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 |
Potential fourth domain: Silk/Protein Mechanics → Cancer Evolution
Nova et al. (2010) (nova2010-silk-hierarchical-mechanics) demonstrated that spider silk achieves its exceptional strength and toughness through a hierarchical two-phase nanostructure: stiff beta-sheet nanocrystals (discrete, size-constrained, H-bonded) embedded in an extensible semiamorphous matrix (continuous, reversible, energy-dissipating). The structural isomorphism with cancer evolution’s dual-regime model (dual-regime-evolution) suggests a fourth cross-domain functor — S: SilkMechanics → CancerEvolution — that would join F, G, and H.
The core structural analogy:
| Silk concept | Cancer concept | Mapping rationale |
|---|---|---|
| Beta-sheet nanocrystal (~3 nm) | DriverMutation (s ≈ 0.4%) | Stiff, discrete, size-constrained for optimal system function |
| Semiamorphous matrix | EpigeneticState / EpigeneticPlasticity | Extensible, continuous, reversible, governs early-response behavior |
| Nanocrystal size | Selection coefficient s | Small values → superior system-level properties (toughness / clonal diversity) |
| H-bond arrays | SubclonalArchitecture | Collective interaction network; failures cascade through the network |
| Stick-slip failure | Clonal interference / incomplete sweeps | Repeated partial failure with energy dissipation; requires nanoconfinement |
| Ultimate tensile strength | OverallSurvival / TherapyResponse durability | System-level failure threshold |
| Toughness (energy to failure) | Clonal diversity / Evolutionary resilience | Capacity to absorb perturbation without catastrophic failure |
The key mechanistic parallel: small nanocrystals are crucial for silk toughness because they enable the semiamorphous matrix to fully unravel before failure — large crystals are brittle and prevent matrix extension. The cancer analogue: small-effect drivers (s ≪ 1) enable sustained clonal diversity because they prevent premature clonal sweeps — large-effect drivers produce brittle, low-diversity populations. In both systems, the confinement of the discrete phase to small size is the design principle that enables the continuous phase to express its full adaptive capacity.
Status: Not yet formalized. Constructing S would require a SilkMechanics domain olog first (per the olog construction discipline: ologs precede functors). Nova et al. (2010) provides the empirical foundation for that olog. The potential functor is noted here as a research direction — its existence would strengthen the cross-domain program by demonstrating that the hierarchical two-phase pattern appears across four domains (ecology, compression, cancer, materials mechanics), suggesting a universal design principle for adaptive hierarchical systems under selective pressure. If this holds, the wiki’s deepest theoretical claim — that clonal evolution IS a hierarchical material in Buehler’s sense — gains a fourth independent domain of support.
Reference: nova2010-silk-hierarchical-mechanics §Relevance to Clonal Evolution.
Potential fifth domain: Fungal Evolution → Cancer Evolution
Naranjo-Ortiz & Gabaldon (2019) (naranjo-ortiz2019-fungal-evolution) provide a comprehensive review of major ecological adaptations and evolutionary transitions across the fungal kingdom, using comparative genomics and phylogenetic reconstruction. The paper’s framework of “evolutionary transitions” — defined as “the acquisition — within a lineage — of a new, sufficiently distinct lifestyle from a previous state” — is structurally parallel to cancer evolutionary transitions (metastatic dissemination, therapy resistance, immune evasion). A fifth cross-domain functor T: FungalEvolution → CancerEvolution would join F, G, H, and the potential silk mechanics functor S.
The key structural parallels:
| Fungal concept | Cancer analogue | Structural isomorphism |
|---|---|---|
| Niche colonization — fungi colonize novel hosts, environments, and ecological niches through adaptive radiations | Metastasis — tumor clones colonize distant anatomical sites | Both involve a population crossing an environmental barrier (host immune system / tissue boundary) and establishing in a new selective regime |
| Lifestyle switching — interconversion between mycoparasitic, entomopathogenic, endophytic, and saprotrophic lifestyles, particularly in Hypocreales | Phenotypic plasticity — transitions between proliferative, invasive, and drug-tolerant states without new mutations | Both involve non-genetic adaptation to novel conditions; the same genotype can express multiple ecological strategies |
| Mycoparasitism — one fungus directly competes with, parasitizes, or eliminates another through chitinases, secondary metabolites, and reactive oxygen species | Clonal competition — subclones compete for resources and space within the tumor microenvironment | Both involve direct interference competition between genetically distinct entities in a shared niche |
| Horizontal gene transfer — facilitated by mycoparasitism through cell wall degradation removing barriers to DNA acquisition | Genomic instability + DNA exchange — via extracellular vesicles, apoptotic bodies, cell fusion, and chromothripsis | Both provide mechanisms for rapid acquisition of adaptive traits outside of vertical inheritance |
| Genome compaction (yeast lifestyle) — convergent reduction of intergenic content, introns, and dispensable metabolic pathways under strong selection for rapid proliferation | Driver/passenger distinction — essential proliferative functions (drivers) are retained and refined; non-essential functions (passengers) accumulate neutrally | Both represent the genomic signature of strong, sustained selection for replicative speed in resource-rich environments |
| Biotrophy vs. necrotrophy — two opposing parasitic strategies with radically different genomic signatures (genome reduction vs. expansion) | Treatment-sensitive vs. treatment-resistant clones — two opposing survival strategies under therapeutic pressure (apoptosis-competent vs. apoptosis-evasive) | Both represent alternative evolutionary solutions to the same ecological challenge (host exploitation) |
What makes this domain particularly valuable for the cross-domain program: Unlike the ecology-invasion domain (which maps a single plant species across three geographic ranges) and the compression-progress domain (which maps an abstract mathematical framework), the fungal evolution domain provides multiple independent evolutionary transitions within a single kingdom — terrestrialization, lichenization, the yeast lifestyle, mycoparasitism, animal pathogenesis, plant pathogenesis — each representing a distinct solution to the problem of niche colonization. This multiplicity enables within-domain comparison: do all evolutionary transitions share common genomic signatures? Do the same principles (genome compaction under strong selection, HGT as a rapid-adaptation mechanism, convergent evolution of similar strategies through different molecular routes) recur across transitions? These within-domain patterns would provide a more rigorous test of cross-domain functor validity than any single-species comparison.
The “evolutionary transition” concept as a formal bridge. The paper operationalizes evolutionary transitions as a structured phenomenon: ancestral state → selective pressure → genomic/phenotypic enabling changes → new ecological opportunities. This four-part structure maps cleanly onto cancer evolutionary transitions: normal cell state → microenvironmental/metabolic/therapeutic pressure → driver mutations + epigenetic remodeling → invasive/metastatic/resistant capabilities. If this mapping holds as a functor, the fungal evolution literature — with its rich comparative genomics of transitions across hundreds of species — becomes a source of testable predictions for cancer evolution.
Status: Not yet formalized. Constructing T requires a FungalEvolution domain olog first (per the olog construction discipline: ologs precede functors). Naranjo-Ortiz & Gabaldon (2019) provides the empirical foundation. The potential functor is noted here as a research direction. If constructed, T would test whether the evolutionary transition framework is truly domain-general — whether the same structural logic that governs how fungi evolve to colonize new niches also governs how cancer clones evolve to colonize new tissues and survive new therapies.
Reference: naranjo-ortiz2019-fungal-evolution §Relevance to Clonal Evolution.
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.
Double-Categorical Enrichment
This page constructs F, G, H as functors between functional ologs (Spivak & Kent, 2012) — categories where all arrows are functions. Lambert & Patterson (2024) (lambert2024-double-functorial-semantics) introduce double ologs, which add a second layer of structure: proarrows (relations) and cells (implications between relations). The functional-olog framework that underlies the present functors is a special case — the vertical-category fragment of a double olog.
What a double-categorical upgrade would change
Arrows → arrows + proarrows. Across the three domain ologs, 13 of 84 arrows (15.5%) are semantically relations, not functions (lambert2024-double-functorial-semantics §Relevance to Clonal Evolution). Examples:
| Domain | Arrow | Category mismatch |
|---|---|---|
| Cancer | metastasizes: Clone × AnatomicalSite → Metastasis | One clone → many sites; many clones → same site |
| Cancer | survive: BottleneckEvent × TumorCellPopulation → TumorCellPopulation | Survivor set is a subset, not uniquely determined |
| Ecology | enables: PhenotypicPlasticity → NicheOccupancy | Causal relation, not mathematical function |
| Ecology | predicts: GeneticDiversity → NicheOccupancy | Probabilistic, acknowledged as non-functional in the olog |
| Compression | decompresses: CancerAsDecompression × CompressedRepresentation → DataSequence | Non-unique inverse — “many decompression paths” |
In a double olog, these become proarrows (horizontal morphisms), distinct from the functional arrows (vertical morphisms). The functors F, G, H would then need to specify not just how functional arrows map, but how proarrows map — e.g., the ecology olog’s predicts proarrow must map to a cancer-domain proarrow expressing that genetic diversity predicts (but does not determine) outcome.
Commutativity conditions → cells. Our 24 commutativity conditions are prose external to the category. In a double olog, each becomes a cell — a 2-morphism that is part of the category itself:
Functional olog (current): Double olog (upgraded):
"Path A must equal Path B" Cell: proarrow₁ → proarrow₂
(English prose, external) (formal 2-morphism, internal)
This matters because cells compose, have well-defined boundaries, and are preserved by double functors. A double functor F: DOlog_Ecology → DOlog_Cancer would carry ecological commutativity conditions into cancer-domain implications automatically — the cross-domain mapping would preserve not just objects and arrows but logical structure.
Functors → double functors. F, G, H are currently 1-functors between 1-categories. Upgraded, each would become a double functor preserving four layers:
| Layer | Current (1-functor) | Upgraded (double functor) |
|---|---|---|
| Objects | Ecological types → Cancer types | Same |
| Arrows | Ecological functions → Cancer functions | F₁: vertical arrows preserved |
| Proarrows | (not represented) | F₁’: horizontal relations preserved |
| Cells | (not represented) | F₂: implications preserved |
The ecology olog’s Condition C1 (Plasticity-Diversity Trade-off) — currently prose — would map via F₂ to a cell in the cancer double olog, formally asserting that the epigenetic-plasticity-compensates-for-ITH relationship is the image of the ecology-domain plasticity-diversity trade-off under the double functor.
Instance data. Lambert & Patterson (2024) define an instance as a cartesian strict double functor D → Rel. This means VAF measurements, clone genotypes, drug responses, and immune infiltration values could be attached to the cancer double olog as formal data, making the olog a queryable knowledge base rather than a pure schema. Queries like “find all clones with KRAS G12D AND CCF > 0.3 AND cisplatin resistance” become internal double-categorical operations (extension + restriction + local product) rather than external SQL.
Partial maps. The cancer olog’s ancestor arrow is explicitly partial — defined only when two clones share a common ancestor. A double olog provides the formal condition: a proarrow is a partial map iff R† ⊗ R ≤ id, equivalent (under tabulators) to the domain leg of the tabulator span being monic (Lambert & Patterson, 2024, Theorem A.1). Other partial arrows (initiates, foundsFromBottleneck) would receive the same formal treatment.
What the upgrade does NOT address
Lambert & Patterson (2024) is an ACT conference paper — 16 pages, no implementation, and explicitly calls out engineering work as future work. It does not address:
- Cross-domain functor construction — the paper’s focus is single-domain representation and querying
- Aggregation — SUM, COUNT, AVG, GROUP BY are not supported
- Temporal dynamics — same static-representation limitation as our current ologs
- Probabilistic enrichment — same deterministic assumption as our current ologs
The upgrade is a formal improvement to the olog infrastructure — it provides the right mathematical language for relations, implications, and queries, but it does not resolve the deeper limitations (stochasticity, dynamics, quantitative parameterization) that affect all category-theoretic representations.
Gap analysis
A full audit of the wiki’s ologs against the double-categorical framework was conducted on 2026-07-06. Key findings:
| Gap | Severity | Detail |
|---|---|---|
| Arrows forced as functions | High | 13/84 arrows across three ologs are semantically relations |
| Commutativity conditions as prose | High | 24 CCs exist outside the formalism |
| Functors at wrong structural level | High | F, G, H are 1-functors, not double functors |
| No instance data | Medium | Ologs are schemas without attached data |
| No tabulators | Medium | Types must be manually defined; no auto-derived subtypes |
| No query mechanism | Medium | Can’t ask “clones with KRAS + CCF > 0.3” |
| No formal partial maps | Low | Prose notes instead of R†⊗R ≤ id condition |
| Dual-regime coupling as prose | Medium | Coherence conditions are paragraphs, not composable cells |
The full audit is recorded in the session context anchor (context-anchor.md, 2026-07-06). Tier 2 (formalize: identify proarrows, express CCs as cells, sketch double-functorial upgrade of F) and Tier 3 (rebuild ologs as double ologs with instance data) are deferred.
Commutativity Conditions as Cells
In a double olog, commutativity conditions become cells — formal 2-morphisms between proarrows that are part of the category itself rather than external prose. Three cells are constructed below, each demonstrating a different structural role.
Cell Type 1 — Implication between composed proarrows: F-C4 (Niche Occupancy Independence)
From the ecology double olog (see ecology-invasion-olog §Double-Categorical Annotation for the proarrow definitions). Define two composite proarrows from PlantPopulation ⇝ NicheOccupancy:
diversityPredicts = hasDiversity ∘ predicts
PlantPopulation → GeneticDiversity (arrow: hasDiversity)
GeneticDiversity ⇝ NicheOccupancy (proarrow: predicts)
plasticityEnables = hasGeneticPlasticity ∘ enables
PlantPopulation → PhenotypicPlasticity (arrow: hasGeneticPlasticity)
PhenotypicPlasticity ⇝ NicheOccupancy (proarrow: enables)
Note: hasDiversity and hasGeneticPlasticity remain arrows (functions) — they compute a unique value. Only predicts and enables are proarrows. The composition arrow ∘ proarrow is the right-whiskering operation in a double category.
Cells:
Cell α: plasticityEnables → ⊤_{PlantPopulation, NicheOccupancy}
"For every plant population, high plasticity implies full niche occupancy."
⊤ is the terminal proarrow — the one that holds for every domain-codomain pair.
Verified for all three populations (Argentina, USA, China).
Cell β: diversityPredicts → plasticityEnables
"When the diversity path predicts full occupancy, the plasticity path also
enables it." Holds for USA and Argentina (both give "full occupancy").
For China: diversityPredicts gives "restricted" and plasticityEnables gives
"full" — false → true in logic, so the implication holds (vacuously).
Stronger relational reading: diversityPredicts ≤ plasticityEnables
(the diversity relation is a subset of the plasticity relation).
Non-implication: plasticityEnables ⇏ diversityPredicts
China: plasticityEnables relates China to "occupied"
but diversityPredicts does NOT relate China to "occupied."
This is the core Geng et al. (2016) finding — plasticity compensates
for absent genetic diversity — expressed as a formal failure of
cell reversal in the double category.
Biological interpretation. Cell α asserts that plasticity alone is sufficient for invasion success. Cell β asserts that diversity never contradicts plasticity (there is no population where diversity predicts occupancy but plasticity doesn’t enable it). The non-implication captures the empirical finding: plasticity works even when diversity doesn’t.
Cell Type 2 — Mechanism-level coherence with constants: Cancer CC3.7 (Dual-Regime Coupling)
From the cancer double olog (see cancer-evolution-olog §Double-Categorical Annotation). Define two proarrows for the dual-regime coupling:
couplesGtoE: DriverMutation ⇝ EpigeneticState
"A driver mutation induces a specific epigenetic change"
— relation, because the same mutation (IDH1 R132) produces different
methylation patterns in glioma vs. cholangiocarcinoma vs. AML.
couplesEtoG: EpigeneticState ⇝ MutationRate
"An epigenetic state modulates the genetic mutation rate"
— relation, because MGMT methylation affects C→T at CpG sites while
chromatin accessibility affects APOBEC at TpCpW contexts — different
loci, different rates, same epigenetic state.
Three specific cells using constant proarrows. The notation ⟨x⟩ denotes the constant proarrow 1 ⇝ Type that picks out element x — the Lambert & Patterson (§2.1) construction for expressing individual facts:
Cell₁: ⟨IDH1 R132⟩ ⊗ couplesGtoE → ⟨hypermethylated⟩
"IDH1 mutation implies genome-wide DNA hypermethylation (G-CIMP phenotype)."
Source: PCAWG Consortium (2020); [[dual-regime-evolution]] §Coupling.
Cell₂: ⟨MGMT-methylated⟩ ⊗ couplesEtoG → ⟨elevated C→T rate⟩
"MGMT promoter methylation implies elevated C→T mutation rate."
Source: PCAWG Consortium (2020); glioblastoma data.
Cell₃: ⟨TP53-null⟩ ⊗ couplesGtoE → ⟨DDR-loss⟩
"TP53 loss implies DNA damage response decoupling."
Source: [[dual-regime-evolution]] §Coupling Between Regimes.
Compositional prediction. Cell₁ then Cell₂ (via the composition couplesEtoG ∘ couplesGtoE) yields: IDH1-mutant tumors should show elevated C→T mutation rates from unrepaired methylation damage. This prediction was IMPLICIT in the prose — the cell composition makes it formally visible as the composite proarrow ⟨IDH1 R132⟩ ⊗ (couplesEtoG ∘ couplesGtoE) → ⟨elevated C→T rate⟩. A researcher can now test: in IDH1-mutant gliomas, is the C→T rate at CpG sites elevated relative to IDH1-wildtype?
Structural non-implication:
Non-implication: couplesGtoE ∘ couplesEtoG ⇏ id_{DriverMutation}
"Genetic → Epigenetic → Genetic does NOT recover the original
mutation. The coupling is not reversible."
This is the formal statement of the profunctor (not functor)
classification of the dual-regime coupling.
Cell Type 3 — Documented failure of functorial preservation: G-CC6 (Passenger subdiagram)
The cancer olog has a proarrow (see cancer-evolution-olog §Double-Categorical Annotation):
accumulates: EvolutionaryTime ⇝ PassengerMutation
"Over time, passenger mutations accumulate in the genome."
— relation, because the same elapsed time accumulates different numbers
of passengers depending on mutation rate, genomic region, and
mutational-signature epoch.
The functor G: DOlog_Compression → DOlog_Cancer maps EvolutionaryTime ↦ CompressorIteration but PassengerMutation has no image under G₀ — there is no “PassengerMutation” object in the compression olog. This categorical gap has a cell-level consequence:
G(accumulates) would need: G(EvolutionaryTime) ⇝ G(PassengerMutation)
= CompressorIteration ⇝ ???
No proarrow exists in the compression double olog with codomain ???
because G(PassengerMutation) = ∅.
The cell G₂(accumulates) cannot be constructed.
Interpretation. In Schmidhuber’s framework, the compressor only stores compressible structure — it never accumulates noise. Evolution must tolerate the accumulation of mutations that don’t affect fitness but get copied anyway (passengers, hitchhikers). This is NOT a failure of the cross-domain analogy to be “complete” — it’s a precise specification of WHERE the domains differ. A formalism that maps everything is propaganda; one that specifies exactly where it fails is science.
Double-functorial upgrade of F
A double functor F: DEcologyOlog → DCancerOlog has four components. The existing 1-functor tables from cross-domain-functors §Functor F (object mapping, arrow mapping) remain the vertical fragment — this section specifies the additional double-categorical structure.
F₀ — Objects
Unchanged from the existing 12-object mapping: PlantPopulation → TumorCellPopulation, Genotype → GenomeState, PhenotypicPlasticity → EpigeneticPlasticity, GeneticDiversity → IntratumorHeterogeneity, Individual → Clone, Phenotype → DrugSensitivity, FounderEvent → BottleneckEvent, EnvironmentPatch → Microenvironment, WaterAvailability → TherapeuticPressure, BioclimaticNiche → MetastaticNiche, NicheOccupancy → Relapse, IntroducedRange → PostTherapyState.
F₁ — Vertical arrows (functions)
Unchanged from existing arrow mapping. The 15 ecology arrows that ARE functions (hasDiversity → hasITH, founds → therapyCreates, expresses → hasPhenotype, hasPlasticity → hasEpigeneticPlasticity, etc.) map via F₁.
F₁’ — Horizontal proarrows (NEW)
| Ecology proarrow | Cancer proarrow | Justification |
|---|---|---|
enables: PhenotypicPlasticity ⇝ NicheOccupancy | enables: EpigeneticPlasticity ⇝ Relapse | ”Plasticity enables full niche occupancy” → “Epigenetic plasticity enables relapse.” New proarrow in cancer olog. |
predicts: GeneticDiversity ⇝ NicheOccupancy | predictsOutcome: IntratumorHeterogeneity ⇝ Relapse | The naive diversity→success arrow. New proarrow in cancer olog. |
invades: PlantPopulation ⇝ NicheOccupancy | relapsesAs: TumorCellPopulation ⇝ Relapse | Population-level invasion→relapse mapping. New proarrow in cancer olog. |
constitutes: Genotype ⇝ PlantPopulation | appearsIn: GenomeState ⇝ TumorCellPopulation | Reverse of genome: Clone → GenomeState. One genome state appears in multiple tumors. New proarrow in cancer olog. |
These four new cancer proarrows are defined and catalogued in cancer-evolution-olog §Double-Categorical Annotation (Proarrow Catalog). They did not exist as named arrows before — the double functor makes the gap visible.
F₂ — Cells (NEW)
| Ecology cell | Cancer cell under F₂ | Status |
|---|---|---|
Cell α: plasticityEnables → ⊤ | epiPlasticityEnables → ⊤ | Testable prediction: Epigenetic plasticity implies full relapse capacity. Test: stratify tumors by epigenetic plasticity (ATAC-seq, methylation array). Do high-plasticity tumors uniformly relapse regardless of ITH? |
Cell β: diversityPredicts → plasticityEnables | ithPredictsOutcome → epiPlasticityEnables | Testable prediction: When ITH predicts relapse, epigenetic plasticity also predicts relapse. In high-ITH tumors, both routes hold; in low-ITH tumors, only the epigenetic route holds. |
Non-implication: plasticityEnables ⇏ diversityPredicts | epiPlasticityEnables ⇏ ithPredicts | The compression-entrenchment prediction: Epigenetic plasticity enables relapse even when ITH does NOT predict it — the monoclonal-but-dangerous regime. Direct test of the ITH-outcome empirical design (docs/superpowers/specs/2026-07-05-ith-outcome-test-design.md). |
Naturality constraint
A double functor requires that F₁’ is natural with respect to F₁: for any proarrow R and arrow f in the ecology double olog, F₁'(R ∘ f) = F₁'(R) ∘ F₁(f) and F₁'(f ∘ R) = F₁(f) ∘ F₁'(R). The ecology-derived cancer cells are NOT free-floating claims — they are structurally constrained by how functions compose with relations. If hasGeneticPlasticity: PlantPopulation → PhenotypicPlasticity maps to a cancer function via F₁, then F₁’(plasticityEnables) must compose correctly with F₁(hasGeneticPlasticity). A violation of naturality would mean the cell carries over compositionally but not structurally — the cross-domain analogy would be broken at a deeper level than commutativity.
What the double functor adds
| Capability | 1-functor (current) | Double functor (upgraded) |
|---|---|---|
| Maps functions | ✓ | ✓ |
| Maps relations | ✗ | ✓ (4 ecology proarrows → 4 cancer proarrows) |
| Carries logical structure | ✗ | ✓ (3 cells → 3 testable cancer predictions) |
| Makes relational gaps visible | ✗ | ✓ (4 new cancer proarrows identified) |
| Constrains predictions structurally | ✗ | ✓ (naturality squares enforce consistency) |
| Identifies failure loci | Only at object/arrow level (C5) | At cell level (G-CC6: no cell for passenger accumulation) |
Revision history
- 2026-07-17 — Added potential fifth domain. Noted fungal evolution → cancer evolution functor (T) as a research direction, based on Naranjo-Ortiz & Gabaldon (2019). The paper’s “evolutionary transitions” framework and multiple independent ecological adaptations (terrestrialization, lichenization, yeast lifestyle, mycoparasitism) provide a richer cross-domain test than single-species comparisons. Status: not yet formalized — requires FungalEvolution domain olog first. Added Naranjo-Ortiz & Gabaldon (2019) to sources and related. (naranjo-ortiz2019-fungal-evolution)
- 2026-07-16 — Added potential fourth domain. Noted silk/protein mechanics → cancer evolution functor (S) as a research direction, based on Nova et al. (2010) spider silk hierarchical mechanics. The structural isomorphism between silk’s two-phase nanostructure (nanocrystals + matrix) and cancer’s dual-regime model (genetic + epigenetic) suggests a fourth cross-domain functor joining F, G, H. Status: not yet formalized — requires SilkMechanics domain olog first. Added Nova et al. (2010) to sources and related. (nova2010-silk-hierarchical-mechanics)
- 2026-07-06 — Added cells and double-functorial F sketch. Constructed 3 cell types (implication, mechanism coherence, documented failure) expressing F-C4, CC3.7, and G-CC6 as formal double-categorical structure. Specified double functor F with F₀/F₁/F₁’/F₂ components, 4 new cancer proarrows, and 3 testable predictions. (lambert2024-double-functorial-semantics,
docs/superpowers/specs/2026-07-06-double-categorical-formalization-design.md)- 2026-07-06 — Added §Double-Categorical Enrichment. Documents the gap between the functional-olog framework (Spivak & Kent, 2012) that F, G, H are built on and the double-categorical framework of Lambert & Patterson (2024). Catalogs 13/84 arrows across three domain ologs that are semantically relations, notes that all 24 commutativity conditions are prose external to the category, and describes what a double-functorial upgrade of F, G, H would entail. Flagged as deferred work (Tier 2 — formalize; Tier 3 — rebuild). (lambert2024-double-functorial-semantics)
- 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)