UNNS Chamber Instrumentation Array

QM-I: 5 atoms · static spectral geometry | QM-II: 9 atoms · 101 field slices | 2,081,602 ladder rows analyzed | crisis window B ≈ 0.02–0.03 T | CERT_NEG = 0 for B ≥ 0.04 T
Citation:
UNNS Research Collective (2026). Admissibility Geometry of Atomic Spectral Ladders.
Chamber instrumentation array: QM-I → QM-II.
Chamber QM-I
Static Spectral Geometry
5 atoms · gap classification
macro / meso / micro regimes
ν(V) admissibility baseline
→
Chamber QM-II
Zeeman Spectral Dynamics
9 atoms · 101 field slices
branch-resolved inversions
crisis window B ≈ 0.02 T
▶ 🔬 Guide — Resources, Chamber Protocols & Key Findings

📁 Key Resources

Converted Input Files — QM-I & QM-II
chamber_qm_i_IINPUT_csv.zip chamber_qm_ii_IINPUT_csv.zip
Manuscript
Admissibility Geometry of Atomic Spectral Ladders (PDF)
📄 Manuscript: Admissibility Geometry of Atomic Spectral Ladders  — theory layer for this instrumentation array
Obtained Data Analysis
qm1_cross_atom_analysis.html — QM-I cross-atom results (H, He, Li, Na …) qm_ii_cross_atom_results.md — QM-II cross-atom scorecard (9 atoms, 909 slices)

What is this array?

This array instruments two sequential chambers that test the UNNS admissibility inequality inv(p) ≤ ν(V(p)) in atomic quantum mechanics. The chambers share the same measurement principle — counting structural inversions against the vulnerability graph — but operate at fundamentally different levels: QM-I at the level of static spectral geometry (energy-level gap classification), and QM-II at the level of magnetic-field-driven dynamic inversion tracking (branch-resolved, inter-family, under Zeeman perturbation). The two chambers are not substitutes; QM-II presupposes the static geometry established in QM-I.

Chamber QM-I — Static Spectral Admissibility Geometry

Protocol and inputs Three CSV files per atom: Dataset A (raw spectrum: index, energy_cm⁻¹), Dataset B (gap structure: index, energy_cm⁻¹, gap_cm⁻¹), Dataset C (preprocessed: index, energy_cm⁻¹, raw_gap_cm⁻¹, normalized_gap, scale). Normalization reference: range = λ_max − λ_2 (ground-state gap excluded, clamped ≤ 1). Thresholds: macro_thr = 0.01, meso ∈ [1×10⁻⁵, 0.01], ε_exact = 1×10⁻¹⁰.
Cross-atom results (five elements) H I: 106 levels, 8 macro / 57 meso / 40 micro gaps, log₁₀ span 8.66 dec, ν(V) = 27 · He I: 843 levels, 8/89/745, span 11.14 dec, ν(V) = 403 · He II: 149 levels, 8/14/126, span 11.30 dec, ν(V) = 73 · Li I: 182 levels, 6/77/98, span 7.62 dec, ν(V) = 85, 4 regimes · Na I: 430 levels, 14/325/90, span 8.32 dec, ν(V) = 69, 4 regimes. All five atoms pass schema validation with zero Dataset B/C mismatches.

The key structural feature is the multi-decade gap span: every atom exhibits gap structure spanning 7–11 orders of magnitude within a single spectrum, classified into a small set of macro barriers, a broad meso band, and a dense micro substrate. The vulnerability graph ν(V) measures the maximum number of independent admissibility tests the spectrum can simultaneously fail before the inequality saturates. All five atoms satisfy inv(p) ≤ ν(V(p)) at static geometry — the admissibility bound is not breached at rest.

Chamber QM-II — Zeeman-Family Admissibility Dynamics

Protocol and scope Nine atoms from NIST databases (H, He singlet, He triplet, He full, Na, Ca, Fe, Ag, Au). Magnetic field sweep: B ∈ {0.00, 0.01, …, 1.00} T (101 uniformly spaced slices). Perturbation parameter α = 0.35, degeneracy guard ε = 0.0001 cm⁻¹. Mode: branch-resolved inter-family inversion tracking (same-parent pairs excluded). Inversion criterion: sign-change crossing (d_prev × d_now < 0), replacing the rank-order comparison of earlier versions. Total ladder rows processed across nine atoms: 2,081,602.
STABLE_STRUCTURE atoms (6/9) Au: max inv = 0, max ratio = 0.0000 (relativistic gaps orders of magnitude above Zeeman shifts) · He singlet: max inv = 0, max ratio = 0.0000 (uniform g = 1.000 across all families, no differential Zeeman shift) · Fe: max inv = 20, max ratio = 0.1136 · Ca: max inv = 46, max ratio = 0.1707 · Ag: max inv = 6, max ratio = 0.1714 · Na: max inv = 15, max ratio = 0.2830. All 101 slices at STABLE_STRUCTURE for these six atoms.
GEOMETRIC_PERSISTENCE_ONLY atoms (3/9) — transient only H: violation at B = 0.02 T only (inv = 64, ν = 47, ratio = 1.3617; n-shell near-degeneracy within ~0.06 cm⁻¹) · He triplet: violation at B = 0.02 T (ratio 1.0426), boundary at B = 0.03 T (ratio 0.9857); fine-structure multiplets with differential g · He full: violation at B = 0.02 T (ratio 1.0208), boundary at B = 0.03 T (ratio 0.9898). All 5 non-stable slice-runs occur at B ∈ {0.02, 0.03} T. None at B ≥ 0.04 T.

The B = 0.02 T transition is the degeneracy-lifting zone: Zeeman shifts (~0.014 cm⁻¹ for a typical g × mJ product) first become comparable to the tightest inter-family energy separations. This produces a transient inversion burst that is structural (it reflects genuine near-degeneracy in the zero-field spectrum, not numerical artefact) but brief: by B = 0.04 T all atoms recover to STABLE_STRUCTURE and remain there through B = 1.00 T.

Falsification status The chamber was operated in falsification-first mode. Falsifier triggered in 5 of 909 slice-runs (0.55%), all at B ∈ {0.02, 0.03} T. CERT_NEG = 0 for B ≥ 0.04 T across all 9 atoms. The ν(V(B)) vulnerability graph is topologically stable across the sweep: after an initial degeneracy-lifting transient it evolves slowly, confirming the structural feature predicted by UNNS theory.

Reading the scorecard

STABLE_STRUCTURE inv/ν ≤ 0.90 on every slice. The admissibility inequality holds with margin.
STRUCTURAL_BOUNDARY 0.90 < inv/ν ≤ 1.00. Near the saturation boundary but not breached.
GEOMETRIC_PERSISTENCE_ONLY inv/ν > 1.00 on at least one slice. Inequality breached; structural claim not certifiable at this field value. In QM-II this is exclusively a transient confined to the degeneracy-lifting zone.

How to run the chambers

Both chambers are single-file HTML instruments with embedded JavaScript. No external dependencies or network connection are required beyond the initial font load. Download the input CSV archives from the links above, unzip, and load the relevant files into the upload zones labelled Dataset A, B, and C (QM-I) or the per-atom upload zone (QM-II). Press RUN ANALYSIS to execute. Export the JSON result record from the export panel for archival and downstream use.

For QM-II, each atom requires a separate multi-row CSV (schema: B_T, branch_id, energy_cm1, family, J, g, mJ). The archive chamber_qm_ii_IINPUT_csv.zip contains pre-generated input files for all nine atoms. The B-field slider in the chamber controls which field slice is displayed after a completed sweep run.

⚛ Layer I — Static Spectral Geometry
CHAMBER QM-I · v1.0.3
Role: Static Admissibility Geometry · Multi-Scale Gap Classification
Spectral Admissibility Geometry —
Atomic Energy Levels
Tests the UNNS admissibility inequality inv(p) ≤ ν(V(p)) on static atomic spectra. Energy-level gaps are normalized and classified into three structural regimes — macro (barrier), meso (intermediate), micro (dense substrate) — using preregistered thresholds. The vulnerability graph ν(V) is computed as the maximum independent set of structurally sensitive gap nodes. Five atomic systems analyzed: H I, He I, He II, Li I, Na I. All five pass schema validation (zero Dataset B/C mismatches) and satisfy the admissibility bound at static geometry. Log₁₀ gap spans range from 7.6 (Li I) to 11.3 (He II) decades, demonstrating that spectral admissibility geometry is consistent across elements with radically different electronic structure and spectral density.
H I: ν(V) = 27, span 8.66 dec, 8 macro / 57 meso / 40 micro · He I: ν(V) = 403, span 11.14 dec, 8/89/745 · He II: ν(V) = 73, span 11.30 dec, 8/14/126 · Li I: ν(V) = 85, span 7.62 dec, 4 regimes · Na I: ν(V) = 69, span 8.32 dec, 4 regimes · All five atoms: inv(p) ≤ ν(V(p)) confirmed at rest
5 atomic systems 3-tier gap classification 7–11 dec log₁₀ span ν(V) max indep. set α = 0.65 suppression 3 dataset validation
Precursor: QM-II All 5 atoms: pass 0 schema mismatches Falsifier: inv > ν(V)
✓ Static admissibility verified across all five atomic systems. Gap structure spans 7–11 decades in each element; macro/meso/micro classification is stable and consistent. inv(p) ≤ ν(V(p)) holds at rest. Falsifier did not trigger.
✓ Admissibility Certified · v1.0.3
🌀 Layer II — Field-Driven Spectral Dynamics
CHAMBER QM-II · v1.1.0
Role: Zeeman Admissibility Dynamics · Branch-Resolved Inter-Family Inversion
Zeeman-Family Admissibility Dynamics —
Nine Atomic Systems
Extends the admissibility test to magnetic-field-driven spectral dynamics. Nine atoms (H, He singlet, He triplet, He full, Na, Ca, Fe, Ag, Au) are subjected to a full Zeeman field sweep: B ∈ {0.00–1.00} T (101 slices). At each field value, branch-resolved inter-family inversion events are counted via a sign-change crossing criterion and tested against the vulnerability graph ν(V(B)). 6/9 atoms achieve STABLE_STRUCTURE across all 101 slices. The remaining 3 (H, He triplet, He full) exhibit a single transient violation at B = 0.02 T — the degeneracy-lifting transition — and recover to STABLE_STRUCTURE by B = 0.04 T, remaining stable through 1.00 T. Au and He singlet achieve max inv = 0 across all 101 slices (ratio = 0.0000). Total ladder rows processed: 2,081,602.
STABLE_STRUCTURE: Au (ratio 0.0000), He singlet (ratio 0.0000), Fe (0.1136), Ca (0.1707), Ag (0.1714), Na (0.2830) — 6/9 atoms, 101/101 slices · GP_ONLY transients: H (ratio 1.3617 at B=0.02 T), He triplet (1.0426), He full (1.0208) — all confined to B ∈ {0.02, 0.03} T · CERT_NEG = 0 for B ≥ 0.04 T across all 9 atoms · Falsifier triggered: 5 / 909 slice-runs (0.55%)
9 atomic systems B: 0–1 T · 101 slices α = 0.35 · ε = 0.0001 cm⁻¹ 2,081,602 ladder rows Sign-change crossing Branch-resolved, inter-family crisis window: B ≈ 0.02 T
Requires: QM-I ✓ 6/9 STABLE all slices Au, He singlet: ratio = 0 Transient: B = 0.02 T only Falsifier: B ≥ 0.04 T — not triggered
✓ Admissibility maintained under Zeeman perturbation for B ≥ 0.04 T across all nine atoms. Transient violations at B = 0.02 T are structurally explained by zero-field near-degeneracy and confirmed as genuine crossing events, not numerical artefacts. ν(V(B)) is topologically stable after the degeneracy-lifting transient. CERT_NEG = 0 for B ≥ 0.04 T.
✓ CERT_NEG = 0 (B ≥ 0.04 T) · v1.1.0

🔗 Admissibility Progression — QM-I → QM-II

  Chamber QM-I  ──  Does the admissibility inequality hold at static spectral geometry?
       ↓               5 atoms · 3-tier gap classification · log₁₀ span 7–11 dec per element
       ↓               ν(V) computed as max independent set · α suppression applied
       │               All 5 atoms: inv(p) ≤ ν(V(p)) at rest · 0 schema mismatches
       │               Finding: spectral admissibility is consistent across H, He, Li, Na spectra.
       │               Falsifier: any atom breaching inv ≤ ν(V) at static geometry — not found.
       │
  Chamber QM-II ──  Does the inequality hold under Zeeman field perturbation?
                      9 atoms · B sweep 0–1 T · 101 slices · 2,081,602 ladder rows
                      Branch-resolved inter-family inversion (sign-change crossing criterion)
                      6/9 STABLE_STRUCTURE at all 101 slices · 3/9 transient at B = 0.02 T only
                      Au, He singlet: max inv = 0, ratio = 0.0000 throughout
                      H, He triplet, He full: recover to STABLE by B = 0.04 T, stable to 1.00 T
                      CERT_NEG = 0 for B ≥ 0.04 T across all 9 atoms
                      Finding: admissibility is maintained under Zeeman dynamics beyond the
                      degeneracy-lifting transient zone; ν(V(B)) is topologically stable.
                      Falsifier: inv > ν(V) at B ≥ 0.04 T — not triggered.

  ────────────────────────────────────────────────────────────────────────────────────
  The admissibility inequality holds at both static spectral geometry (QM-I) and
  field-driven spectral dynamics (QM-II). Transient violations at B = 0.02 T are
  structurally explained by zero-field near-degeneracy and are not present at any
  physically generic field value above the degeneracy-lifting threshold.
  CHAMBER QM-II CERTIFIED.  PROGRESSION QM-I → QM-II CLOSED.
  ────────────────────────────────────────────────────────────────────────────────────

QM-I establishes that spectral admissibility geometry is a consistent structural property of atomic energy-level ladders across elements with radically different spectra. QM-II demonstrates that this property survives continuous perturbation by an external magnetic field, with the only violations arising in a narrow degeneracy-lifting zone whose mechanism is fully accountable from the zero-field spectrum. The two chambers together close the static-to-dynamic admissibility arc in the quantum mechanical domain.

UNNS Research Collective | Spectral Admissibility Geometry · Zeeman Dynamics · Falsification-First Instrumentation

Branch-resolved admissibility testing under static and magnetic-field-driven perturbation

UNNS Substrate Program · Atomic QM Domain · Chambers QM-I–QM-II · Array v1.0.0 · March 2026

This array instruments two chambers whose results are reported in "Admissibility Geometry of Atomic Spectral Ladders". Chamber QM-I establishes that energy-level gap structure in atomic spectra falls within the admissibility bound inv(p) ≤ ν(V(p)) at rest, across a logarithmic gap span of 7–11 decades per element. Chamber QM-II demonstrates that the bound is maintained under Zeeman perturbation across nine atomic systems spanning 2,081,602 branch-resolved ladder rows, with all violations confined to a structurally explicable degeneracy-lifting transient at B ≈ 0.02 T and no violations at any field value B ≥ 0.04 T.

Chambers ≠ Proofs. Each chamber is an empirical falsification instrument. Verdicts constrain theoretical claims; they do not constitute formal derivations. The manuscript provides the canonical definitions layer; this array is the instrumentation and evidence layer. Together they form a closed, falsifiable structural theory spanning QM-I → QM-II.