IndisputableMonolith.LedgerUniqueness
LedgerUniqueness assembles lemmas showing that potentials and reachability relations remain unique up to additive constants on connected discrete components. Modelers of causal graphs in Recognition Science cite these results to guarantee deterministic ledger evolution from local recognition rules. The module structures its content as a sequence of inductive uniqueness statements built directly on imported reach-set lemmas from Potential.
claimFor a discrete reach set $N$ generated by the relation Reaches, any two potentials $V_1, V_2$ satisfying the affine condition IsAffine agree up to a constant: $V_1(x) - V_2(x) = c$ for all $x$ in the component. Similar uniqueness holds inside balls (inBall) and on $n$-step reaches (ReachN).
background
The module sits inside the Recognition Science development and imports three core modules. Recognition supplies the foundational T1 statement that nothing cannot recognize itself, which forces the existence of recognition processes and the associated ledger. Potential contributes dependency-light T4 uniqueness lemmas on discrete reach sets, while Causality.Basic supplies the underlying causal primitives. Local definitions include ReachN (n-step forward reachability), inBall (membership in a finite causal ball), Reaches (the transitive closure), and IsAffine (linearity of the potential along causal links).
proof idea
The module is organized as a collection of targeted uniqueness theorems rather than a single overarching proof. Each theorem proceeds by induction on the size of the reach set: base cases use the empty or singleton reach, and inductive steps apply the imported T4 lemmas from Potential together with the causal axioms from Causality.Basic to propagate the constant difference across one additional reach step. Auxiliary predicates such as unique_on_reachN and unique_up_to_const_on_component are introduced to factor the argument.
why it matters in Recognition Science
LedgerUniqueness supplies the concrete uniqueness infrastructure required by the T4 step of the forcing chain, ensuring that the ledger potential is rigidly determined once a single reference value is fixed on each connected component. It therefore feeds directly into any later theorem that reconstructs global spacetime geometry or conserved quantities from local recognition data. Because used_by is currently empty, the module functions as an internal supporting layer whose results are expected to be invoked once higher-level ledger reconstruction theorems are formalized.
scope and limits
- Does not establish uniqueness across disconnected components.
- Does not extend the result to continuous or continuum limits.
- Does not incorporate measurement or observer-dependent effects.
- Does not address stability under small perturbations of the reach relation.