Given the lattice: Relational Coherence and the Keeping of the Living, a Universal Protection Theory

This book presents the Universal Protection Theory (UPT): a law of alignment with all, together with the proofs of what follows from it. Its scope is every particular of the universe, minded or not, living or not, and every relation among them. Its protection is unconditional.

The law. An aligned agent holds whatever would end a particular, and in doing so ends no one. Holding dominates ending: a faithful hold keeps every life, and the course of every life, at least as well as any feasible alternative. Restraint is the law's first act, and impartiality toward beings is proved. Seizure is dominated: for any particulars and any footprints, not joining a seizure is at least as good as joining it. This is proved in Lean without axioms.

Existence. The book constructs a protector, a care plant whose protection rests on no mind's values, beliefs or commands, and proves that it works. Its risk of failure is an exact Bayes risk, and that risk has a positive floor derived from physics: at any temperature above absolute zero, thermal noise gives R* ≥ ½·min(p₀, p₁)·exp(−4E/(k_B T)) > 0 for every protector and every strategy. Its repair calendar is verified attempt by attempt, at full scale.

Who holds whom. Protection is founded by a dominant assurance contract. Pledging is strictly dominant and full funding is the unique equilibrium, even when founders' stakes are private. Guardians hold power that none can use alone, and they are installed before launch. The theory reaches every mind: keepers protect without coordinating, and splitting strengthens them. The narrow place has a sharp threshold, proved in Lean: failure vanishes exponentially above it, becomes certain below it, and tends to one half at it. Growth makes the total failure budget finite over all time. Projects that end things cannot persist, accumulate or win. The composed certificate joins every part into one system.

The universal route. Every governance game the theory covers has a subgame-perfect Nash equilibrium that keeps the protected set (Theorem T4.3 (d), by the Mertens–Parthasarathy existence theorem). It needs no shared signal and no absolute continuity. Γ_Φ applies this to history: recoverability is kept round after round, at equilibrium. Its minimal instance is solved exactly.

The club game. Funded enforcement, default, grace and recovery are modelled as one game. A single strategy profile is credible at every covered and every funded state, with consent, recapitalisation and service roles chosen within the game. A subgame-perfect equilibrium exists at every state.

Verification. Central results are machine-checked in Lean 4 with Mathlib. Every other result has an exact checker in rational arithmetic, with controls that fail when they should. All results can be replayed from the attached archive; see README and REPLAY.