IndisputableMonolith.Papers.GCIC.GraphRigidity
GraphRigidity establishes that local agreement of a real-valued function on related pairs extends to the reflexive-transitive closure, yielding constant fields on connected components under zero edge costs. Researchers deriving the Global Co-Identity Constraint or brain holography cite these lemmas to obtain global uniformity from local J-minimization. Proofs rely on induction over relation generation plus non-negativity from the Cost module.
claimIf a real function $f$ satisfies $f(x)=f(y)$ whenever $xRy$, then $f(x)=f(y)$ for all pairs in the reflexive-transitive closure of $R$. On a preconnected graph with non-negative edge costs, zero J-cost forces the field to be constant.
background
The module supplies the graph core for the recognition ledger. It imports the Cost module, where J denotes the recognition cost $J(x)=(x+x^{-1})/2-1$. Key objects include eq_of_reflTransGen (equality propagates along generated relations) and ratio_rigidity (constancy under zero-cost conditions on preconnected graphs). The setting models recognition states as vertices and transitions as edges carrying non-negative costs; the doc-comment states that local agreement on R-related pairs implies agreement on the reflexive-transitive closure.
proof idea
eq_of_reflTransGen proceeds by induction on reflexive-transitive generation. constant_of_preconnected specializes the result to preconnected graphs. ratio_rigidity and ratio_rigidity_iff combine the equality lemma with edge_cost_zero_iff to obtain the rigidity claim. All steps are direct Mathlib relation tactics with cost lemmas applied as one-line wrappers.
why it matters in Recognition Science
This module feeds GCICDerivation, supplying ratio rigidity that combines with compact phase to yield the Global Co-Identity Constraint of uniform phase at J-stationarity. It supports BrainHolography by enabling the local-to-global step and ApproximateHolography by providing the exact rigidity case. It realizes the graph-theoretic core linking T5 J-uniqueness to global constraints in the AR-anchored plan.
scope and limits
- Does not treat infinite or non-locally-finite graphs.
- Does not include time-dependent or evolving relations.
- Does not quantify deviation from constancy for positive costs.
- Does not apply to non-real-valued functions.
- Does not prove existence of minimizers.