A Meta-Science Test: Is the d1/d2/d3 Grammar More Primitive Than Mathematics?
Pre-registered experiment H_PRIMITIVITY: 15 IRDME-tested domains as nodes, three layers encoding formal mathematical dependencies, structural coupling class, and FPL confirmation outcome. Key finding: mathematical formalization does not significantly predict whether the law confirms (r=0.37, p=0.21 ns). A potential fourth boundary condition identified.
The question
The Functional Proximity Law is organized around a three-layer grammar: d1 (declared coupling), d2 (structural coupling), d3 (behavioral coupling). This grammar appears consistently across molecular biology, software systems, neuroscience, formal mathematics, and ecology.
H_PRIMITIVITY asks: does the law's applicability depend on whether a domain is highly mathematical? If so, formalization level would predict FPL confirmation strength.
Test design
15 IRDME-tested scientific domains as nodes, three meta-layers:
d1 - formal_dependency: which domains formally require each other's mathematical tools. Top hub: formal_mathematics (degree 20).
d2 - structural_grammar_type: which domains share the same class of d1/d2/d3 coupling structure. Software and formal math share import/declaration => structural => co-change. Biology and neuroscience share physical binding => functional => co-expression.
d3 - law_confirmation_coupling: which domains confirmed FPL with similar r-values. Strong (r>0.70): software, mol_bio, neuroscience, sys_bio, formal_math, comp_ling. Medium (0.50-0.70): hardware, network_topo, phys_chem. Lower (0.40-0.50): ecology, proj_mgmt, climate. Denied: finance, psychiatry, math_attr.
Four hypotheses pre-registered (hash cfb38b83, commit 3e609f2, 2026-05-21T04:04 UTC) before analysis.
Results
| Hypothesis | Verdict | Key number | |---|---|---| | h1: FPL holds at meta-science level (r(d1-d2) > r(d1-d3)) | DENIED | r(d1-d2)=0.212, r(d1-d3)=0.368 - reversed | | h2: mol_bio or sw_eng is top hub in law_confirmation | DENIED | Top hub is sys_bio; mol_bio ranks #4 | | h3: formal_math is #1 hub in formal_dependency | CONFIRMED | Degree 20 vs sys_bio at 10 | | h4: r(formalization, confirmation) not significant | PARTIAL | r=0.368, p=0.206 |
1 CONFIRMED / 2 DENIED / 1 PARTIAL
What the results mean
h3 CONFIRMED validates the layer construction. Note: this is partially a consequence of design - formal_math was always likely to dominate d1 by degree. It confirms correct graph construction, not that the law holds here.
h4 PARTIAL is the primary finding. Mathematical formalization level does not significantly predict FPL confirmation strength (r=0.368, p=0.21). Molecular biology (r=0.97) depends on physics/chemistry, not formal math. Financial markets uses stochastic calculus but the law fails there. Ecology (r=0.43) is one of the least formalized domains yet confirms.
Statistical note: n=15 is a small sample. p=0.21 is inconclusive under low power, not evidence of absence. The correct claim: the FPL captures structure not fully reducible to formalization level alone.
h2 DENIED: systems biology is the top hub in law_confirmation - the structural integrator bridging strong-confirmed and medium-confirmed clusters.
BC4 - edge semantic type mismatch
Existing boundary conditions: BC1 (Finance) - relational regime mismatch; BC2 (Psychiatry) - intervention reality shapes classification; BC3 (Mathematics) - layer resolution mismatch.
BC4 candidate: Meta-graph edges encode institutional and epistemic dependencies - which field uses which mathematical tools, which coupling class two fields share - rather than physical, computational, or biological interaction mechanisms. This is not a resolution mismatch. It is a mismatch in what the edges mean. Object-graph edges represent direct interactions between entities within a domain. Meta-graph edges represent abstract structural relationships between entire knowledge regimes.
Three graph levels where the law behaves differently: object graphs (molecules, code, neurons) - confirmed; structural similarity graphs (cross-domain coupling patterns) - partially valid; meta-graphs (disciplines as nodes, epistemic dependencies as edges) - unstable.
Summary
The FPL is sensitive to interaction regime, not to formalization intensity. It confirms in formal mathematics, biology, and ecology at very different positions in the mathematical hierarchy. The law does not confirm at the meta-graph level - a scale-level failure: the edge semantics in a meta-science graph are categorically different from the edge semantics in object-level graphs where the law was derived.
Pre-registration: https://github.com/vladi160/preregistrations/commit/3e609f2. Data: science_formalization_spectrum.json.