Executable mathematical exposition

UIM Theorem Frontier

An executable paper about what survives representation

Choose what information a graph representation keeps, choose a target property, and compute whether that information is sufficient to determine the property.

The question

A representation keeps some structure and discards the rest. When is the retained information enough to determine a mathematical property?

UIM feature-fiber criterion

P = p ∘ e ⇔ P is constant on every fiber of e

Here: e is the chosen representation, P is the target property, and p is a rule that uses only the representation value.

Fiber: the set of source graphs that collapse to the same representation value. One fiber containing different values of P is an exact obstruction.

Build the representation

Choose the finite universe, the information retained by e, and the property P you want that representation to determine.

Domain Locked scientific domain pack: graphs-simple-labeled-v1 Finite scope Exhaustive only within the declared finite scope. Maximum / exact n Browser-safe exact enumeration limit: n ≤ 6. Target predicate

Representation coordinates

Add or remove exact features. Changing this tuple changes the mathematical fiber partition.

Vertex count n(G) number of vertices Edge count m(G) number of undirected edges Sorted degree sequence degseq(G) sorted nondecreasing degree sequence Triangle count t(G) number of 3-cycles Connected-component count c(G) number of connected components Exact adjacency bitstring a(G) exact labeled adjacency bitstring in fixed lexicographic edge order; for n >= 1, the bitstring length uniquely determines n

Current representation e(G) = (…)

Demonstration presets

Compute the fibers

The browser enumerates every graph in the declared scope, groups graphs with identical representation values, and tests whether the target predicate is constant inside each fiber.

Theorem Frontier: TF-1 exact obstruction · TF-2 bounded factorization · TF-3 registered general theorem.

Inspect the result

Counts are computed first. Interpretation follows from the exact fiber computation.

Objects tested—

Distinct fibers—

Non-singleton fibers—

Largest fiber—

—

Strongest justified claim

Generalization boundary

Why this conclusion is valid

Reproducible experiment

This experiment record identifies the exact domain, scope, representation, predicate, and scientific registry versions used in the computation.

Experiment ID Experiment link

Fiber explorer

Inspect actual equivalence classes induced by the selected representation.

Page 1

Representation value	Fiber size	Predicate values

Property spectrum

Hold the representation fixed and ask which enabled predicates it can determine.

What this instrument does—and does not claim

It does: compute exact finite representation fibers, generate exact obstruction witnesses, certify bounded factorization after exhaustive enumeration, and enforce registered theorem boundaries.

It does not: turn finite search into an unbounded theorem, treat one preserved predicate as full equivalence, solve open mathematical problems, or use AI to manufacture mathematical conclusions.

Mathematical provenance

Unified Informational Mathematics: UIPO Foundations, Formal Semantics, Analytic Realization, and Recovery Theorems Primary mathematical source for the feature-fiber and recovery/transfer framework.
UIM Theorem Frontier Technical Guide Technical guide for the executable exposition, experiment workflow, theorem-frontier statuses, reproducibility, and logical boundaries.

JavaScript is required for the executable experiment. The mathematical rule remains: a target predicate factors through a representation exactly when it is constant on every representation fiber. This page does not infer a general theorem from finite search alone.