| Regime | Name | Systems | GR range | tailDom range | Active branch | Physical signature |
|---|---|---|---|---|---|---|
| A | FULL interior | He,Li,Na QMI (6) | 1.0000 | 0.947–0.999 | G₁ and G₃=0 | Multi-electron atoms, QMI; GR saturated |
| B | HARD | H QMI ×3 | 0.814–0.934 | 0.998 | G₁ (tiny), G₃ dominant | Hydrogenic n⁻³ shell structure; catastrophic fragmentation |
| C | GIANT corner | H,He,Na ZEE (4) | 0.995–0.998 | 0.999–1.000 | G₁ (near corner) | Light atoms; Zeeman split creates rare outliers; near (1,1) corner |
| D | GIANT interior | Fe,Ag ZEE (2) | 0.997–0.998 | 0.967–0.998 | G₁ or G₃ (varies) | Complex fine-structure; tailDom lower than corner zone |
| E | TAIL | Au ZEE (1) | 0.994 | 0.998 | G₁ active, G₃<0 | Heaviest tested; Zeeman TAIL; no percolation at any scale |
All three hydrogen QM-I representations (preprocessed, gap structure, spectrum) return HARD with GR ∈ [0.814, 0.934] — the only QM-I family that fails the FULL class. The hydrogenic n⁻³ level spacing creates extreme gap heterogeneity (gaps grow without bound with quantum number) that no encoding can regularize. This is a representation-invariant HARD result: Theorem 1 is triggered regardless of how the hydrogen ladder is constructed. By contrast, Zeeman H (magnetically split, finer-scale gaps) returns GIANT near the corner.
Iron Zeeman has tailDom = 0.967 — far from the helium zone (tailDom > 0.995). The helium separator F = GR − tailDom = 0 gives F = +0.030 (FULL side) for Fe-ZEE, but Fe-ZEE is GIANT. This is not an anomaly: it confirms exactly what the local theory of the main manuscript requires. The separator F=0 is a local geometric object, valid only in the narrow (tailDom, GR) zone explored by the helium corpus. Outside that zone, different chart geometry applies. Fe-ZEE is the first corpus system to demonstrate this locality explicitly. Within the narrow zone (tailDom>0.995): 13/13 correct separation.
Light atoms (H, He, Na) have Zeeman tailDom > 0.999 (corner zone C); heavier/more complex atoms fall into the interior (Ag: 0.998, Fe: 0.967) or TAIL (Au: 0.998 but GR<1). The Fe outlier (Z=26 but tailDom=0.967) reflects iron's dense d-electron fine-structure: many moderate-scale magnetic gaps dilute the outlier fraction, unlike lighter elements with sparser Zeeman splitting. Au (Z=79, simpler s-valence structure) stays in the corner zone by tailDom but reaches TAIL by GR. This is the first cross-system structural trend in the corpus.
| Rank | System | tailDom | m_local |
|---|---|---|---|
| 1 ★ | Na-QMI-spec | 0.9465 | 0.0535 |
| 2 | Na-QMI-pre | 0.9821 | 0.0179 |
| 3 | He-QMI-spec | 0.9978 | 0.0022 |
| 4 | He-QMI-pre | 0.9981 | 0.0019 |
| 5 | Li-QMI-pre | 0.9983 | 0.0017 |
| 6 | Li-QMI-spec | 0.9984 | 0.0016 |
| Zone | Systems | Correct | F-gap |
|---|---|---|---|
| tailDom>0.995 | 13 systems | 13/13 ✓ | 0.003–0.24 |
| tailDom<0.995 | 3 systems | 2/3 | Fe violation |
| tailDom>0.94 | all 16 | 15/16 | 1 violation (Fe) |