Six Birds Foundations IV: A Catalog of Layer-Agnostic Structural Laws

Abstract

Foundations IV states and proves 52 structural laws about layers of description. A layer is a set of states read through a quotient, which fixes the distinctions the layer can make. Each law is a theorem with its own hypotheses, stated without reference to what the layer is made of, so that it applies to numbers, physical systems, cells, markets, languages or programs whenever its hypotheses hold. Most laws are instances of one criterion, descent: a readout on states descends through a quotient exactly when it is constant on the quotient's classes, and when it is not, a pair of states that the quotient merges and the readout separates witnesses the failure; the repair either refines the quotient or coarsens the readout. Most laws are elementary consequences of this criterion, and the contribution is the catalog itself: uniform statements with explicit hypotheses, proofs and scope. The laws give descent-style counterparts of notions such as access and hidden predictive structure, stability and residuals, transport and local-to-global gluing, symmetry and its breaking, records, objectivity, probability, causality, composability, information loss, complementarity, entropy, limits and completeness, and the boundary between formal closure and physical realisability. A few laws use other arguments: a Loewner-order inequality (No-Needles Stability), a trace squeeze (Duality Fixity, whose domination hypothesis is shown to hold exactly when the confined readout vanishes almost everywhere), a connectedness argument for phase boundaries, and finite counterexamples showing that, for abstract realisability predicates stable under reducts, formal closure and realisability need not imply each other in either direction. All laws are proved here with two qualifications: the Hiddenness Normal Form is conditional on an explicit source hypothesis, motivated by but not verified for the saturated-SAT construction of an earlier paper, and one equivalence of the Closure–Reality Boundary is imported from the paper on audited operational realisability. Special cases and analogies from mathematics, the sciences and engineering follow each law and are marked as such. The laws are formalized in Lean 4 with Mathlib; for Duality Fixity the formalization treats an abstract ledger rather than Hilbert-space measures. The catalog is not exhaustive.