants producing visibly different work may share an
organization and differ in conditions. This blocks any inference from
dissimilarity to independence, and it is the reason the diagnostic must compare
arrangements rather than participants.
7.4 Design Enrichment Refines the Class
This subsection states the monotonicity result that gives
Thesis ? formal content.
For admissible design families $\Qq_1\subseteq\Qq_2$ using the same response
metrics on shared conditions, the induced exact-equivalence partitions satisfy
$\Pi_{\Qq_2}\preceq\Pi_{\Qq_1}$: the partition induced by the larger family
refines that induced by the smaller.
This result was proved for the companion topological project and is used here
without reproof. Its consequence is that non-identifiability is never a terminal
fact. A richer design family can only refine the class, so the equivalence class
is a report on the current design rather than a discovery about the object.
Section 9 shows that this has a counterfactual payoff: a refined
class carries a narrower ambiguity band.
8. The Set-Valued Lift
This section contains the results original to this paper. Its objective is to
determine when the non-identification of Section 8
propagates to the capability object, and when it does not.
The preceding results concern discrimination among hypotheses or latent spaces:
questions about which model generated the observed data. The diagnostic’s object
is different in kind. A reachable viable repertoire is a set of
trajectories under conditions that were not realized. Non-identification
of the generating model is neither necessary nor sufficient for
non-identification of that set, and no transferred result settles which obtains.
8.1 The Reachability Functional
This subsection introduces the construction that makes the question tractable.
For a counterfactual condition $C’$, define
$$f_G^{C’}(\tau)
=\sup_{a\in\A(C’,h)}
K_G!\left(\Nn_\delta(\tau)\cap\V(C’,h)\mid h,a,C’\right),$$
and
$$\Pp(G;C’)
=\left{\tau\in\V(C’,h):f_G^{C’}(\tau)\geq\epsilon\right}.$$
Equation (2) writes the repertoire as a superlevel set. This
is the move on which everything below depends: a question about equality of sets
becomes a question about how far a scalar functional can move across an
equivalence class.
$$\Pp^{\cap}(C’)=\bigcap_{G’\in[G]}\Pp(G’;C’),
\qquad
\Pp^{\cup}(C’)=\bigcup_{G’\in[G]}\Pp(G’;C’).$$
A trajectory in the core is reachable under every organization consistent with
the evidence, so its reachability is entailed. A trajectory outside the hull is
reachable under none. Trajectories between them have their status left open.
These are the sharp bounds: no procedure using only on-regime evidence narrows
them, since every member of the class is consistent with everything observed.
8.2 Maximal Failure Under Off-Regime Surgery
This subsection establishes the first of two brackets on the problem.
The model class $\Gfam_d$ is closed under off-regime surgery when for every
$G_1,G_2\in\Gfam_d$ the splice
$$K_{G_{12}}(\cdot\mid h,a,C)
\begin{cases}
K_{G_1}(\cdot\mid h,a,C), & C\in\Cc_{\mathrm{obs}},\
K_{G_2}(\cdot\mid h,a,C), & C\notin\Cc_{\mathrm{obs}},
\end{cases}$$
again lies in $\Gfam_d$.
If $\Gfam_d$ is closed under off-regime surgery and
$C’\notin\Cc_{\mathrm{obs}}$, then for every $G\in\Gfam_d$,
$$\Pp^{\cap}(C’)=\bigcap_{G_2\in\Gfam_d}\Pp(G_2;C’),
\qquad
\Pp^{\cup}(C’)=\bigcup_{G_2\in\Gfam_d}\Pp(G_2;C’).$$
Fix $G$ and any $G_2\in\Gfam_d$, and form the splice $G_{12}$ with
$G_1=G$. It agrees with $G$ at every $C\in\Cc_{\mathrm{obs}}$, hence
induces identical observation laws there under every design, hence lies in
$[G]{\Qq_0,\Cc{\mathrm{obs}}}$ by Definition ?. It
agrees with $G_2$ at $C’$, so $\Pp(G_{12};C’)=\Pp(G_2;C’)$. Since
$G_2$ was arbitrary, the equivalence class realizes every repertoire
realizable in the whole model class.
Under surgery closure, on-regime evidence constrains the counterfactual
repertoire exactly as much as no evidence at all. This is a genuine failure
mode and not a technicality: it obtains whenever the model class permits
off-regime behavior to be detached from on-regime behavior.
The result is nonetheless informative. It localizes the entire question in a
structural property of the model class, and it shows that any positive result
must be purchased by a restriction tying off-regime to on-regime behavior.
8.3 The Ambiguity Band
This subsection establishes the paper’s central positive result.
$$\eta(C’)
=\sup_{G_1,G_2\in[G]}\ \sup_{\tau\in\T}
\left|f_{G_1}^{C’}(\tau)-f_{G_2}^{C’}(\tau)\right|.$$
For every $G’\in[G]$,
$$\left{\tau\in\V(C’,h):f_G^{C’}(\tau)\geq\epsilon+\eta(C’)\right}
\subseteq
\Pp(G’;C’)
\subseteq
\left{\tau\in\V(C’,h):f_G^{C’}(\tau)\geq\epsilon-\eta(C’)\right},$$
and consequently
$$\Pp^{\cup}(C’)\setminus\Pp^{\cap}(C’)
\subseteq
\left{\tau\in\V(C’,h):
\left|f_G^{C’}(\tau)-\epsilon\right|<\eta(C’)\right}.$$
Write $\eta=\eta(C’)$. Suppose $f_G^{C’}(\tau)\geq\epsilon+\eta$. By
Definition ?, $f_G’^C’() f_G^C’()-
$, so $(G’;C’)$ by (2). This gives the
left inclusion. Conversely suppose $\tau\in\Pp(G’;C’)$, so
$f_{G’}^{C’}(\tau)\geq\epsilon$. Then
$f_G^{C’}(\tau)\geq f_{G’}^{C’}(\tau)-\eta\geq\epsilon-\eta$, giving the right
inclusion. Both hold for every $G’\in[G]$ and therefore pass to the
intersection and the union of Definition ?. A trajectory in
$\Pp^{\cup}\setminus\Pp^{\cap}$ lies in the outer set and outside the inner
set, which is (4).
The interpretation is the paper’s central point. Non-identification of the
organization does not spread uncertainty across the trajectory space. It
confines uncertainty to a band around the reachability threshold, of width
controlled by $\eta(C’)$. Outside that band, membership in the counterfactual
repertoire is entailed by the evidence despite the organization being
unidentified.
The bounds in (3) are computable from a single reference
organization together with a bound on $\eta$, which is what makes them usable.
They are outer approximations to the sharp core and hull rather than the sharp
bounds themselves.
8.4 Robust Capability Comparison
This subsection derives the form the diagnostic actually requires.
The diagnostic compares an assessed arrangement with a baseline. Let $C^m$ and
$C^b$ be the assessed and baseline conditions, with deviations $\eta_m$ and
$\eta_b$.
If
$$\left{\tau:f_G^{C^m}(\tau)\geq\epsilon-\eta_m\right}
\subseteq
\left{\tau:f_G^{C^b}(\tau)\geq\epsilon+\eta_b\right},$$
then $\Pp(G’;C^m)\subseteq\Pp(G’;C^b)$ for every $G’\in[G]$.
By Proposition ?, $\Pp(G’;C^m)$ is contained in the left
set of (5) and $\Pp(G’;C^b)$ contains the right set. Compose the
inclusions.
A capability disadvantage is identified across the entire equivalence class
whenever the gap between the assessed and baseline reachability functionals
exceeds the combined ambiguity. This is a signal-to-ambiguity condition. The
organization need not be identified; what must be established is that the
comparison survives the residual ambiguity.
Corollary ? states a sufficient condition. A disadvantage may be
identified through structure this construction does not use, and its failure is
therefore not a finding of no disadvantage.
8.5 Counterfactual Distance Bounds the Ambiguity
This subsection establishes the second bracket, quantifying $\eta$.
Proposition ? leaves $\eta(C’)$ unquantified, and
Proposition ? shows it can reach its maximum. Two assumptions
control it.
For $C\in\Cc_{\mathrm{obs}}$, equality of observation laws under every design
in $\Qq_0$ implies equality of trajectory kernels: if
$\mathcal L(Z\mid G_1,C,q)=\mathcal L(Z\mid G_2,C,q)$ for all $q\in\Qq_0$,
then $K_{G_1}(\cdot\mid h,a,C)=K_{G_2}(\cdot\mid h,a,C)$ for all
$a\in\A(C,h)$.
There exists $L<\infty$ such that for every $G\in\Gfam_d$, every $h$,
every action feasible under both conditions, and every $C_1,C_2\in\Cc$,
$$\left|K_G(\cdot\mid h,a,C_1)-K_G(\cdot\mid h,a,C_2)\right|_{\TV}
\leq L,\rho_\Cc(C_1,C_2).$$
Under Assumptions ? and ?,
$$\eta(C’)\leq 2L\cdot\dist!\left(C’,\Cc_{\mathrm{obs}}\right),
\qquad
\dist(C’,\Cc_{\mathrm{obs}})=\inf_{C\in\Cc_{\mathrm{obs}}}\rho_\Cc(C’,C).$$
Fix $G_1,G_2\in[G]$ and $\varsigma>0$, and choose
$C’’\in\Cc_{\mathrm{obs}}$ with
$\rho_\Cc(C’,C’’)\leq\dist(C’,\Cc_{\mathrm{obs}})+\varsigma=:r_\varsigma$.
Assumption ? gives
$K_{G_1}(\cdot\mid h,a,C’’)=K_{G_2}(\cdot\mid h,a,C’’)$ for every feasible
$a$. Write $A_\tau=\Nn_\delta(\tau)\cap\V(C’,h)$, which is measurable. For
any feasible $a$, inserting the on-regime condition and applying the triangle
inequality,
$$\begin{aligned}
\left|K_{G_1}(A_\tau\mid h,a,C’)-K_{G_2}(A_\tau\mid h,a,C’)\right|
&\leq
\left|K_{G_1}(A_\tau\mid h,a,C’)-K_{G_1}(A_\tau\mid h,a,C’’)\right|\
&\quad+\left|K_{G_1}(A_\tau\mid h,a,C’’)-K_{G_2}(A_\tau\mid h,a,C’’)\right|\
&\quad+\left|K_{G_2}(A_\tau\mid h,a,C’’)-K_{G_2}(A_\tau\mid h,a,C’)\right|\
&\leq Lr_\varsigma+0+Lr_\varsigma=2Lr_\varsigma,
\end{aligned}$$
where the outer terms use Assumption ? together with the fact
that total-variation distance dominates the difference of two measures on any
single measurable set. The action set $\A(C’,h)$ is common to both
organizations, and for real functions $u,v$ with $|u(a)-v(a)|\leq c$
pointwise one has $\left|\sup_a u-\sup_a v\right|\leq c$. Applying this to
(1) gives
$\left|f_{G_1}^{C’}(\tau)-f_{G_2}^{C’}(\tau)\right|\leq 2Lr_\varsigma$ for
every $\tau$. Taking suprema over $\tau$ and over the class, then letting
$\varsigma\downarrow0$, gives (6).
The ambiguity band grows at most linearly in the distance from the counterfactual
condition to the observed regime. Counterfactuals near the observed regime are
nearly identified; distant ones are not.
Assumption ? is the structural restriction whose absence
Proposition ? exploits, and the two results bracket the
problem. Without a tie between off-regime and on-regime behavior the lift fails
completely; with a uniform tie it degrades gracefully with distance. The lift
therefore admits an informative intermediate answer.
8.6 Consequences for the Surrounding Framework
This subsection records four implications. Each is conditional on the results
above and none is an empirical claim.
Baseline admissibility acquires an identification criterion.
The companion diagnostic constructs a family of technically feasible
arrangements and applies an independent admissibility filter.
Proposition ? supplies a reason, internal to identification,
to prefer baselines near the observed regime: distant baselines carry
proportionally wider ambiguity. This licenses no selection by convenience and
does not override the requirement that admissibility be defended independently.
It records a cost that distant baselines incur.
Design enrichment has a counterfactual payoff.
By Proposition ?, enlarging the design family refines the
equivalence class. A smaller class has a smaller $\eta$, hence a narrower
band. Enrichment therefore tightens counterfactual bounds and not merely
on-regime discrimination. Choosing which probes to add is an experimental-design
problem, and the expected-information criterion supplies the natural objective
(Lindley, 1956; Chaloner & Verdinelli, 1995); computing it for a set-valued
target is unresolved.
Existence and magnitude separate formally.
Corollary ? supports a comparison without point-identifying
either repertoire and without valuing any foreclosed trajectory. This is the
formal counterpart of Thesis ?.
The unfalsifiability objection acquires a determinate negative.
When the assessed functional dominates the baseline functional by more than the
combined ambiguity, no member of the class exhibits a disadvantage, and the
framework returns a supported negative rather than an unresolved status. This is
the formal counterpart of Thesis ?.
9. Partial Identification Through Documented Condition Change
This section states a route to a capability claim that bypasses latent
identification entirely. Its objective is to supply a result that survives the
failure of Section 9, since Proposition ? shows
that failure is possible.
Removing feasible actions cannot create actor-controlled reachable branches: if
$\A(C^m,h)\subseteq\A(C^b,h)$, then every trajectory reachable and
actor-controlled under $C^m$ is reachable and actor-controlled under $C^b$.
Suppose $\A(C^m,h)\subsetneq\A(C^b,h)$ is directly documented, and suppose
Assumption ? holds. Then
$$\Pp^{\mathrm{aut}}(G;C^m)\subseteq\Pp^{\mathrm{aut}}(G;C^b)$$
for the actor-controlled sub-repertoires, without point-identifying $G$ and
without valuing any foreclosed trajectory.
Every actor-controlled branch reachable under the contracted arrangement is
witnessed by an action surviving the contraction, since the action set under
$C^m$ is contained in that under $C^b$ and the kernels are those of the same
$G$. Assumption ? forbids the appearance of new
actor-controlled branches under the contracted arrangement.
The content lies entirely in the two conditions, and both are demanding.
Monotonicity fails under substitution.
Constraint provokes adaptation. A foreclosed route can produce a new strategy, a
coalition, a change of domain, or a relation that opens branches previously
unavailable. This is not an edge case, and a companion project in the same
programme treats exactly that substitution as its central mechanism. The tension
is real: there, constraint reliably changes the response set, which is what
Assumption ? denies here. The assumption is plausible over
short horizons and for narrowly specified action sets, and implausible over long
horizons where adaptation has time to operate.
Documentation is required, not inference.
The contraction must be directly documented. Where feasibility is inferred from
behavior, preference adaptation corrupts the measurement
(Elster, 1983), and it does so worst under sustained constraint,
which is the condition the diagnostic targets. Behavioral inference of
feasibility is therefore inadmissible for this purpose. The restriction narrows
the applicable domain sharply and is a cost of the result rather than a
refinement of it.
10. A Worked Design and Its Limits
This section specifies what an applicable study would require. Its objective is
to state requirements, not to demonstrate that they can be met.
The most tractable design uses directly logged access restrictions. Where a
controller records access terms, permission grants and revocations, allocation
decisions, and prices, the contraction of the feasible action set is observable
without inference, which is what Proposition ? requires.
Such a study would proceed in dependency order. Declare the domain, model class,
carrier, horizon, viability standard, resolution, and threshold before examining
any outcome. Declare the observation-design family and the channel, since the
equivalence class is indexed to both. Establish the observed condition regime and
the counterfactual condition of interest, together with the metric under which
their distance is measured. Report the layer statuses of Table 1
separately. Where Section 9 is used, state the model-class
restriction supporting Assumption ? and report sensitivity to
plausible alternatives. Where Section 10 is used, exhibit the
documentation and defend Assumption ? over the stated horizon.
Three limits apply and none is incidental. The design requires records held by
the party whose conduct is under assessment, which is the practical problem
Section 7 identifies. The Lipschitz constant is a
modeling commitment and no procedure for estimating it is proposed here. And
both $f_G^{C’}$ and $\eta$ are population objects; their estimation from
finite samples is untreated, so the results are identification results rather
than inference procedures.
11. Open Questions Across Registers
This section collects unresolved questions by the register in which each would
be answered.
11.1 Formal Questions
Do model classes of interest satisfy Assumption ?, or are they
closed under off-regime surgery in the sense of Definition ??
The positive half of Section 9 is valid in form but vacuous in
application if realistic classes fall on the wrong side of this divide.
How loose is (3) relative to the sharp core and hull of
Definition ?? The bound may be weak when the supremum defining
$\eta$ is attained far from the threshold.
The construction bounds the repertoire component alone. Does an analogous lift
hold for autonomy, direction, exit, attribution, and the other enriched
capability coordinates the companion diagnostic admits?
Proposition ? is stated for binary hypotheses under equal priors.
What is the corresponding ceiling for a model class of many candidates, and does
the design-family adaptation preserve it?
11.2 Methodological Questions
Both $f_G^{C’}$ and $\eta$ are population objects. What estimators exist,
what are their finite-sample properties, and does estimation error compound with
the ambiguity band or absorb into it?
Expected information gain supplies a criterion for choosing probes
(Lindley, 1956; Chaloner & Verdinelli, 1995). What is the corresponding
objective when the target is a set and the quantity to be reduced is $\eta$?
11.3 Normative and Institutional Questions
Can Thesis ? be sustained as a claim specific to constitutively
unobservable objects, or does it reduce to a general argument for discovery
obligations?
What follows for justificatory burden when a layer of Table 1 is
not merely unresolved but demonstrably unresolvable under every available
design?
11.4 Empirical Questions
No empirical case supports any identification claim in this paper. The priority
questions are whether any domain supplies documented condition changes
accessible to a complainant; whether a Lipschitz constant can be defended in any
applied setting; and whether a determinate negative has ever been produced. A
framework that never returns a negative finding in practice would be
unfalsifiable regardless of its formal capacity to do so, and
Section 13 records this as a defeat condition.
12. Limits and Revisable Research Program
This section consolidates the contribution, states what is transferred and what
is new, and fixes the conditions under which the paper should be narrowed or
withdrawn.
12.1 Contribution and Its Provenance
The methodological contribution is the six-layer decomposition of
Table 1 with its reporting rule and three prohibited promotions.
The negative results of Section 8 are transferred with
adaptation from two companion projects and are not claimed as new. What is added
there is the adaptation to a design family and the explicit statement of both
inferential directions for the generativity setting.
The results of Section 9 are original to this paper: the
reachability-functional construction, the core and hull bounds, the
surgery-closure collapse, the ambiguity-band sandwich, the robust-comparison
corollary, and the extrapolation bound. Their significance is that they answer a
question the transferred results leave open, since a set-valued target under
counterfactual conditions is not a hypothesis-discrimination problem.
The philosophical contribution is the four theses, of which
Theses ? and ? are load-bearing. The
political-economic contribution is the structural account of where the usable
records sit and the distinction between incidental and constitutive opacity.
The paper establishes no estimator, no general non-identifiability theorem, and
no valuation method, and it classifies no observed relation.
12.2 Defeat Conditions
The formal contribution loses support if realistic model classes prove closed
under off-regime surgery or admit no uniform Lipschitz constant, since the
positive half would then be vacuous in application; if the bounds prove too
loose to support any comparison; or if Assumption ? fails
generically, removing the bypass route.
The methodological contribution loses support if the layers cannot be evaluated
separately even in the weak sense the reporting rule requires, which is that
success at one entails success at none.
The normative contribution loses support if Thesis ? cannot be
distinguished from a general discovery argument.
The practical contribution loses support if no domain supplies documentation
both adequate for Proposition ? and accessible to a
complainant, or if no determinate negative is ever produced where the framework
is applied.
A narrower paper survives most of these. A rigorous account of what finite
traces cannot establish, combined with the layer decomposition and the reporting
rule, would still discipline the surrounding programme and would still answer
the unfalsifiability objection. The retreat should be stated explicitly rather
than performed by weakening claims silently.
12.3 Deferred Questions
The diagnostic itself, its comparison construction, and its benefit-through
conditions belong to a companion project. Propagation across relations belongs
to another. Legal doctrine, evidentiary burden, remedy, and valuation for
adjudication belong to a further project with its own verified sources; this
paper supplies only the inferential structure such an account would need, and
its use of the lost-chance analogue extends no further than the structural point
recorded in Section 6.
The resulting position is conservative. Latent generativity is frequently
unidentified, and this paper establishes that trace evidence alone shows neither
identity nor independence. It also establishes that this failure need not
propagate to the capability object: uncertainty about the organization confines
uncertainty about the counterfactual repertoire to a band around the
reachability threshold, and a capability comparison survives whenever the gap
exceeds that band. Whether real domains satisfy the conditions under which this
holds is unresolved, and the paper’s most useful outcome may often be an
explicit statement of which layer failed and why.
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