Model sanity, metric sensitivity and analytic benchmarks

D(A,B)

Weighted shortest path

Minimum total relational length over all paths from A to B.

1-Lipschitz in L1 edge-weight perturbations, so Γ_A ≤ 1 by construction.

R_eff(A,B)

Effective resistance

(e_A − e_B)ᵀ L⁺ (e_A − e_B) on the weighted Laplacian with conductance c_e = 1/w_e.

Aggregates every parallel route, not just the geodesic. Electrical language is an analogy for the linear algebra; no circuit and no physics is implied.

E_glob

Global efficiency

Mean of 1/d(i,j) over all ordered node pairs.

A whole-network diffusion-style summary. It has no A→B interpretation and no physical units.

Γ_A ≤ 1 under shortest path with L1 weight cost

Weighted shortest-path distance is 1-Lipschitz with respect to L1 perturbations of the same edge weights, so |ΔD| ≤ Σ|Δw| and the adjacency gain cannot exceed 1. This is a property of the chosen graph metric. It is not a physical law, it is not a speed limit, and it is neither evidence for nor against physical Adjacency Theory. Γ > 1 under this metric would indicate an implementation error, never a discovery.

Seeded perturbation sweep finds zero violations

Thousands of deterministic seeded perturbation trials on the baseline universe produce zero Γ > 1 events. This confirms the implementation matches the proof; it discovers nothing about nature.

Gain is metric-relative

Γ_A is only bounded when D and C_A are measured in the same units on the same edge set. Under the coupling convention d = −L·ln(R) with cost measured in coupling space, or under a different observable such as effective resistance, the bound does not transfer and must be re-derived. Cross-metric gain figures are not comparable.

Normalized structural sensitivity is a ratio, not an amplifier

A ~10% reduction of one local edge changes global observables only modestly in raw units, but the fractional observable change divided by the fractional whole-network L1 modification is large simply because the denominator spans every edge. This is normalized structural sensitivity. It is not energy gain, free energy, amplification, or anything superluminal.

Adjacency Debt is non-negative

Debt_A = C_path − ‖R_final − R_initial‖₁ ≥ 0 by the triangle inequality. Direct and monotonic transformations have zero debt; detours and reversals create positive debt. It measures history inefficiency in relational-state space and is not a physical or energetic debt.

Step count cannot decide continuity

Monotonic multi-step transformations cost exactly the same as the direct one-step transformation between the same endpoints, so the number of stored intermediate relational states is a bookkeeping choice of the transformation record, not an observable. The absence of intermediate graph states does not demonstrate physical discontinuity.

No hysteresis in a reversible static rule

The current transformation rule has no state or history dependence, so a round trip R → R' → R leaves exactly zero residual. We report zero hysteresis. Any nonzero figure would require a genuinely history-dependent rule and a measurable round-trip residual.

Does any physical system realise a relational-distance observable?

Everything above concerns a graph. Whether any physical structure supports an observable that behaves like D(A,B) = F(R), whether identity can be defined independently of spatial state, and whether a transition operator has any physical referent all remain entirely open. No result on this page bears on them.

Which observable, if any, would be the physically meaningful one?

Shortest path, effective resistance and global efficiency disagree about how strongly a local relational edit matters. Nothing in the mathematics selects one of them as privileged. A physical theory would have to supply that selection before any of these numbers could be interpreted.