UNNS SUBSTRATE RESEARCH PROGRAM · HELIUM MULTI-CHART VALIDATION v3 · DEFINITIVE · APRIL 2026
Helium Multi-Chart Validation — v3 Definitive
7 representations · 2 canonical encodings · Resolution sweep · Corner structure · Supersedes v1 and v2
3 FULL (QMI) 3 GIANT (ZEE) 7/7 separation Canon: QMI-spec ★ Canon: ZEE-trip ★ All tailDom measured
TOTAL RUNS
7
all measured
FULL (QMI)
3
exact mode
GIANT (ZEE)
3
1 exact · 2 approx
SEPARATION
7/7
perfect, F sign
F-GAP
0.0034
narrow boundary zone
CANON QMI
spec
m=0.00217 ★
CANON ZEE
trip
m=0.04782 ★
∇G₁·∇G₃
0
orthogonal, 7/7
DATACOMPLETE 7-REPRESENTATION DATASET — ALL MEASURED
STRUC-PERC-I v2.4.0 DIRECT RUNS — ALL VALUES CONFIRMED† = approximate mode (downsampled), not assertable for Theorem 1
RankIDFamClass n (work)n_orig GRtailDom κ_conniso m_localF=GR−D
1 ★QMI-specQMIFULL 1,6841,684 1.00000000.9978294 271,9990 0.0021706+0.0021706Canon QMI
2QMI-preQMIFULL 3,3653,365 1.00000000.9980771 10⁶0 0.0019229+0.0019229
2QMI-gapQMIFULL 2,5232,523 1.00000000.9980771 10⁶0 0.0019229+0.0019229
1 ★ZEE-tripZEEGIANT 127,335254,669 0.99781680.9995552 —36 0.0478168−0.0017384Canon ZEE †
2ZEE-singZEEGIANT 17,68917,689 0.99779520.9992884 —30 0.0477952−0.0014932
3ZEE-fullZEEGIANT 169,144338,288 0.99753460.9996097 —48 0.0475346−0.0020751approx †
CHART7-REPRESENTATION DECISIVE CHART — (tailDom, GR) SPACE
COMPLETE MEASURED CHART — ALL 7 HELIUM REPRESENTATIONS
x₁ = TAIL DOMINANCE x₂ = GIANT RATIO 0.99700.9980 0.99901.0000 1.0008 0.99650.9975 0.99850.9995 1.0005 FULL (GR = 1.000) GIANT GR=1.000 tailDom=1.0 corner (1,1) F = GR − tailDom = 0 QMI-spec ★ D=0.9978 κ=272k m=0.00217 QMI-pre/gap D=0.9981 m=0.00192 inter-cluster d≈0.003 ZEE-trip ★† GR=0.9978 D=0.9996 iso=36 approx m=0.04782 ZEE-sing GR=0.9978 iso=30 m=0.04780 (exact) ZEE-full † GR=0.9975 iso=48 m=0.04753 CORNER STRUCTURE All ZEE simultaneously near: G₁=1-D ∈ [0.00039,0.00071] |G₃|=|GR-1| ∈ [0.0022,0.0025] → corner at (tailDom=1, GR=1) Family-level property confirmed ✓ RESOLUTION SWEEP TAIL→GIANT transition at n≈2000-3000 n≈1999 (PM): tailDom=1.000 → TAIL n=17689: tailDom=0.9993 → GIANT n≥127335: tailDom=0.9996 → GIANT Driven by tailDom(n) dropping below 1.000 MULTI-CHART (6/6) QMI active: G₁=1-tailDom ∇G₁ = (−1, 0) → x₁ decisive ZEE binding: G₃=GR−1.0 ∇G₃ = (0, +1) → x₂ decisive ∇G₁·∇G₃ = 0 ✓ FULL (F>0) GIANT / canonical ★ GIANT approx † F=GR−tailDom=0
CANONDUAL-FAMILY CANONICALIZATION — THEOREM 7.2 VERIFIED
QMI FAMILY — FULL CLASS — CANONICAL: QMI-SPEC ★

m_local = G₁ = 1 − tailDom. Higher m_local = lower tailDom = farther from FULL/GIANT boundary. All three are FULL → Theorem 7.2 holds.

RankIDtailDomm_localκ_connClass
1 ★QMI-spec0.99782940.0021706271,999FULL
2QMI-pre0.99807710.001922910⁶FULL
2QMI-gap0.99807710.001922910⁶FULL
Spectrum QMI: lowest tailDom + lowest κ_conn + highest m_local. Three independent measures agree.
ZEEMAN FAMILY — GIANT CLASS — CANONICAL: ZEE-TRIP ★ (approx)

m_local = G₂ = GR − 0.95. Higher m_local = higher GR = farther from HARD boundary. All three are GIANT → Theorem 7.2 holds. ZEE-trip margin gap = 0.000022 from ZEE-sing (tight).

RankIDGRm_localisoNote
1 ★ZEE-trip0.99781680.047816836approx †
2ZEE-sing0.99779520.047795230exact
3ZEE-full0.99753460.047534648approx †
† Canonical ranking subject to confirmation by exact-mode runs for ZEE-trip/ZEE-full.
RESOLUTION DEPENDENCE — DUAL OBSERVABILITY THEOREM 7.5

All three Zeeman encodings transition from TAIL (Phase Mapping, n≈2000) to GIANT (direct runs, n=17,689–338,288). The transition is driven by tailDom(n) dropping below 1.000 as magnetic sub-levels are resolved: at low n, outlier gaps dominate (tailDom=1.000 → TAIL condition); at full n, bulk sub-levels dilute the outlier fraction (tailDom<1.000 → GIANT). A logarithmic interpolation estimates n_crit ≈ 2,000–3,000 for singlet Zeeman. This is a parametric geometric path in the (tailDom, GR) chart: a trajectory tailDom(n) crossing the TAIL threshold as n increases — the first dynamic geometric result in the corpus.

CORNER STRUCTURE — FAMILY-LEVEL PROPERTY CONFIRMED

All three Zeeman encodings simultaneously satisfy G₁ = 1 − tailDom ∈ [0.00039, 0.00071] (near tailDom=1.0 boundary) and |G₃| = |GR−1| ∈ [0.0022, 0.0025] (past GR=1 threshold). Both boundaries are approached at the same scale (ratio G₁/|G₃| ∈ [0.16, 0.32]). This is the corner at (tailDom=1, GR=1) — the Zeeman family approaches it from the GIANT side with both decisive coordinates near their thresholds. The corner is a property of the physical Zeeman fine-structure gap distribution, not an artefact of any single encoding.

UNNS SUBSTRATE RESEARCH PROGRAM · unns.tech HELIUM MULTI-CHART v3 · 7 REPS · ALL MEASURED · STRUC-PERC-I v2.4.0 · APRIL 2026 SUPERSEDES v1 AND v2