The Equation Is Not a Bridge: A Genealogical Account of Physical Law

Abstract

Physical equations are commonly pictured as bridges: mass is related to energy, matter to geometry, preparation to probability. The relata are tacitly treated as already constituted, and the law is asked to connect them. This paper argues that the picture reverses the actual dependency. Let K be a carrier of physically admissible antecedents, let πᵢ: K → Xᵢ be terminal readings, and let J = (πᵢ)ᵢ: K → ∏ᵢXᵢ be their joint map. If the readings are freely recombinable, J is surjective. Any exact non-trivial cross-law F(π₁(k), …, πₙ(k)) = 0 whose zero-set is a proper subset of the product therefore proves that the readings are not jointly free. The law constrains the realised joint image; it cannot be what first joins independently constituted quantities. That conclusion is elementary. The stronger result concerns fundamentality. A terminal law fixes at most an image in a product of readings. It does not determine the carrier, its fibres, or the action of further continuations. Non-isomorphic continuation systems can have the same terminal image and satisfy exactly the same equations while differing in a fibre that a later response distinguishes. Hence an exact law can still begin too late. A representation is complete only if its induced equivalence is a congruence for every continuation it claims to govern. The paper separates two logically independent audits. Horizontal compatibility asks whether the joint image is a proper subset of the product. Vertical sufficiency asks whether a terminal reading identifies antecedents that an admitted continuation can still distinguish. A representation can pass either audit and fail the other. The argument is developed through three cases. The relativistic mass shell shows that energy, momentum, and invariant mass are non-coextensive readings of one four-momentum; a pair of massless photons can have positive composite mass. The Einstein equation defines a compatibility relation between stress-energy and geometry, not a function from matter to metric; vacuum gravitational degrees make the distinction explicit. Trace normalisation shows how an exact quantum reading can erase an active scale even when the full unnormalised carrier splits as state and scale. The resulting position is distinguished from Humean, governing-law, laws-as-constraints, semantic, operationalist, and ontic-structuralist accounts. It treats neither objects nor laws as the unexamined base. The antecedent is instead the least continuation-sufficient carrier: the quotient that preserves all and only the distinctions that some admitted continuation can still make effective. Physical laws are then terminal constraints on its projections, not bridges laid between independent pieces of reality. The mathematical ingredients are largely established; their role here is to prove a dependency reversal, not to present familiar representation theorems as new.