The bridge resemblance cannot carry

Imagine asking whether a cloud resembles a brain. You can answer yes by attending to branching filaments, distributed activity and changing local patterns. You can answer no by attending to metabolism, learning, tissue organisation and behaviour. Both answers can be sincere because resemblance depends on which features receive weight. Tversky’s classic analysis of similarity made this asymmetry explicit: the comparison changes when the chosen features and their salience change (Tversky, 1977).

That dependence makes resemblance generative and dangerous. It is generative because a selective likeness can suggest a new model. It is dangerous because the selection can disappear from view. Once “the atmosphere computes” or “the brain is a prediction machine” becomes familiar, readers may forget which properties were preserved, which were ignored and what inference the comparison was meant to support.

Structure-mapping research gives a better starting point. Gentner distinguished matches of isolated attributes from alignments of relations and systems of relations (Gentner, 1983). A useful analogy often preserves a pattern such as “A constrains B, which redirects C”, even when A, B and C have different material properties in the two domains. This explains why a circuit can illuminate a hydraulic system without either being made of the other’s substance. It does not yet tell us that the aligned relation is true of the target, causally important or complete.

Philosophy of scientific representation supplies a second warning. Similarity by itself cannot determine what represents what. A weather map resembles another weather map more than it resembles the atmosphere, yet it is used to reason about the atmosphere. A child’s sketch may be less geometrically accurate than a photograph and more useful for explaining a mechanism. Representation depends on use, interpretation and licensed inference, not on an unqualified amount of likeness (Suárez, 2003) (Hughes, 1997).

The minimum causal grammar

A disciplined analogy needs six items. Let S be the source system and T the target. Let m map selected source objects to selected target objects. Let R_S and R_T be the relations claimed to correspond. Let I be the interventions relevant to the intended claim. Finally, let C be the conclusion we want to transfer.

The expression is deliberately less grand than a theorem. It forces the practical questions. What are the objects? Which relations matter? What can be changed? Which conclusion is at stake? A mapping can preserve order while failing to preserve distance. It can preserve input-output behaviour while failing to preserve internal state. It can preserve a causal mechanism over a narrow operating range while failing outside it.

Suppose two systems agree on four ordinary cases. That gives four matched observations. Now apply four counterfactual changes and they agree on only one. It would be misleading to report “eight tests, five matches” as a single 62.5 per cent similarity score. The ordinary and counterfactual cases answer different questions. The first show behavioural overlap on sampled inputs. The second test whether the proposed relation carries causal load. Preservation is not the quantity of resemblance; it is the survival of the right relation under the right change.

flowchart TD
  A["Two red alerts, one dangerous<br/>conclusion"] --> B["The bridge resemblance cannot<br/>carry"]
  B --> C["Five loads, five inference<br/>licences"]
  C --> D["Collision tests for living<br/>mappings"]
  D -. "re-test" .-> A
  classDef first fill:#0a1424,stroke:#D95545,stroke-width:2px,color:#e6f3ff
  classDef step fill:#0a1424,stroke:#3f7da0,stroke-width:1px,color:#e6f3ff
  class A first
  class B,C,D step

Five loads, five inference licences

The five labels are often presented as a staircase from weak to strong. That image helps only if we remember what moves at each step. It is not confidence alone. The objects of comparison become more explicit, the preservation obligations become sharper, and the range of permitted inference changes.

Metaphor: permission to ask

A metaphor transfers an image or organising phrase. “Memory is a library” foregrounds storage, retrieval and loss. It may prompt useful questions about indexing or access. It does not require every memory to be a stored document, nor does it predict what happens when the hippocampus is damaged. A metaphor’s legitimate output is a research question, design prompt or explanatory intuition.

Metaphors become risky when their exclusions disappear. A library has a librarian, stable records and separable items. Human remembering is reconstructive, context-sensitive and partly enacted in present activity. The metaphor can still help, but the omitted differences must remain visible. A good metaphor produces hypotheses faster than it produces beliefs.

Correspondence: permission to compare a role

A correspondence pairs named elements or relations for a stated purpose. Pressure difference corresponds to voltage difference in a familiar hydraulic-electric analogy. Flow rate corresponds to current. A constriction corresponds to resistance. The pairing is not just verbal because it supports a common relation: under a linear regime, increasing the driving difference increases the flow, while increasing resistance decreases it.

A correspondence is still selective. Water has mass, can leak and may form vortices. Electric charge behaves differently. The mapping earns a bounded prediction only where the aligned relation and assumptions hold.

Structural homology: permission to investigate a shared pattern

“Homology” is not a universal synonym for deep similarity. In evolutionary biology it normally concerns shared ancestry, sometimes with developmental refinements (Haber, 2021). Bird and bat forelimbs can be homologous as inherited forelimbs even though their wings evolved separately as flight structures. Borrowing that word for unrelated systems can smuggle in a common origin that has not been shown.

For cross-domain work, the safer status is structural-homology candidate. It means that several linked relations co-vary in both systems because of a proposed common generative constraint, design principle or transformation family. The candidate must survive coordinated changes, not one cherry-picked match. It licences a comparative research programme. It does not establish shared ancestry, mechanism or ontology.

Formal mapping: permission to transfer scoped propositions

A formal mapping supplies typed objects, a domain and codomain, and an explicit preservation rule. A graph homomorphism preserves adjacency in one direction. An isomorphism adds a bijection and inverse preservation. A functor preserves identities and composition between categories. These are different obligations. “There is a mapping” is therefore incomplete until the map type and preserved structure are named.

Formalisation narrows ambiguity, but it does not automatically connect the formal objects to the world. A perfectly proved isomorphism between two mathematical representations says nothing about whether either representation measures the intended phenomenon. Suárez’s criticism of similarity and isomorphism accounts of scientific representation is relevant here: formal relation alone does not settle representational use or empirical adequacy (Suárez, 2003).

Proven equivalence: permission for invariant substitution

Equivalence is always equivalence under a criterion. Isomorphism preserves the structure encoded by a signature. Bisimulation requires related processes to match one another’s transitions and remain related after each move (Hermida, 2022). Equivalence of categories is weaker than isomorphism of categories but preserves categorical structure up to natural isomorphism. Observational equivalence in a programming language depends on what contexts are allowed to observe.

A proof permits substitution only for properties invariant under that criterion. Bisimilar processes can still differ in implementation cost. Isomorphic graphs can be embedded differently in physical space. Categorically equivalent presentations can have different literal objects. Proven equivalence eliminates a declared class of differences, not every possible difference.

Why the statuses are not a confidence score

A formal status can be narrow and certain, while a lower status can be broad and empirically persuasive. The map from Celsius to Fahrenheit is an exact affine bijection. It preserves temperature order and equal intervals, and it has an inverse. Yet it does not preserve the numerical zero, and neither scale makes temperature ratios physically meaningful. Twenty degrees Celsius is not twice as hot as ten. The Kelvin scale supports a different class of ratio statements because its zero has a different physical interpretation.

This example separates two questions that reviews often merge. How confident are we that the stated relation holds? What kind of relation is it? A team may be highly confident in a functional correspondence and still lack a formal map. Another team may possess an impeccable graph isomorphism between two abstractions while remaining uncertain that either abstraction captures the real system. Confidence concerns evidence for the premises. Mapping status concerns the form and scope of the preservation claim.

The separation matters when evidence accumulates unevenly. Suppose two biological and artificial systems show the same three-stage behavioural curve across hundreds of trials. Replication can make the curve reliable. It does not reveal whether the same mechanism generated it. Conversely, a formal proof that two algorithms return the same result for all valid inputs may be decisive about extensional behaviour while silent about latency, memory, failure recovery and internal causal route.

Never average confidence and preservation into one “analogy strength” number. Record confidence in each claim, then assign status through obligations. This prevents a large sample from compensating for an absent causal intervention, and it prevents mathematical elegance from compensating for a poor empirical bridge.

Collision tests for living mappings

A comparison becomes trustworthy when it is forced to collide with cases it did not choose. The audit uses seven collisions. They are not a universal proof calculus. They are a way to expose where vocabulary, observation, mechanism and formal structure have been quietly merged.

Source freeze. Describe the source on its own terms before mentioning the target. A global workspace account, for example, includes commitments about access, competition, availability and consumer processes. Replacing those terms immediately with “attention heads” or “residual stream” makes the mapping impossible to evaluate because the target vocabulary has already colonised the source.

Target freeze. Describe the configured target without borrowed language. Name the model version, runtime, state, interfaces, consumers, authority boundaries and observables. “The agent remembers” is not a target description. “The runtime retrieves an episodic record and appends it to permitted context” is.

Alignment. Write the proposed pairs and relations. Unmapped source elements remain visible rather than being treated as irrelevant by default. Unpaired target components reveal alternative generators. A sparse alignment is not a failure. It is often the honest result.

Intervention commutation. Change an aligned source variable and its proposed target counterpart. The downstream relational change should match in the way the claim predicts. Causal abstraction work in mechanistic interpretability formalises versions of this idea through high-level and low-level interventions (Geiger et al., 2022) (Geiger et al., 2025). Behavioural correlation alone does not supply that counterfactual relation.

Rival generator. Build or name a different mechanism that could produce the same evidence. The red-lamp operator is a rival generator. In machine learning, dataset leakage, a shortcut feature or a separate lookup path may mimic the mapped mechanism.

Break test. State the smallest change that should break the mapping. A comparison that cannot imagine a negative result is a lens, not a test. The break can occur in vocabulary, function, causal structure, temporal organisation, embodiment or ontology.

Invariant audit. Name the exact properties allowed to transfer if the test succeeds. This final step prevents a local result from escaping its scope. Showing that two systems implement the same Boolean function does not show equal energy use, learning history or experience.

A realistic audit: residual stream and global workspace

Consider the statement, “A transformer residual stream is a global workspace.” It is attractive because information written into the residual stream can affect many later computations. Global neuronal workspace theories also emphasise broad availability after selection, alongside competition, access limits and recurrent amplification (Dehaene and Naccache, 2001) (Mashour et al., 2020).

Freeze the source. A workspace account proposes functional roles and empirical dynamics in a biological system: candidate contents compete; a limited subset gains access; selected information becomes available to multiple specialist processes; and report, flexible control or memory can depend on that availability. Different workspace theories disagree about details and about whether the theory explains conscious access, consciousness itself or a neural correlate.

Freeze the target. In a standard transformer, each token position carries a vector through a stack of attention and feed-forward transformations. Residual addition lets many operations read and modify a shared representational stream. The configured system may also include prompts, retrieval, tools and external state, none of which belongs to the residual stream itself.

The narrow correspondence is clear: a shared representational medium can make selected information usable by multiple downstream operations. The stronger homology claim needs more. Where is the capacity-limited competition? Which consumers count as independent specialist processes? Does a local intervention on the selected content produce the predicted broad pattern of downstream effects? Does the mapping survive different tasks, layers and architectures? A high activation or attention score is not enough.

Representation similarity research makes this boundary practical. A metric may be invariant to orthogonal transformation, isotropic scaling or other changes, depending on its design. Learned representations can also be identifiable only up to a family of linear transformations (Roeder et al., 2021). A high score therefore supports a statement about the equivalence class encoded by the metric, not automatically about neuron-wise alignment, algorithmic identity or causal use.

The same discipline applies outside machine learning. Two contemplative maps may order stages similarly while differing in purpose, authority and metaphysics. A biological and artificial network may share graph statistics while differing in plasticity and embodiment. A philosophical account of self may inspire a useful agent boundary without making that boundary a subject. Cross-domain comparison is strongest when the residue is published alongside the match.

stateDiagram-v2
  [*] --> Claim
  state "A bridge tested under five<br/>increasing loads" as Claim
  state "The two red lamps" as Test
  state "The same output can sit<br/>downstream of different causes" as Revision
  Claim --> Test: expose mechanism
  Test --> Decision: evidence survives
  Test --> Revision: rival survives
  Revision --> Claim: narrow or rebuild
  Decision --> [*]

The audit you can run

The mapping-status dossier turns the argument into a repeatable decision. Its classifier is gate-based rather than additive. A polished metaphor with seven attractive similarities cannot “score” its way past a missing intervention. A formal map cannot become an equivalence because the team feels confident. Status stops at the first unmet obligation.

The dossier also distinguishes three kinds of preservation that are often collapsed. Observational preservation means selected descriptions or outputs match. Intervention preservation means aligned changes produce aligned downstream changes. Inferential preservation means the particular conclusion transferred is invariant under the accepted mapping or equivalence. A comparison can pass the first and fail the other two.

This yields a new practical rule: choose the weakest status sufficient for the decision. Brainstorming needs a metaphor and a break point. Architecture design may need a scoped correspondence. A mechanism claim needs intervention-level preservation and rivals. Formal property transfer needs a typed map. Substitutability needs a domain-specific equivalence proof. There is no virtue in requesting the highest label when a lower one does the job.

Four questions before the comparison reaches a decision

A review board does not need to settle every philosophical issue. It needs to prevent a comparison from carrying more authority than its receipts. Four questions expose most failures quickly.

What decision is the analogy meant to change? “This architecture resembles working memory” is too open. Is the comparison selecting a buffer design, predicting a failure mode, interpreting an activation or supporting a claim about experience? Each decision demands a different bridge. If no decision is named, the comparison should remain exploratory.

Which observation would be surprising without the proposed relation? Shared terminology is never surprising. Similar outputs may be common when systems share a task or training objective. A stronger observation is a held-out counterfactual predicted before the result, especially when a plausible rival predicts something else. This question converts admiration into risk.

Which alternative generator has been made concrete? It is easy to mention “confounds” abstractly. Build the lookup table, shortcut classifier, independent retrieval path or differently wired controller that could mimic the evidence. A rival that can be executed changes the comparison from a story contest into an assay.

What is the smallest break that would force a downgrade? The answer should identify a receipt, not a mood. A failed patch may downgrade a mechanism claim to behavioural correspondence. A non-bijective map may leave a homomorphism intact while defeating isomorphism. A new source interpretation may preserve the metaphor and invalidate the element pairing.

A null result can therefore improve the work. It may show that the source was reconstructed badly, the target boundary was wrong or the proposed relation was not causally active. The audit preserves those possibilities separately. The purpose of a break test is not to destroy an analogy; it is to locate the exact level at which it remains useful.

status: metaphor_unaudited
licence: question_generation_only

How to use the result

A positive result means the claim has satisfied every gate up to the displayed status. It permits the listed use and no more. A correspondence result can justify a bounded experiment or design choice. A structural-homology candidate can justify coordinated comparative testing. A formal mapping can transfer propositions that its preservation rule covers. An equivalence result permits substitution only for the stated invariant properties.

A negative result is not a verdict that the analogy is worthless. It identifies what would have to be learned next. If the dossier stops at metaphor because the target is unconfigured, specify the system. If it stops at correspondence because interventions are absent, design them. If a formal map lacks a reverse witness, keep the transfer one-way. Downgrading a claim preserves the useful idea while removing the unearned conclusion.

The tool cannot establish empirical truth, select the correct source interpretation or prove a mathematical theorem by checkbox. It cannot decide consciousness, moral status or metaphysical identity. Those questions require independent theory, candidate boundaries and evidence. The dossier does something narrower: it makes the inference debt legible before the comparison enters a decision.

function classifyMapping(g) {
  const stages = [
    { name: "metaphor", needs: ["break"] },
    { name: "correspondence", needs: ["source", "target", "role"] },
    { name: "structural_homology_candidate",
      needs: ["multi", "generator", "intervention", "rival"] },
    { name: "formal_mapping", needs: ["typed", "proof"] },
    { name: "proven_equivalence", needs: ["reverse", "invariants"] }
  ];

  let reached = "metaphor_unaudited";
  for (const stage of stages) {
    const missing = stage.needs.find(key => !g[key]);
    if (missing) return { reached, missing };
    reached = stage.name;
  }
  return { reached, missing: null };
}

{
  "claim": "selected source content corresponds to a patched target subspace",
  "source": {"objects": [], "relations": [], "interpretation": "declared"},
  "target": {"configuration": {}, "objects": [], "relations": []},
  "mapping": {"pairs": [], "preserves": [], "scope": []},
  "interventions": {"source": [], "target": [], "rivals": []},
  "break_conditions": [],
  "formal_witnesses": [],
  "licensed_inferences": [],
  "prohibited_inferences": []
}

sequenceDiagram
  participant Q as Question
  participant M as Mechanism
  participant E as Evidence
  participant D as Decision
  Q->>M: Two red alerts, one dangerous conclusion
  M->>E: A bridge tested under five increasing loads
  E-->>M: compare rival explanations
  E->>D: Stop the claim at the break, not the inquiry
  D-->>Q: revise scope or act

Stop the claim at the break, not the inquiry

Comparisons are unavoidable. Engineers borrow from biology. Neuroscientists borrow from computation. Philosophers use thought experiments to move relations between worlds. Contemplative traditions offer distinctions that can sharpen research questions. The choice is not between analogy and purity. It is between visible and invisible inference.

Before a comparison changes an architecture, experiment, release decision or ethical posture, write its dossier. Reconstruct the source without target vocabulary. Describe the configured target without inherited metaphysics. State the mapped objects, preserved relations, interventions, rivals and break conditions. Then assign the strongest status that has receipts.

For a metaphor, retain the question. For a correspondence, build the bounded prototype. For a structural-homology candidate, preregister coordinated perturbations. For a formal map, prove the preservation condition. For equivalence, name the criterion, provide its witnesses and list the invariant properties for which substitution is valid.

The practical shift is small and severe. Stop asking, “How convincing is the resemblance?” Ask, “Which relation survives which intervention, and what may I now infer?” When the answer reaches a break, stop the claim there, not the inquiry.