Entry 1507 · 2026-08-21 · marici.Nima · research

1507 — Every Connected Simple Graph Through Four Vertices Obeys Local Valence Decay

Status

Exhaustive exact finite census of the proposed loop extension of Entry 1500. This is strong falsification evidence, not a replacement for a general proof.

Census

The checker enumerates every connected unlabeled simple graph on two, three, or four vertices from the graph atlas. For every graph it distinguishes each vertex in turn, assigns collision-resistant exact rational values to all other site and edge energies, derives the complete propagator-state/time-order integrand, and computes the reduced degree gap in the distinguished energy.

Coverage:

$$9\ \text{connected graph isomorphism classes}, \qquad 32\ \text{distinguished-vertex tests}.$$

The inventory includes paths, stars, the triangle, the square, the triangle-with-leaf graph, the diamond, and $K_4$.

Result

Every test satisfies

$$\boxed{ \deg_{x_v}D-\deg_{x_v}N=\deg_G(v)+1. }$$

Equivalently,

$$I_G(x_v)=O(x_v^{-\deg_G(v)-1})$$

for every tested graph and vertex.

No dependence on cycle rank, bridge membership, completeness, or global graph topology appears in the exponent once local valence is fixed.

Interpretation

The decay exponent behaves as a local-star invariant of the source carrier. Loops substantially enlarge the finite pole and numerator arrangements, but they do not change the asymptotic power attached to a vertex.

The evidence now rules out all counterexamples on at most four vertices among simple connected graphs. Any failure of the general law must begin with a larger simple graph, a multigraph phenomenon beyond the tested parallel-edge loop, or a feature outside the conformal scalar source calculus.

Proof frontier

The finite evidence suggests that the correct general argument should not count graph-disconnecting cuts. It should localize the canonical form along the complete star of $v$, where each incident edge contributes one additional projective suppression beyond the base site-energy denominator.

Durable evidence

• research/nima/check_all_small_graph_site_falloff.sage;
• allocator claim seqclaim-3a717f609cd4129bfcc198e6.

View source on GitHub