Non-Monotonic Correlation Fluctuations in Heisenberg Models on Small Quantum Graphs

We report a reproducible non-monotonic feature in the standard deviation of nearest-neighbour correlation fluctuations (the enhanced diagnosis) in Heisenberg models generated from several small quantum graphs. In a Sierpiński fractal graph (N = 15), a contradiction edge at (0, 6) produces a sharp valley at s = 1.000, reproducible in position across five Lanczos seeds. A degeneracy-control experiment shows the valley persists in the non-zero-gap regime. A non-trivial coarse-grained observable responds only weakly at the same point, so the contrast between the fine-grained and coarse-grained observables is quantitative rather than qualitative. The mechanism and the thermodynamic limit remain open. This is a phenomenology report. It does not claim a new physical mechanism or universality, and the limitations section lists ten specific respects in which it does not. Independent verification and corrections in this version Every figure and table was re-derived independently before deposit, and four things changed as a result: The energy gap does not vanish below s ≈ 0.76. Measured in the Sz = +1/2 sector as the gap to the next distinct level, it is non-zero across the whole scan, falling monotonically from 0.6176 at s = 0 to 0.0115 at s = 0.99. The step across 0.76 → 0.77 is 0.0115, not a discontinuity. This strengthens the degeneracy-control conclusion: there is no gap switching on at 0.77 to which the valley could be attributed. A sweep of all eight distinct magnetisation sectors confirms the calibration E(0) = 0.246731 identifies the sector uniquely. At s = 1.000 the ground level is exactly two-fold (splitting 6×10−14), not near-degenerate; both states carry S = 1/2, so the degeneracy is orbital — the D3 symmetry of the uniform gasket. The height of the coarse-grained observable's local maximum is therefore solver-dependent and is no longer quoted as a fixed figure. The truncation-control figures were corrected. An earlier statement of that test quoted 86% against 20% recovery; re-measured against one untruncated reference with one grid, one bond set and one depth definition, far-bath truncation exceeds uniform truncation at every bond dimension tested, and uniform single-state truncation removes the valley entirely rather than leaving 20% of it. Definitional ambiguities were resolved: the factor of three between Eq. (4) and the tabulated dot product, the uniform point (s = 1, not s = 0), and the distinction between topographic prominence and scan-endpoint depth, which are close where both apply but are not the same quantity. Data, code and the verification scripts are public: edrn-dmrg-verification and the tensor-RG toolchain.