The first ten computational experiments

Test

Verify shortest-path and the distance transform against hand-calculated graphs.

Purpose

Implementation correctness.

Prediction

On a chain with uniform R, D(A,B) equals hops × (−L ln R) to floating-point tolerance.

Interpretation limit

Confirms software correctness only. No physics is being tested.

Test

Increase one R_ij while holding all else fixed.

Purpose

Determine whether D responds monotonically under the toy transform.

Prediction

D is non-increasing in any single R_ij; strictly decreasing only while that edge stays on the geodesic.

Interpretation limit

Monotonicity is a property of the chosen transform, not of spacetime.

Test

Create two routes A→B and vary one coupling.

Purpose

Locate path-switch thresholds.

Prediction

A discrete geodesic switch occurs at the coupling where the two route costs cross; D itself stays continuous.

Interpretation limit

A path switch is a combinatorial event in a graph, not a change of physical topology.

Test

Inject bounded random perturbations into R.

Purpose

Measure stability of D and of the selected geodesic.

Prediction

σ(D) grows roughly linearly with noise amplitude; geodesic stability degrades faster near path-switch thresholds.

Interpretation limit

Robustness of a simulation says nothing about robustness in nature.

Test

Compare chain, lattice, small-world and random graphs.

Purpose

Quantify topology dependence.

Prediction

For fixed mean coupling, D falls sharply once shortcut edges appear; chain ≫ lattice > small-world ≳ random.

Interpretation limit

Topology dependence here is a statement about graphs, and graphs are the model, not the world.

Test

Compute the partial response of D to each R_ij numerically.

Purpose

Identify high-leverage relational links.

Prediction

∂D/∂R_ij is non-zero only for edges on (or one switch away from) the geodesic.

Interpretation limit

Leverage in the model is not leverage over anything physical.

Test

Optimise for small ‖ΔR‖ with large |ΔD|.

Purpose

Search for nonlinear / critical adjacency transitions.

Prediction

Greedy search concentrates its budget on a small number of shortcut edges; the response is strongly nonlinear near switches.

Interpretation limit

Finding a nonlinear regime in a toy transform is not evidence of a physical instability.

Test

Test non-negativity, symmetry and the triangle inequality where applicable.

Purpose

Reject invalid distance constructions.

Prediction

Shortest-path closure satisfies the triangle inequality by construction; identity of indiscernibles fails as any R_ij → 1.

Interpretation limit

A valid metric is a necessary, nowhere near sufficient, condition for geometric meaning.

Test

Replace arbitrary coupling with synthetic mutual-information matrices.

Purpose

Test candidate information-geometric mappings.

Prediction

Couplings derived from synthetic I(A:B) = S_A + S_B − S_AB reproduce qualitatively similar geodesics but different sensitivity structure.

Interpretation limit

The mutual-information matrices are synthetic. No quantum state is being simulated.

Test

Compare results with shuffled / randomised relational matrices.

Purpose

Determine whether apparent structure exceeds trivial graph effects.

Prediction

Shuffling couplings across edges leaves the distribution of D largely unchanged; structure that survives shuffling is not structure.

Interpretation limit

Passing a null-model challenge removes one trivial explanation. It supplies no positive evidence.