WILL Relational Geometry
An attempt to derive the laws of physical reality from the most elementary geometric principles — without introducing new postulates, axioms, or arbitrary choices.
A Note Before the First Equation
As humans, we often project ourselves onto observed phenomena, obscuring the true nature of reality behind our anthropic tendencies. In order to prevent such misleading practices, we must establish extreme methodological constraints. These constraints are not statements about the nature of reality, but the strict rules of logical purity for constructing a theory.
Without strict methodology, any theoretical construct risks that excessive freedom of interpretation will result in a framework that tells us more about ourselves than about the nature of the Universe.
The philosophical reasoning behind these particular methodological constraints might deserve its own dedicated publication and can be challenged, but this research's main domain lies within physical reality that can be empirically tested; therefore, empirical alignment is the ultimate test of the methods presented.
Five Principles, One Constraint
Before any geometry, any equation, any derivation — these five rules govern what this research permits itself to assume. Nothing is added; complexity is only removed.
Physics by the refusal to import unjustified assumptions. No postulates, no free parameters — laws are derived by removing hidden assumptions, not by adding new ones.
Read in paperA fundamental theory must proceed from the minimum possible number of ontological assumptions. Every new entity or construct bears the burden of proof.
Read in paperAll physical quantities must be defined by their relations. Absolute properties reintroduce metaphysical artefacts and contradict the foundational insight of relationalism.
Read in paperEverything must be expressed in the simplest form possible. Any unjustified complexity risks reintroducing metaphysical artefacts and contradicts the insight of Epistemic Hygiene.
Read in paperEvery mathematical phrase carries its own ontological statement. One symbol, one physical idea. The number of symbols must equal the number of independent physical ideas — no more.
Read in paperThese five rules are not independent preferences. They form a single filter: admit only what is geometrically or energetically necessary. Everything else is a historical accident of how physics has been told.
What is Energy?
Across all domains of physics, one empirical fact persists: in every closed system there exists a quantity that never disappears or arises spontaneously, but only transforms in form. This invariant is observed under many guises — kinetic, potential, thermal, chemical — yet all are interchangeable, pointing to a single underlying structure.
Crucially, this quantity is never observed directly, but only through differences between states: a change of velocity, a shift in configuration, a transition of phase. Its value is relational, not absolute: it depends on the chosen frame or comparison, never on an object in isolation.
Moreover, this quantity provides continuity of causality. If it changes in one part of the system, a complementary change must occur elsewhere, ensuring the unbroken chain of transitions. Thus it is the bookkeeping of causality itself.
Where to Go Next
Four ways into the research, depending on how you prefer to read.
The complete derivation tree — every step from Axiom 0 to the unified field equation, with each node linked into the paper.
WILL Part I — Relational Geometry. The full 60-page derivation chain, archived on Zenodo.
A model trained strictly on the paper. Ask whether this work meets your criteria before committing to the full read.
Falsifiable consequences of Relational Geometry — the claims this framework puts at empirical risk.