A number does not carry its meaning

Imagine two thermometers that both display 37. One measures degrees Celsius. The other maps electrical resistance onto an arbitrary 0 to 100 scale. The displays match, but no shared fact about temperature follows until the units, calibration curve, sensor range and target quantity are aligned. This is obvious because the word “temperature” already tells us what the instrument is supposed to track.

Consciousness measures are harder. The target is not directly observed in all cases, and the score is usually several transformations away from the system. A perturbation acts on a chosen site. Sensors record a response at a chosen spatial and temporal grain. Preprocessing removes some variation and retains other variation. A statistic compresses the retained pattern. A reference population connects score ranges to provisional labels. A theory then explains why the statistic should bear on consciousness rather than on arousal, responsiveness, signal quality or another correlated property.

Validity belongs to an interpretation and use of a score, not to the bare score itself. This follows the construct-validity tradition, which treats measurement as a network of theoretical and empirical commitments rather than a successful naming ceremony (Cronbach and Meehl, 1955; Messick, 1995).

Three properties are often collapsed at this point. Reliability asks whether a procedure gives sufficiently stable results when the relevant conditions are repeated. Construct validity asks whether the interpretation fits a network of predicted relations and survives discriminating tests against nearby constructs. Transport validity asks whether that interpretation remains licensed after the population, substrate, boundary or causal environment changes. A score can be highly reliable in people, clinically useful within a defined human population and still lack a warranted machine interpretation. None of those achievements is diminished by saying so. They answer different questions.

This distinction changes experimental order. Before searching for an artificial system that crosses a human threshold, specify the source claim in full: which intervention, which recordings, which reference conditions, which target phenomenon and which decision the score supported. Only then build the target measurement. Otherwise the most visually persuasive machine result will quietly redefine the source measure around whatever the machine happens to expose.

The smallest useful formalism

Let a source measurement be written as:

X is the source system. B is the system boundary. P is the perturbation. R is the recorded response. G is the spatial and temporal grain. N is the normalisation. f is the statistic. The target score uses the same pattern of symbols with subscript T.

A numerical match, q_S ≈ q_T, is informative only if the transformations are commensurable. The same function name is not enough. A cortical pulse and an activation overwrite may inject different energies, touch different state variables and interact with different recovery dynamics. A scalp EEG channel and a transformer feature are not automatically comparable units. Normalising by channel count can conceal a growth effect that remains visible under another denominator.

The useful question is which transformations the comparison is supposed to tolerate. Let A be the set of allowed changes of representation, such as relabelling channels, changing basis, resampling time or moving between equivalent implementations. An invariant statistic gives the same answer after a change in A. An equivariant statistic changes in a known way that can be corrected. A fragile statistic changes unpredictably. The source and target need not share every physical unit, but the claimed relation must survive the transformations declared irrelevant.

Consider four binary channels ordered as A, A, B, B. Run-length coding compresses the repeated blocks. Reorder the same channel states as A, B, A, B and the compressed string grows. The system state did not change, only the presentation order did. That is not necessarily a defect if adjacency is part of the construct. It is a defect if the paper describes the result as basis-free complexity. A transfer study should publish a sensitivity surface across grain, basis, perturbation strength and normalisation, not only its best-looking scalar.

flowchart LR
  A["Opening case"] --> B["A number does not carry its<br/>meaning"]
  B --> C["Four spans, four different loads"]
  C --> D["Break the bridge on purpose"]
  D -. "re-test" .-> A
  classDef first fill:#0a1424,stroke:#007D68,stroke-width:2px,color:#e6f3ff
  classDef step fill:#0a1424,stroke:#3f7da0,stroke-width:1px,color:#e6f3ff
  class A first
  class B,C,D step

Four spans, four different loads

The four homologies are ordered because later claims depend on earlier ones. They are not stages of increasing “similarity”. Each asks a different counterfactual question. Could the quantity be reproduced with a different ruler? Could the score be reproduced by another mechanism? Could the source phenomenon be absent while the mechanism remains? Could the same observation support a different conclusion in the target domain?

1. Quantity homology: does the ruler preserve its object?

Quantity homology holds when the source and target procedures preserve the relevant mathematical object under stated transformations. For a perturbational measure, this includes the intervention type and strength, the recorded variables, the temporal window, spatial grain, thresholding, compression method and normalisation. It also includes the reference class. A score of 0.5 relative to healthy human EEG is not automatically the midpoint of an architecture-independent scale.

The key test is not “did we use the same code?” It is “would a legitimate change of representation preserve the comparison?” If one researcher records neurons, another records cortical parcels and a third records model features, their results need an invariance argument. Otherwise, channel count or basis choice may dominate the score.

Quantity transfer can succeed at several strengths. The weakest form preserves a computable object, such as a binary response matrix. A stronger form preserves rank order across systems. Stronger still is threshold invariance, where the same boundary separates independently established conditions. The strongest decision-level form preserves the consequences of crossing that boundary. Researchers often demonstrate the first and speak as though they have the fourth. A machine study should state the achieved level directly. “The statistic can be calculated” is useful, but it is not “the clinical cutoff transfers”.

2. Mechanism homology: would the same intervention matter for the same reason?

Mechanism homology concerns causal organisation. A biological complexity measure may receive its relevance from a balance of differentiation and integration in a system whose parts exert effective influence on one another. A machine can generate a similarly complex trace through recurrence, through independent random sources, through a lookup table, or through a controller outside the measured boundary. Those generators are observationally confusable until an intervention separates them.

Reviews of brain-complexity measures already warn that differentiation-only metrics can be inflated by independent generators or noise, while correlation-only metrics can reward stereotyped global activity (Sarasso et al., 2021). The transfer problem magnifies that warning because artificial systems offer many new ways to manufacture the surface pattern.

A causal fingerprint needs more than one ablation. Vary perturbation site, timing and amplitude. Cut candidate connections. Swap a stored state. Reset one subsystem while preserving the others. Repeat the intervention from several initial states. Then compare the topology of spread, the decay profile, the dependence on prior state and the system’s recovery. Two substrates can differ physically yet share a theory-relevant fingerprint. Conversely, one architecture can change fingerprint when a scheduler, cache or external service is moved across the boundary.

Necessity tests also need controls. If removing recurrence collapses the score because all activity collapses, the experiment has not isolated recurrence’s distinctive role. A matched control should preserve overall activity, response energy and task difficulty while changing the proposed organisation. Sufficiency is harder still. Adding a recurrent loop that raises complexity may show that recurrence contributes to the statistic, not that the loop supplies the phenomenon. Mechanism homology is earned when a preregistered family of interventions distinguishes the proposed generator from matched rivals.

3. Phenomenon homology: is the measure still about the same thing?

A measure is designed for a target phenomenon, but the phenomenon may be narrower than its public label. PCI has been interpreted as an index of the brain’s capacity to sustain consciousness, not a complete readout of what a person experiences. Other tests may target conscious access, metacognitive availability, recurrent sensory processing or wakeful responsiveness. Moving the measure into a machine requires a precise counterpart.

This is where metaphysical commitments enter. A computational functionalist may hold that the right causal organisation is sufficient for consciousness across substrates. A biological naturalist may require specific living processes. An emergentist may require a particular scale and dynamics. An idealist, neutral monist, panpsychist, dual-aspect theorist or process philosopher may place the explanatory burden differently. The audit does not settle these views. It requires the researcher to state which view makes the phenomenon portable and what result would count against that portability.

A phenomenon claim should therefore be embedded in a small nomological network. Which independent observations should covary if the target property is present? Which nearby properties should dissociate? In humans, a proposed consciousness test may be checked against reports, behaviour, state transitions, lesions, anaesthesia and converging measures, while recognising that none is an infallible oracle. In machines, there is no accepted ground-truth population. The best available strategy is triangulation under declared theory: use indicators produced by different methods, seek predicted dissociations and include systems designed to satisfy one indicator while violating another.

Generated self-report cannot validate the same architecture that produced the report unless leakage and prompting alternatives are controlled. Task success cannot validate phenomenal capacity when the task can be solved feed-forward. A broad broadcast signature cannot validate subject unity if two disconnected controllers produce it through a shared log. These are discriminant-validity tests in the spirit of multi-method measurement (Campbell and Fiske, 1959). They do not prove experience. They stop one operational proxy from defining the phenomenon by itself.

Phenomenon homology is not biological resemblance. It is an explicit argument that the property targeted in the source has a defensible counterpart in the target. If the source target is “capacity for human-like conscious experience”, an artificial score does not inherit that target by using the same statistic. If the target is merely “rich, integrated perturbation response”, the transfer may succeed, but the conclusion is structural rather than phenomenal.

4. Inference homology: does the source evidence licence the target claim?

Inference homology is the final and most demanding span. It asks whether the empirical relation learned in the source domain remains valid in the target domain. Causal transportability research frames this as a licensing problem. Source and target differ in specified mechanisms; an inference travels only when those differences do not block the causal query, or when target observations can repair the difference (Pearl and Bareinboim, 2014).

For a consciousness measure, the source graph may contain biological variables such as cortical bistability, recurrent effective connectivity, anaesthetic action, report capacity and lesions. The target graph may contain activation routing, external state, scheduler timing, deterministic replay, sampling temperature and tool orchestration. A common score node does not make the graphs equivalent. The path from score to consciousness may pass through source-specific causes.

Three transport cases clarify the burden. In the first, source and target differ only in variables known not to affect the causal query. Direct transfer may be justified. In the second, they differ in an observed variable, such as delay or perturbation strength, and target data can adjust the estimate. In the third, the source relation depends on an unobserved or target-absent mechanism. No amount of matching on the score repairs the gap. The study must either measure the missing relation, narrow the conclusion or abstain.

Artificial calibration is possible, but it cannot be created by relabelling the top-scoring machines as positive cases. Useful anchors must be independent of the score. They might include theory-predicted causal signatures, negative-control architectures, blinded human judgements about predeclared behavioural properties or outcomes from interventions not used to construct the measure. Measurement-invariance work offers a parallel warning: equal-looking scores do not guarantee equal latent meaning across groups when item relations shift (Meredith, 1993). The cross-substrate case is more demanding because both the “items” and the candidate construct may change.

Contemporary proposals for AI consciousness indicators make the theoretical dependency explicit. Butlin and colleagues derive indicators from theories and use them to inform credences, particularly when a theory has computational implications (Butlin et al., 2026). That is more disciplined than importing a biological cutoff. It still leaves each theory’s assumptions, indicator validity and target implementation open to challenge. Bayne and colleagues similarly emphasise that tests for consciousness outside consensus cases require validation strategies rather than simple extension (Bayne et al., 2024).

Break the bridge on purpose

A transfer claim becomes informative when it survives a designed attempt to break it. The most useful experiment holds the score steady while changing the proposed reason for the score, or holds the mechanism steady while changing the ruler. This turns similarity into a discriminating test.

A four-cell break protocol

Use four paired tests before treating a biological measure as machine evidence. First, keep the target fixed and move the ruler through plausible grains, bases and normalisations. This tests quantity. Second, keep the score approximately fixed and replace the generator with replay, independent noise, a lookup table or an external controller. This tests mechanism. Third, preserve the proposed organisation while removing a theory-linked property, such as broad availability, recurrent dependence or unified control. This tests the phenomenon bridge. Fourth, hold the target observation fixed while changing a source-target difference in the causal graph. This tests inference.

The pairs should be chosen before the headline result is known. Record a lowering condition for every span: how much scale sensitivity defeats quantity, which rival result defeats mechanism, which dissociation weakens the phenomenon claim and which unmeasured difference blocks transport. A positive test then has a clear meaning. A negative test is not a failed paper. It identifies the exact bridge span that needs a better measure or a narrower claim.

A configured-system example

Consider Vela-4, a synthetic research agent built to make the transfer problem inspectable. It has a language model that proposes actions, a recurrent controller that tracks task state, a typed tool gateway, an authoritative world-state service and an append-only evidence ledger. The model weights are not called memory. Runtime controller state, external world state, retrieved knowledge and decision receipts are recorded separately.

A researcher perturbs one controller variable, records the next 30 state transitions and computes a PCI-inspired score. The score falls when recurrence is disabled and rises when the controller is allowed to coordinate several state channels. This is interesting evidence about the configured system. What does the four-span audit permit?

stateDiagram-v2
  [*] --> Claim
  state "The four-span bridge" as Claim
  state "The anatomy of a measurement claim" as Test
  state "The smallest useful formalism" as Revision
  Claim --> Test: expose mechanism
  Test --> Decision: evidence survives
  Test --> Revision: rival survives
  Revision --> Claim: narrow or rebuild
  Decision --> [*]

The audit that refuses to average

Many assurance tools convert heterogeneous evidence into a weighted score. That is attractive because it yields one sortable number. It is dangerous here. Strong quantity homology cannot compensate for a missing phenomenon. Ten replications of a statistic cannot repair an invalid target construct. An elegant mechanism cannot transport a human clinical cutoff without a reference class.

The four-homology audit uses a weakest-span rule. Each span can be failed, untested, partial or supported. The tool then reports the strongest claim level that all necessary spans permit. It does not estimate the probability that a system is conscious. It records what an observation can currently be used to say.

The status labels are categorical for a reason. “Partial” does not mean half a homology and must not be converted into 0.5. It means a named subclaim is supported while a necessary test remains open. “Failed” means an observed result violates a declared requirement. “Untested” means no relevant evidence exists. “Supported” always carries a scope, version and reference class. This vocabulary prevents uncertainty from being disguised as arithmetic precision.

The weakest-span rule is deliberately asymmetric. Evidence at a later span can motivate work on an earlier one, but it cannot repair it. A persuasive philosophical argument for substrate independence does not make two statistics commensurable. A clean causal intervention does not create an independent target phenomenon. A target calibration study cannot rescue a measure that changes arbitrarily with channel ordering. The audit preserves the dependency structure of the claim.

How to apply the audit

1. Write the source claim before looking at the target score. Name the original population, intervention, observable, reference class and interpretation.
2. Freeze the candidate boundary. List weights, runtime state, external world state, retrieved knowledge, tools and evidence ledgers separately.
3. Rebuild the quantity. Justify units, grain, sampling, thresholding and normalisation. Run basis, scale and noise sensitivity checks.
4. Construct a matched rival generator. Preserve the observed score while removing or replacing the proposed causal organisation.
5. Name the phenomenon and theory. State whether the target is access, capacity, valence, subject unity, report, experience or another property.
6. Draw the source-target difference graph. Mark which mechanisms differ and what target observations or interventions could repair the inference.
7. Publish the permitted claim and prohibited inferences. Record what a positive result allows, what a negative result means and what remains outside the test.

For a positive result, report the narrowest newly licensed claim. Quantity support permits score comparison under the tested transformations. Mechanism support permits a causal-organisation claim under the intervention family. Phenomenon support permits theory-conditional relevance to the named property. Inference support permits transport only across the declared source-target differences. For a negative result, report which span failed and the smallest repair experiment. The tool cannot establish consciousness, settle a metaphysical theory or convert missing target labels into evidence.

Run the audit before and after the experiment. The pre-run forces assumptions and break conditions into the open. The post-run records which evidence changed the status. Exported JSON can be versioned beside code, model configuration, perturbation manifest and raw traces. This creates a revision trail in which a later architecture change cannot silently inherit an earlier conclusion.

type Status = "failed" | "untested" | "partial" | "supported";
type Homology = "quantity" | "mechanism" | "phenomenon" | "inference";

type Audit = {
  measure: string;
  target: string;
  spans: Record<Homology, Status>;
  assumptions: string[];
  breakConditions: string[];
  prohibitedInferences: string[];
};

const claimOrder = [
  "no cross-system comparison",
  "descriptive quantity comparison",
  "mechanism-relative structural evidence",
  "phenomenon-relevant conditional evidence",
  "transported inference under declared assumptions"
] as const;

function permittedClaim(a: Audit): typeof claimOrder[number] {
  const s = a.spans;
  if (s.quantity !== "supported") return claimOrder[0];
  if (s.mechanism !== "supported") return claimOrder[1];
  if (s.phenomenon !== "supported") return claimOrder[2];
  if (s.inference !== "supported") return claimOrder[3];
  return claimOrder[4];
}

// Invariant: no higher span can compensate for a failed lower span.

sequenceDiagram
  participant Q as Question
  participant M as Mechanism
  participant E as Evidence
  participant D as Decision
  Q->>M: Opening case
  M->>E: The four-span bridge
  E-->>M: compare rival explanations
  E->>D: Make every gap in the transfer bridge visible
  D-->>Q: revise scope or act

Make every gap in the transfer bridge visible

The usual question is, “Does this machine obtain a consciousness-associated score?” The better question is, “Which evidential relation survives from the biological source to this configured target?” The first invites threshold hunting. The second forces a boundary, an intervention family, a target phenomenon and a transport licence.

This reframing also separates two legitimate research programmes. One programme studies architecture-independent structural properties such as perturbational richness, recurrence or broadcast. It can make progress without using the word consciousness. The other asks whether those properties bear on consciousness under a stated theory and measurement bridge. Both matter. Confusion begins when a result from the first is announced as though the second had already succeeded.

A high score should therefore change one decision only: whether the system deserves a more discriminating experiment. It should not by itself change a welfare classification, public label or claim of subjectivity. When quantity homology holds, compare the measurements. When mechanism homology also holds, make a mechanism-relative claim. When phenomenon homology is argued under a declared theory, treat the result as conditional evidence about that phenomenon. When inference homology is supported by target calibration and explicit source-target assumptions, transport the inference within that scope.

The practical decision is to stop asking one number to carry four kinds of evidence. Every report should state the highest permitted claim and the prohibited next claim. “The target shows a quantitatively comparable perturbation response” is different from “the target implements the causal organisation proposed by theory.” Both are different from “the target may instantiate the relevant phenomenon,” and all are different from “the source evidence transfers.”

The framework may produce more abstentions than a scorecard. That is a feature. A negative quantity result means the cross-system comparison is not yet meaningful. A negative mechanism result points to a false friend or an underspecified intervention. A negative phenomenon result says the measure has become a structural diagnostic rather than a consciousness test. A negative inference result says the target lacks the calibration needed to reuse the source conclusion.

None of these negatives proves that an artificial system is not conscious. None proves that consciousness requires biology. They locate the missing work. The bridge becomes scientifically useful when its gaps are visible, because visible gaps can be turned into experiments rather than hidden inside a headline.