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Six Birds Verified IV: Tolerance and Surrogates
When merged states give outputs that are close but not equal, what single value should the class report?
In plain words
A coarse description often lumps states that give outputs which are close but not equal. A thermometer of finite resolution or a cluster of measured profiles are everyday cases. One would like to report one value per lump that is close to every output in it. It is tempting to just declare close things equal. But closeness is not transitive: a is close to b and b to c without a being close to c. This paper replaces that shortcut with exact finite mathematics.
Part I shows that a single reported value exists exactly when each lump has an allowed common center, a value close to every output in it. The outputs need not be close to each other: 1 can serve 0 and 2. The best error is a finite minimax, at least half the lump's spread and at most the full spread when each lump may use one of its own outputs as a center. Both bounds are sharp. There is often no single minimal fix: on three points two incomparable smallest repairs exist. A natural looking bound on one audit by another is shown false and replaced by a sharp weight floor inequality.
Part II starts from one graded closeness relation and builds four different summaries of it: a shape made of cliques, its connected pieces, a matrix algebra, and a path distance. These are separate constructions with different properties, not views of one object. The four cycle and the complete graph on four points share every connected piece profile, yet differ in their loops. All results are exact, with Lean 4 formalizations.
What it shows
- Approximate reporting works exactly when every class has an allowed common center.
- The exact error lies between half the spread and the full spread, and both ends occur.
- There is no single minimal repair: on three points two incomparable ones exist.
- Four common summaries of one closeness relation are genuinely different objects.
What it does not claim
It makes no physical, statistical or learning theoretic claim. The ingredients, such as the 1 center problem, tolerance spaces and persistence, are classical and credited; what is new is their exact typing at a coarse graining interface.
Cite
Tsiokos, I. (2026). Six Birds Verified IV: Tolerance and Surrogates. Zenodo. https://doi.org/10.5281/zenodo.23097917