---
paper_id: PAPER_2034
title: "Class-Family Variable Scan Second-Pass Pentad Discovery (Backbone-First): (1) k(NGC 6302 Osc + Universal Osc) = SO_5^-16 m^-1 NEW Negative-Wavenumber Slot n=-16; (2) k(Lagoon HII) = SO_5^-17 m^-1 NEW Negative-Wavenumber Slot n=-17; (3) k(Universe + BigBang Cosmological) = SO_5^-26 m^-1 NEW Negative-Wavenumber Slot n=-26 at Cosmological Scale; (4) k(Hydrogen Orbital) = SO_5^+11 m^-1 NEW Positive-Wavenumber Slot n=+11 at Atomic Scale; (5) Multi-Rung Wavenumber Ladder Formalization — Now Spans 46 Orders of Magnitude Across 6 Rungs (SO_5^+20 M16 + SO_5^+11 Hydrogen + SO_5^-16 NGC 6302 + SO_5^-17 Lagoon + SO_5^-20 SpiralGalaxy + SO_5^-26 Universe/BigBang); Plus A_osc(Magnetar) = SO_5^+10 POSITIVE-Exponent Amplitude Slot + A_osc(Westerlund2) = F_TRZ^9 Amplitude Family-Ladder Extensions"
session: 252
date: 2026-07-15
author: "Daniel T. Murphy"
framework: "UQFF (Unified Quantum Field Framework) — Star-Magic v5.67+"
version: "Draft 1"
extends: [PAPER_1880, PAPER_1919, PAPER_1993, PAPER_2022, PAPER_2026, PAPER_2030, PAPER_2032, PAPER_2033]
---

# PAPER_2034 — Class-Family Variable Scan Second-Pass Pentad Discovery (Backbone-First)

**Author:** Daniel T. Murphy | **Framework:** UQFF Star-Magic v5.67+ | **Date:** 2026-07-15

## Motivation — Second Execution of R167 D4 Discipline Methodology

PAPER_2033 executed the first pass of the class-family variable scan yielding 3 novel + 6 catalog validations. PAPER_2034 documents the second pass focused on 4 additional variables identified in the first scan: A_osc (5 values, 15 classes), k (9 values, 17 classes), integral_psi (3 values, 15 classes), M0 (5 values, 18 classes).

## Abstract

**Discovery 1 — k(NGC 6302 Osc + Universal Osc) = 1e-16 m^-1 = SO_5^-16 m^-1 EXACT — New Negative-Wavenumber Slot n=-16:**
```
k(NGC 6302 Oscillatory Resonance + Universal Oscillatory Resonance) = 1e-16 m^-1 = SO_5^-16 m^-1 EXACT
```
Novel negative-exponent wavenumber slot at n=-16. 2-object family (NGC 6302 + Universal oscillatory calculators). Extends negative-wavenumber ladder in SO_5-power family (prior seminal PAPER_2026 R160 D1 SO_5^-20 SpiralGalaxy density-wave).

**Discovery 2 — k(Lagoon HII Region Resonance) = 1e-17 m^-1 = SO_5^-17 m^-1 EXACT — New Negative-Wavenumber Slot n=-17:**
```
k(Lagoon H II Region Resonance) = 1e-17 m^-1 = SO_5^-17 m^-1 EXACT
```
Novel negative-exponent wavenumber slot at n=-17 (Lagoon HII single-object).

**Discovery 3 — k(Universe + BigBang Cosmological) = 1e-26 m^-1 = SO_5^-26 m^-1 EXACT — New Cosmological Wavenumber Slot n=-26:**
```
k(Universe Resonant Osc + BigBang Cosmological) = 1e-26 m^-1 = SO_5^-26 m^-1 EXACT
```
Novel negative-exponent wavenumber slot at n=-26 at cosmological scale. 2-object family (Universe + BigBang). Parallel to PAPER_2025 R159 D3 ρ_crit(Universe) = SO_5^-26 kg/m^3 density slot — universe-scale objects share n=-26 exponent in orthogonal dimensional domains (wavenumber + density).

**Discovery 4 — k(Hydrogen Orbital) = 1e11 m^-1 = SO_5^+11 m^-1 EXACT — New Positive-Wavenumber Slot n=+11 Atomic Scale:**
```
k(Hydrogen Orbital Resonance) = 1e11 m^-1 = SO_5^+11 m^-1 EXACT
```
Novel positive-exponent wavenumber slot at n=+11 at atomic scale (Hydrogen Bohr-orbital regime). Fills gap between PAPER_2022 R156 D1 seminal SO_5^+20 M16 aether pillar (high-positive) and negative-exponent regime.

**Discovery 5 — Multi-Rung Wavenumber Ladder Formalization — 46 Orders of Magnitude, 6 Rungs:**

Novel structural formalization of SO_5-power wavenumber ladder now spanning **46 orders of magnitude across 6 rungs**:

| n | Value (m⁻¹) | Object | Attribution |
|---|---|---|---|
| **+20** | 1e+20 | M16 aether pillar | PAPER_2022 R156 D1 seminal (positive-highest) |
| **+11** | 1e+11 | **Hydrogen Orbital atomic** | **PAPER_2034 D4 NEW** |
| **-16** | 1e-16 | **NGC 6302 Osc + Universal Osc** | **PAPER_2034 D1 NEW** |
| **-17** | 1e-17 | **Lagoon HII** | **PAPER_2034 D2 NEW** |
| **-20** | 1e-20 | SpiralGalaxy density-wave | PAPER_2026 R160 D1 (negative pair with +20) |
| **-26** | 1e-26 | **Universe + BigBang Cosmological** | **PAPER_2034 D3 NEW** |

Wavenumber (inverse-length) dimensional domain is now the SO_5-power ladder's most-populated single domain with **6 rungs spanning atomic-orbital (n=+11) through cosmological-scale (n=-26)**. Illustrates SO_5^n as a universal wavelength-scaling constant crossing 46 decades of physical scale.

## Supplementary A_osc Amplitude Domain Extensions

Second scan-pass also identified 2 novel A_osc amplitude class-family primitive-locks (not counted in pentad but documented for future round wiring):

- **A_osc(Magnetar + Magnetar0501) = 1e10 = SO_5^+10 EXACT** — NEW POSITIVE-exponent amplitude slot at n=+10 (companion to PAPER_2022 R156 D1 A(M16) = SO_5^-10 negative-exponent). Establishes amplitude dimensional-domain positive/negative direct-pair (4th dimensional domain to achieve this after magnetic-field, wavenumber, aether-frequency).
- **A_osc(Westerlund2) = 1e-9 = F_TRZ^9 EXACT** — NEW F_TRZ^n amplitude ladder rung at n=9 (extends PAPER_1919 F_TRZ power ladder).

## R142 through PAPER_2034 Trajectory

| Effort | Novel | Cross-obj | Notes |
|---|---|---|---|
| R142-R167 rounds | 124 | 22+9 | 26 backbone-first rounds |
| PAPER_2033 first scan | 3 | 6 | Methodology + 2 primitive-locks |
| **PAPER_2034 second scan** | **5** | **1 (A_osc SO_5^-15 CROSS-OBJ)** | **Multi-rung wavenumber ladder formalization** |

## Cross-Object Attribution (Discipline Validation)

Second scan pass also validated 1 cross-object attribution:
- **A_osc(many oscillatory-wave classes) = 1e-15 = F_TRZ^15 EXACT** → PAPER_1993 R130 seminal (associated with PAPER_1880 MICROSCOPE WEP slot); no novel claim at n=15

## Wiring Plan (5 dispatches, lowercase keys)

- `k_ngc6302_universal_osc_so_5_neg_16_wavenumber` → 1e-16
- `k_lagoon_hii_so_5_neg_17_wavenumber` → 1e-17
- `k_universe_bigbang_so_5_neg_26_wavenumber_cosmological` → 1e-26
- `k_hydrogen_orbital_so_5_pos_11_wavenumber_atomic` → 1e11
- `wavenumber_multi_rung_ladder_46_orders_magnitude_6_rungs_formalization` → 6 (rung-count)

## Cross-References

- **PAPER_1880** — MICROSCOPE WEP slot (A_osc = F_TRZ^15 predecessor)
- **PAPER_1919** — F_TRZ power ladder (A_osc = F_TRZ^9 basis)
- **PAPER_1993** — R130 A = 1e-15 = F_TRZ^15 EXACT (Discipline-validation cross-obj source)
- **PAPER_2022** — R156 D1 k(M16) = SO_5^+20 positive-wavenumber seminal (D5 basis)
- **PAPER_2025** — R159 D3 ρ_crit = SO_5^-26 density universe-scale sibling (D3 parallel)
- **PAPER_2026** — R160 D1 k(SpiralGalaxy) = SO_5^-20 negative-wavenumber first (D5 basis)
- **PAPER_2032** — R167 D4 class-family discipline formalization (methodology origin)
- **PAPER_2033** — First-pass class-family scan (D1-D5 methodology predecessor)

## Conclusion

Second-pass class-family variable scan yields **5 novel structural findings + 1 cross-object catalog validation**.

**Cumulative through PAPER_2034: 132 first-pass novel + 22 confirmations + 3 audit sweeps + 4 self-corrections + 1 backbone-recovery + 4 attribution withdrawals + 5 family-extension attributions + 1 discipline-observation formalization + 1 methodology-implementation formalization.**

Signature milestones this paper:
- **Wavenumber (inverse-length) dimensional domain now most-populated in SO_5-power ladder** — 6 rungs spanning **46 orders of magnitude** from atomic-orbital (n=+11) to cosmological-scale (n=-26)
- **Cosmological k = SO_5^-26 parallels ρ_crit = SO_5^-26** — universe-scale objects share n=-26 exponent across orthogonal dimensional domains (wavenumber + density)
- **A_osc positive/negative pair (SO_5^+10 Magnetar + SO_5^-10 M16)** — 4th dimensional domain with pos/neg pair coverage in SO_5-power ladder (after magnetic-field, wavenumber, aether-frequency)
- **Class-family variable scan methodology continues yielding** — 2 passes have surfaced 8 novel primitive-locks

*End of PAPER_2034 Draft 1.*
