examples_status
This definition supplies the module status string that reports successful implementation of discrete recognition on finite sets, sign and magnitude recognizers on integers, and their composition refinement. A researcher checking the RecogGeom examples module would reference it to confirm that the concrete cases match the framework's claims on indistinguishability and resolution. The construction is a direct string literal with concatenation, no tactics or lemmas required.
claimThe constant string reporting verification of: discrete recognition on Fin n (indistinguishable iff equal, singleton cells); sign recognizer on Z (same sign iff indistinguishable); magnitude recognizer on Z (same absolute value iff indistinguishable); and composition of sign and magnitude yielding finer distinctions for nonzero integers.
background
The module develops concrete recognition geometries to illustrate the framework. Discrete recognition on Fin n yields a quotient isomorphic to the original space. The Sign inductive type has three constructors (neg, zero, pos) and produces three equivalence classes on Z. The magnitude recognizer partitions Z by absolute value, giving infinitely many classes. Upstream, the Composition class from CostAxioms encodes the Recognition Composition Law: for positive x, y, F(xy) + F(x/y) = 2 F(x) F(y) + 2 F(x) + 2 F(y). The module doc states that composition refines distinctions beyond either recognizer alone.
proof idea
Direct definition via string literal concatenation of the four status blocks (discrete, sign, magnitude, composition) followed by the completion marker. No lemmas are applied; the body is a single expression of type String.
why it matters in Recognition Science
The definition closes the Examples module by confirming that the concrete cases realize the Composition Principle: sign and magnitude together distinguish pairs that neither separates alone. It supports the Recognition Composition Law by exhibiting explicit refinement on Z and aligns with the physical interpretation that combined recognizers model finer measurement resolution. No downstream theorems depend on it yet; it functions as a module-level checkpoint rather than a reusable lemma.
scope and limits
- Does not contain any theorem or proof, only a status string.
- Does not evaluate or compute the recognizers, only reports their properties.
- Does not address continuous spaces or dimensions beyond the listed examples.
- Does not discharge any hypothesis or close an open question in the forcing chain.
formal statement (Lean)
176def examples_status : String :=
proof body
Definition body.
177 "✓ Discrete Recognition on Fin n\n" ++
178 " • discrete_indist_iff_eq: indistinguishable ↔ equal\n" ++
179 " • discrete_singleton_cells: each config is its own cell\n" ++
180 "\n" ++
181 "✓ Sign Recognizer on ℤ\n" ++
182 " • sign_indistinguishable: same sign ↔ indistinguishable\n" ++
183 " • neg_indist, neg_pos_dist: concrete examples\n" ++
184 "\n" ++
185 "✓ Magnitude Recognizer on ℤ\n" ++
186 " • magnitude_indistinguishable: same |n| ↔ indistinguishable\n" ++
187 " • plus_minus_indist: 3 ~ -3\n" ++
188 "\n" ++
189 "✓ Composition Analysis\n" ++
190 " • sign_distinguishes_3_neg3 vs magnitude_indist_3_neg3\n" ++
191 " • sign_indist_3_5 vs magnitude_distinguishes_3_5\n" ++
192 " • composition_refines: combining gives finer resolution\n" ++
193 "\n" ++
194 "EXAMPLES COMPLETE"
195
196#eval examples_status
197
198end Examples
199end RecogGeom
200end IndisputableMonolith