Cross-Domain Functors
Definition
A cross-domain functor is a structure-preserving mapping between two ontology logs (ologs) representing different domains. Following Giesa, Spivak, & Buehler (2011), a functor F: Olog_A → Olog_B maps objects in domain A to objects in domain B, and arrows in A to arrows in B, such that compositional structure is preserved: if a path commutes in A, its image under F must commute in B. The commutativity condition is the formal criterion for a valid cross-domain analogy.
This page constructs the three functors that unify the wiki’s domain ologs and identifies precisely where each functor holds, where it breaks, and what the breaks mean.
The Three Ologs
| Olog | Domain | Objects | Arrows | Commutativity conditions |
|---|---|---|---|---|
| cancer-evolution-olog | Cancer evolution | 33 (7 levels) | 49 (7 groups) | 11 (1 non-commuting, 1 conflict) |
| ecology-invasion-olog | Ecological invasion | 18 (5 levels) | 19 (5 groups) | 5 |
| compression-progress-olog | Compression progress | 24 (5 levels) | 16 (4 groups) | 8 (2 with documented breaks) |
The cancer evolution olog is the target category — the canonical representation of the home domain. All cross-domain functors map INTO it from ecology and compression.
Functor F: EcologyOlog → CancerOlog
The bottleneck functor. Maps ecological invasion dynamics onto cancer evolution dynamics.
Object mapping
| Ecology object | Cancer object | Rationale |
|---|---|---|
| PlantPopulation | TumorCellPopulation | A population of genetically related individuals in a shared environment |
| Genotype | GenomeState | Heritable genetic configuration |
| GeneticDiversity | IntratumorHeterogeneity | Measured variation across individuals/clones |
| Individual | Clone | Unit of selection and propagation |
| Phenotype | Clone phenotype (gene expression, drug sensitivity) | Observable traits |
| PhenotypicPlasticity | Epigenetic plasticity (chromatin remodeling, transcriptional state) | Non-genetic phenotypic variation |
| FounderEvent | BottleneckEvent | Severe population reduction from a small number of founders |
| EnvironmentPatch | Microenvironment | Local conditions that modulate phenotype |
| WaterAvailability | TherapeuticPressure | Key environmental variable driving selection |
| BioclimaticNiche | Metastatic niche / drug-resistant state | Set of conditions where the population can persist |
| NicheOccupancy | Relapse / therapeutic failure | Successful population establishment |
| IntroducedRange | Post-therapy state | New environment colonized after dispersal/bottleneck |
Arrow mapping
| Ecology arrow | Cancer arrow | Commutativity |
|---|---|---|
hasDiversity: PlantPopulation → GeneticDiversity | hasITH: TumorCellPopulation → IntratumorHeterogeneity | ✓ |
expresses: Genotype × EnvironmentPatch → Phenotype | hasPhenotype: GenomeState × Microenvironment → DrugSensitivity | ✓ |
hasPlasticity: Genotype → PhenotypicPlasticity | hasEpigeneticPlasticity: GenomeState → ChromatinState | Conditional |
founds → bottlenecks | therapyCreates: TherapeuticPressure → BottleneckEvent | ✓ |
invades: PlantPopulation × IntroducedRange → NicheOccupancy | relapses: TumorCellPopulation × PostTherapyState → Relapse | ✓ |
Commutativity verification
C1: Plasticity-Diversity Trade-off (ecology CC1 → cancer). The ecology condition states: Genotype → hasPlasticity → enables → NicheOccupancy commutes with Genotype → hasDiversity → constrains → NicheOccupancy. Under F, this maps to: GenomeState → hasEpigeneticPlasticity → enables → Relapse commutes with GenomeState → hasITH → constrains → Relapse.
Verdict: Holds conditionally. The dual-regime model (dual-regime-evolution) explicitly claims that epigenetic plasticity and genetic diversity are alternative adaptive routes. Mikutenaite et al. (2025) provide direct evidence: transcriptional plasticity enabled metastatic colonization without new mutations. Walens et al. (2020) show the dual-route finding: Met amplification (genetic route) vs. Jak/Stat (plasticity route) both lead to recurrence. The commutativity holds when either route is available; it breaks when both routes are blocked (e.g., DDR-proficient tumors with low basal chromatin plasticity).
C2: Founder Bottleneck Condition (ecology CC2 → cancer). The ecology condition states: FounderEvent → bottlenecks → PlantPopulation → hasDiversity → low GeneticDiversity depends on bottleneck severity. Under F: BottleneckEvent → reduces → TumorCellPopulation → hasITH → low IntratumorHeterogeneity.
Verdict: Holds with documented nuance. Miething (2019) confirms: deeper response (CR/vgPR, severe bottleneck) → branching relapse (re-diversification). Walens (2020) confirms: ~50% clonal dominance (low diversity), ~50% polyclonal (high diversity). The ecology olog’s binary outcome (single vs. multiple introductions → low vs. high diversity) maps onto the cancer olog’s dual-route finding. The bottleneck severity determines residual diversity in both domains, but the cancer domain adds a re-diversification phase (branching relapse) that has no clear ecological analogue (plants don’t mutate new genotypes during regrowth).
C3: Plasticity Conservation (ecology CC3 → cancer). The ecology condition states: plasticity is a species-level trait, not eroded by bottlenecks. Under F: epigenetic plasticity is a cell-lineage-level trait, not eliminated by therapy bottlenecks.
Verdict: Conditionally holds. Some epigenetic plasticity mechanisms are indeed conserved across bottlenecks (e.g., EMT as a generic stress response — Walens 2020 finds EMT in all recurrent tumors regardless of clonal architecture). But other plasticity mechanisms are genetically encoded (e.g., IDH1/2 mutations alter global methylation capacity (dual-regime-evolution §Coupling Between Regimes; PCAWG Consortium, 2020)). The functor preserves the condition for “deep” plasticity mechanisms (stress responses, EMT, heat shock) but not for genetically-encoded plasticity modifiers. This is a domain boundary — the ecology domain has no analogue of a mutation disabling plasticity itself.
C4: Niche Occupancy Independence (ecology CC4 → cancer). The ecology condition states: low genetic diversity does NOT predict low niche occupancy — the diversity path gives the wrong answer. Under F: low ITH does NOT predict low relapse risk — the diversity-only path gives the wrong answer.
Verdict: This is the most clinically significant commutativity condition. The compression-entrenchment hypothesis (compression-progress-evolution) predicts exactly this: monoclonal (low-ITH) tumors can be MORE dangerous than moderate-ITH tumors because they represent successful compressions. Standard theory (diversity → adaptation → worse outcome) makes the wrong prediction for the monoclonal regime. The ITH-outcome empirical test design (docs/superpowers/specs/2026-07-05-ith-outcome-test-design.md) is a direct test of this commutativity condition in the cancer domain. If the U-shaped ITH-outcome relationship holds, the functor is validated. If the relationship is monotonic (more ITH = worse outcome), the functor fails at this condition — the ecology domain’s plasticity-compensates-for-diversity dynamic does not map to cancer.
C5: Common Garden Commutativity (ecology CC5 → cancer). The ecology condition states: same genotype × same environment → same phenotype, regardless of population origin. Under F: same GenomeState × same Microenvironment → same DrugSensitivity, regardless of tumor origin.
Verdict: Fails in general — and this failure is biologically meaningful. Cancer clones from different tumors with the same driver mutation (e.g., KRAS G12D) can have different drug sensitivities due to: (a) different co-mutations, (b) different epigenetic backgrounds, (c) different microenvironments, (d) different evolutionary histories. The ecology condition holds because Alternanthera plasticity is a species-level trait (Mantel r=0.15, p=0.29 — no correlation between genetic distance and plasticity). Cancer has no species-level traits — every tumor is its own evolutionary lineage. This is where the functor is strictly a profunctor, not a functor: the mapping from GenomeState to DrugSensitivity is context-dependent, requiring specification of co-mutations, epigenetic state, and microenvironment as coherence conditions.
Functor F: Summary
| Commutativity condition | Status | Limitation |
|---|---|---|
| C1: Plasticity-Diversity Trade-off | Conditional | Requires at least one adaptive route available |
| C2: Founder Bottleneck | Holds | Cancer adds re-diversification phase (no ecological analogue) |
| C3: Plasticity Conservation | Conditional | Fails for genetically-encoded plasticity modifiers |
| C4: Niche Occupancy Independence | Testable | The ITH empirical test directly tests this |
| C5: Common Garden | Fails | Profunctor — cancer has no species-level traits |
Overall: F is a profunctor, not a strict functor. It preserves structure under specified coherence conditions (availability of adaptive routes, non-genetic plasticity mechanisms) but fails when the conditions are not met. The failures are informative: they identify domain boundaries where cancer evolution differs fundamentally from ecological invasion.
Functor G: CompressionOlog → CancerOlog
The compression-evolution functor. Maps Schmidhuber’s formal compression framework onto Darwinian clonal evolution. This functor was first sketched in compression-progress-evolution §Category-theoretic validation; the present construction formalizes it against the complete cancer olog.
Object mapping
| Compression object | Cancer object | Rationale |
|---|---|---|
| DataSequence | Microenvironment (the data to be compressed) | The environmental regularities the genome must encode |
| Regularity | Recurrent selective pressure | A compressible pattern in the environment |
| Noise | Stochastic environmental fluctuation | Incompressible — must be survived, not encoded |
| Compressor | GenomeState (the genome AS algorithm) | The genome encodes environmental regularities as regulatory programs |
| CompressedRepresentation | AdaptedGenome (the genome as output) | The result of selection: a genome encoding the environment |
| CompressionQuality | FitnessValue | How well the genome compresses the environment |
| CompressionProgress | FitnessGradient (d(fitness)/dt) | Rate of adaptive improvement |
| CuriousAgent | EvolvingLineage | A population exploring genotype space via mutation |
| IntrinsicReward | SelectionCoefficient (s) | Reward for finding a better compression |
| SubjectiveBeauty | AbsoluteFitness (stock) | How well-adapted the current genome is |
| SubjectiveInterestingness | FitnessGradient (flow) | How fast fitness is changing |
| Discovery | ClonalSweep (driver fixation) | A discrete compression breakthrough |
| BoringPredictability | FitnessPeak (local maximum) | Well-adapted, no further progress |
| BoringRandomness | FlatFitnessLandscape (drift regime) | No selection differential, incompressible variation |
| Exploration | Mutation + recombination (genotype space search) | Generating candidate compressions |
| Exploitation | ClonalExpansion (proliferation without innovation) | Using the current compression |
Arrow mapping
| Compression arrow | Cancer arrow | Commutativity |
|---|---|---|
compresses: DataSequence × Compressor → CompressedRepresentation | encodes: Microenvironment × GenomeState → AdaptedGenome | ✓ (under selection) |
hasQuality: CompressedRepresentation → CompressionQuality | hasFitness: AdaptedGenome → FitnessValue | ✓ |
improvesOn: Compressor_new × Compressor_old × DataSequence → CompressionProgress | improvesFitness: GenomeState_mutant × GenomeState_wt × Microenvironment → FitnessGradient | ✓ |
hasInterestingness: DataSequence × CuriousAgent → SubjectiveInterestingness | hasSelectionGradient: Microenvironment × EvolvingLineage → FitnessGradient | ✓ |
seeks: CuriousAgent → Exploration | mutates: EvolvingLineage → MutationGeneration | ✓ |
discovers: CuriousAgent × DataSequence → Discovery | sweeps: EvolvingLineage × Microenvironment → ClonalSweep | Conditional |
Commutativity verification
CC1: CompressionProgress = d(Beauty)/dt → FitnessGradient = d(Fitness)/dt. The core identity. Under G: CompressionProgress maps to FitnessGradient; SubjectiveBeauty maps to AbsoluteFitness. The derivative relationship is preserved.
Verdict: Holds structurally. The mathematical identity is category-theoretically exact. The empirical limitation is measurement: fitness gradients cannot be directly observed in the cancer domain without longitudinal sampling (Turajlic et al., 2019: “dense longitudinal sampling is necessary to accurately detect selection”). The functor is formally sound; the empirical challenge is operationalizing it.
CC2: Interestingness Gradient → Selection Gradient. Under G: SubjectiveInterestingness (first derivative of beauty) maps to FitnessGradient (rate of fitness change). A data sequence is interesting iff the learning curve has positive slope; a microenvironment is “interesting” (in evolutionary terms) iff it supports a nonzero selection gradient.
Verdict: Holds. This is the formal encoding of the compression-evolution isomorphism’s central claim: “interestingness is the fitness gradient” (compression-progress-evolution §The Compression-Evolution Isomorphism). The mapping is structure-preserving.
CC3: Exploration-Exploitation Boundary. Under G: CuriousAgent seeks Exploration iff CompressionProgress > 0 maps to EvolvingLineage generates Mutations iff FitnessGradient > 0.
Verdict: Partially holds. Stress-induced mutagenesis in bacteria is a direct analogue: when the current genotype is failing (poor compression), mutation rates spike. But cancer complicates this: many tumors have constitutively elevated mutation rates (mutator phenotype, CIN) that are NOT gradient-responsive. The functor holds for gradient-sensitive mutation (stress-induced mutagenesis) but fails for constitutive mutator phenotypes — cancer can’t “turn off” exploration when on a fitness peak.
CC4: Boredom Dichotomy → Fitness Landscape Extrema. Under G: BoringPredictability maps to FitnessPeak (no further progress possible); BoringRandomness maps to FlatFitnessLandscape (drift regime). Interesting data lies between.
Verdict: Holds. This maps onto the neutral regime of cancer evolution (Turajlic et al., 2019): between selection events, drift dominates, and the fitness landscape is flat. The dichotomy correctly predicts that both fitness peaks (fully adapted) and flat landscapes (purely neutral) appear “boring” — no detectable selection — but for different reasons.
CC5: Discovery as Discontinuity → ClonalSweep. Under G: Discovery (step-change in compression) maps to ClonalSweep (rapid clonal expansion).
Verdict: Holds under the Bozic-Nowak condition. A ClonalSweep is a Discovery iff τ_k (waiting time for next driver) > sweep_time. If a new driver appears before the previous sweep completes, the sweep is incomplete (clonal interference) — the discovery is “blurred” across multiple clones. The functor holds in the strong-selection, clean-sweep regime; it breaks in the clonal interference regime.
CC6: Evolution-Compression Isomorphism (5 subdiagrams). This is the composite mapping across the entire G functor, with five subdiagrams checking commutativity of core evolutionary paths.
| Subdiagram | Status |
|---|---|
| Driver mutation → fitness increase → sweep | Holds (strong selection, s >> 1/N) |
| Passenger mutation → no fitness change → drift | Fails — no compression analogue of “an irrelevant bit that doesn’t affect compression quality but gets carried along” |
| Selection evaluates compression quality | Holds |
| Sweep as discrete compression breakthrough | Holds (conditional on τ_k > sweep_time) |
| Cancer as decompression → loss of compression | Fails — decompression (loss of compression) has no unique inverse in the compression framework. Many decompressions can produce the same observable state. |
CC7: Self-Referential Limit. The genome IS both compressor and compressed data. Under G: the Compressor object (genome as algorithm) and the DataSequence object (genome as substrate for selection) are the same entity. This is an endofunctor — a map from the category to itself — not a functor between distinct categories.
Verdict: This is the root limitation of the compression-evolution analogy. Schmidhuber’s framework assumes compressor ≠ data. Evolution violates this: the genome that encodes the environment IS also the entity being evaluated by selection. Categorically: the functor G requires an endofunctor structure that Buehler’s olog framework (designed for cross-domain functors between distinct categories) does not natively support. This is not a refutation — it’s a precise specification of where the framework needs extension.
CC8: Bottleneck as Forced Decompression. Under G: a BottleneckEvent in the cancer domain maps to a forced reset of the Compressor. Shallow bottleneck → partial decompression (existing compression repairable); deep bottleneck → complete decompression (renewed exploration required).
Verdict: Holds. This is the compression-progress resolution of the bottleneck paradox (population-bottleneck §Resolution via Compression-Progress). The Miething-Walens dual-route evidence maps onto the shallow/deep decompression dichotomy.
Functor G: Summary
| Aspect | Status |
|---|---|
| Core isomorphism (fitness = compression quality) | Holds |
| Interestingness = fitness gradient | Holds |
| Exploration-exploitation boundary | Conditional — fails for constitutive mutators |
| Discovery = sweep | Conditional — requires τ_k > sweep_time |
| Passenger mutations | No analogue — categorical gap |
| Decompression inverse | No unique inverse — categorical gap |
| Self-referential genome | Endofunctor — beyond Buehler’s framework |
| Bottleneck = forced decompression | Holds |
Overall: G is a functor on the subcategory of fitness-affecting mutations under strong selection, with two documented categorical gaps (passengers, decompression inverse) and one fundamental structural limitation (endofunctor).
Functor H: EcologyOlog → CompressionOlog (via CancerOlog as span)
The mapping from ecology to compression is constructed as a span through the cancer olog: rather than composing functors (which would require an inverse G⁻¹ that does not exist — G is not an isomorphism), we define H by specifying object and arrow mappings directly, using the cancer olog as the intermediate representation that establishes the correspondence.
Formally: given functors F: EcologyOlog → CancerOlog and G: CompressionOlog → CancerOlog, a span from ecology to compression is a set of pairs (E, C) such that F(E) and G(C) refer to the same or isomorphic cancer objects. For each ecology object E, H(E) is the compression object C whose image under G is F(E) (or the closest available — where no exact match exists, the mapping is approximate and flagged). This is not function composition — it is correspondence via shared target category.
Key composite mapping
| Ecology | Cancer (via F) | Compression (via span) |
|---|---|---|
| PhenotypicPlasticity | EpigeneticPlasticity | Exploration (non-genetic search) |
| GeneticDiversity | IntratumorHeterogeneity | CompressorDiversity (multiple candidate compressions) |
| FounderEvent | BottleneckEvent | ForcedDecompression |
| NicheOccupancy | Relapse | SuccessfulCompression (finding a viable compression) |
| IntroducedRange | PostTherapyState | NewDataSequence (new environment to compress) |
Composite commutativity
H1: Plasticity-Exploration Commutativity. Under H: PhenotypicPlasticity → enables → NicheOccupancy (ecology CC1) maps to Exploration → discovers → SuccessfulCompression (compression). The ecology claim that plasticity compensates for low diversity maps onto the compression claim that exploration (searching for new compressions) can succeed even when the current compressor is poor.
Verdict: Holds. This is the deepest structural convergence across all three domains: plasticity, epigenetic adaptation, and exploration are the same arrow in three different categories. Each domain provides independent evidence for the same structural claim.
H2: Bottleneck-Severity Commutativity. Under H: FounderEvent → bottlenecks → low GeneticDiversity maps to ForcedDecompression → resets → Compressor → renewed Exploration. The ecology finding that bottleneck severity determines residual diversity maps onto the compression finding that decompression severity determines subsequent exploration.
Verdict: Holds. Both domains predict the same structural outcome: severe bottlenecks/resets → more exploration → more diverse outcomes. The cancer domain adds the mechanism-level distinction (Met amp vs. Jak/Stat) that the other two domains lack.
The Unified Framework
flowchart TD subgraph Ecology["Ecology Invasion Olog"] PP["PlantPopulation"] --> GD["GeneticDiversity"] PP --> PL["PhenotypicPlasticity"] FE["FounderEvent"] --> PP PL --> NO["NicheOccupancy"] GD --> NO end subgraph Cancer["Cancer Evolution Olog"] TCP["TumorCellPopulation"] --> ITH["IntratumorHeterogeneity"] TCP --> EP["EpigeneticPlasticity"] BE["BottleneckEvent"] --> TCP EP --> RL["Relapse"] ITH --> RL end subgraph Compression["Compression Progress Olog"] CA["CuriousAgent"] --> CQ["CompressionQuality"] CA --> EX["Exploration"] FD["ForcedDecompression"] --> CA EX --> SC["SuccessfulCompression"] CQ --> SC end Ecology -->|"F (profunctor)"| Cancer Compression -->|"G (functor on subcategory)"| Cancer Ecology -.->|"H (span via Cancer)"| Compression
The unified three-domain framework. Solid arrows: within-domain structure. Bold arrows between domains: functors (F, G) and composite mapping (H). F is a profunctor (conditional, requires coherence conditions). G is a functor on the subcategory of fitness-affecting mutations under strong selection. H is a derived composite mapping.
What This Means
The framework is not a metaphor — it’s a mathematical structure
The 75 objects, 84 arrows, and 24 commutativity conditions across the three ologs are not prose analogies. They are formal claims about which paths in one domain must produce the same result as which paths in another domain. Each commutativity condition is falsifiable: find a case where the paths diverge, and the functor breaks at that point. The breaks that have already been identified (F-C5 Common Garden, G-CC6 Passenger Subdiagram, G-CC7 Endofunctor) are not weaknesses — they are the precise specification of domain boundaries. A framework that claims to map everything is propaganda. A framework that specifies exactly where it fails is science.
The framework generates testable predictions
| Functor | Condition | Prediction | Test |
|---|---|---|---|
| F | C4: Niche Occupancy Independence | ITH-outcome relationship is non-monotonic (U-shaped) | ITH empirical test design |
| G | CC3: Exploration-Exploitation | Constitutive mutator tumors should show different ITH-outcome relationship than gradient-sensitive tumors | Stratify by DDR/mutator status |
| G | CC5: Discovery = Sweep | τ_k > sweep_time predicts sweep probability; measurable from longitudinal VAF data | Cohort with ≥2 time points |
| H | H1: Plasticity-Exploration | Epigenetic plasticity should predict relapse in monoclonal tumors better than genetic ITH | Matched genetic + epigenetic data |
The framework has been surprised by its own data
A formalism that only confirms what its builders already believed is not a scientific instrument — it’s a mirror. The following commutativity conditions were expected to hold (based on the prose analogies that motivated the olog construction) but were found to FAIL when checked against the empirical corpus. These failures are the framework’s most valuable output: they identify where the intuitive analogy is wrong.
Disconfirmation 1: Bottlenecks reduce diversity (F-C2, ecology→cancer). The naive functorial mapping from ecology to cancer predicts: BottleneckEvent → reduces → TumorCellPopulation → hasITH → lower ITH. The ecology evidence (Geng 2016) is clear: founder bottlenecks reduce genetic diversity. The cancer evidence (Miething 2019) contradicts this: deeper therapeutic responses (CR/vgPR, more severe bottlenecks) produce branching relapses with HIGHER clonal diversity than shallower responses (PR). The bottleneck paradox — that severe bottlenecks can increase rather than decrease diversity — was discovered by the data, not predicted by the functor. Status: Functor prediction falsified. The ecology→cancer mapping requires a re-diversification arrow (not present in the ecology olog) that captures post-bottleneck exploration. This failure produced the bottleneck paradox concept (population-bottleneck §The Bottleneck Paradox) and the compression-progress resolution (shallow decompression → repair; deep decompression → renewed exploration).
Disconfirmation 2: Neutral evolution is clinically boring (G-CC4, compression→cancer). The compression framework predicts: neutral evolution → flat fitness landscape → no selection gradient → BoringRandomness. A “boring” data sequence in Schmidhuber’s framework is one that yields no compression progress — it is ignored by the curious agent. The naive mapping to cancer: neutral evolution is clinically boring — nothing is being learned, nothing is changing. This prediction fails. Neutral evolution in cancer generates passenger mutations and standing subclonal diversity that, while selectively neutral NOW, becomes the substrate for adaptation when the environment changes (therapy, metastasis, immune pressure). A tumor undergoing neutral evolution is “boring” in Schmidhuber’s sense (no current compression progress) but is building adaptive potential that is clinically significant. Status: Functor prediction falsified for the mapping of “boring” → “clinically insignificant.” The compression framework’s concept of “boring” does not distinguish between harmless stasis and dangerous latent potential. This failure is documented in compression-progress-evolution §Limitations: “Evolution cannot ‘ignore the noise.’”
Disconfirmation 3: Same genotype predicts same drug sensitivity (F-C5, ecology→cancer). The ecology olog’s Common Garden condition (CC5) states: same genotype × same environment → same phenotype. This holds for Alternanthera: plasticity is a species-level trait independent of population origin (Mantel r=0.15, p=0.29). The naive mapping to cancer: same driver mutation × same drug → same response. This is categorically false. KRAS G12D in lung adenocarcinoma predicts different drug sensitivity than KRAS G12D in colorectal cancer, due to different co-mutations, different epigenetic backgrounds, different tissue contexts (McGranahan & Swanton, 2017; clonal-evolution §Pan-cancer heterogeneity). Status: Functor prediction falsified. Cancer has no species-level traits — every tumor is its own evolutionary lineage with a unique history that modulates drug response. This failure produced the profunctor classification for F: context-dependent mappings requiring co-mutations, epigenetic state, and microenvironment as coherence conditions.
What these failures mean. The framework has falsified three of its own predictions. This is evidence that it CAN falsify predictions — addressing the concern that it only confirms what its builders already believed (see outputs/review-2026-07-05-cross-domain-functors.md §DA-C3). A formalism that never surprises you is a mirror. A formalism that surprises you — by making a prediction you believed, then showing it to be false when you checked — is a scientific instrument.
The framework has edges where it breaks
These are documented, not hidden:
- F-C5 (Common Garden): Cancer has no species-level traits — every tumor is its own lineage. The functor is a profunctor here.
- G-CC6 (Passenger Subdiagram): No compression analogue of neutral mutations. A categorical gap.
- G-CC7 (Endofunctor): The genome’s self-reference is beyond Buehler’s framework. Requires categorical extension.
- G-CC3 (Constitutive Mutators): Cancer can’t turn off exploration. The compression analogue of “curiosity off” has no cancer equivalent for mutator-phenotype tumors.
The framework is incomplete
The three ologs formalize WHY (compression drives adaptation), HOW (dual genetic/epigenetic regimes couple), and VALIDATION (functorial commutativity). What’s missing:
- No quantitative parameterization. The commutativity conditions are qualitative (holds/breaks/conditional). A quantitative extension would specify parameter regimes where each condition holds (e.g., “F-C4 holds when epigenetic plasticity > threshold θ and genetic diversity < threshold δ”).
- No dynamics. Ologs are static — they represent structure, not temporal trajectories. A dynamical extension (time-indexed categories, stochastic enrichment) would capture the temporal dimension of clonal evolution.
- No experimental validation. The four testable predictions above are derived from the functors but none have been tested. The ITH test design is the closest to implementation.
- Incomplete domain coverage. The cancer olog has 33 objects — many more could be added (spatial structure, metabolic constraints, cell-cell interactions, microbiome). Each addition creates new arrows and potential commutativity conditions.
Limitations
- Static representation. Ologs capture structure, not dynamics. Clonal evolution is inherently temporal (clone frequency dynamics, sweep times, waiting times). The arrows represent functions, not processes — they say what maps to what, not how fast or through what mechanism.
- Deterministic assumption. Arrows are functions (unique outputs for given inputs). Evolution is stochastic (drift, mutation timing, environmental fluctuations). A probabilistic extension (Markov categories, stochastic functors) would be more appropriate for full evolutionary modeling.
- Qualitative commutativity. The current verification is qualitative (holds/fails/conditional). A quantitative extension would specify parameter regimes, effect sizes, and statistical tests for each condition.
- Buehler’s framework is thin. The olog methodology (Buehler et al., 2011) uses only the elementary fragment of category theory — categories, subcategories, functors. Universal constructions (limits, colimits, adjunctions, natural transformations) are not used here but may be needed for deeper structure (e.g., the endofunctor limitation, the profunctor coupling).
- Construction bias. All three domain ologs and all three functors were constructed by the same system (LLM-assisted researcher) with mutual knowledge. The ologs were built with the cross-domain mappings already in mind. A stronger test would construct each olog independently — by different researchers, from primary literature — and then search for functorial mappings without prior knowledge of which objects would correspond. The current functors may reflect construction bias (ologs designed to make the mappings work) rather than discovered structure. An independent replication protocol is proposed in
outputs/review-2026-07-05-cross-domain-functors.md§R10. - Confidence. All three ologs and the functors between them have confidence: medium. The objects, arrows, and commutativity conditions are derived from the wiki’s 39-source corpus, but the formalization is the wiki’s synthesis — none of the source papers constructed ologs or verified functorial commutativity in these terms.
Revision history
- 2026-07-05 — Major revision. Fixed G⁻¹ notation (replaced with span construction via cancer olog). Added §Disconfirmatory Evidence with 3 cases where the framework falsified its own predictions. Added construction bias limitation. Updated cancer olog counts (immune objects added). (cancer-evolution-olog,
outputs/review-2026-07-05-cross-domain-functors.md)- 2026-07-05 — Page created. Constructs the three cross-domain functors (F: Ecology → Cancer, G: Compression → Cancer, H: Ecology → Compression via Cancer) unifying the wiki’s three domain ologs. For each functor, specifies object mappings, arrow mappings, and commutativity verification with documented breaks and limitations. Derives four testable predictions from functor structure. Defines the unified three-domain framework. Confidence: medium — formal structure is rigorous but not empirically validated. (cancer-evolution-olog, ecology-invasion-olog, compression-progress-olog)