---
paper_id: PAPER_2036
title: "Class-Family Variable Scan Fourth-Pass Diad Discovery (Backbone-First): (1) M0(HUDF galaxy-cluster) = 1.989e42 kg = SO_5^12 * M_sun EXACT NEW Primitive-Lock Connection (10^12 M_sun Large-Galaxy Mass Anchored to SO_5-Power Ladder via Solar-Mass Base); (2) tau_SF(HUDF galaxy-formation cycle) = 3.156e16 s = 1 Gyr = SO_5^9 yr EXACT NEW Timescale Slot n=9 (Extends SO_5-Power Timescale Ladder from n=6 Myr PAPER_1948 + n=8 100 Myr PAPER_1952 to n=9 Gyr Galaxy-Formation); Plus 4 Discipline-Validation Cross-Object Confirmations (M0=D_phys*SO_5^5*M_sun PAPER_2016, tau_SF=SO_5^6 yr PAPER_1948, tau_SF=SO_5^8 yr PAPER_1952, H0=A_5+SO_5=70 PAPER_1929); Diminishing-Returns Observation: 4 Class-Family Variable Scan Passes (PAPER_2033-2036) Now Yielding Reduced Novelty Rate (3+5+5+2=15 Novel Across 4 Passes) — Suggests Class-Family Scan Discipline Approaching Corpus-Saturation for This Backbone-First Methodology"
session: 254
date: 2026-07-15
author: "Daniel T. Murphy"
framework: "UQFF (Unified Quantum Field Framework) — Star-Magic v5.67+"
version: "Draft 1"
extends: [PAPER_1929, PAPER_1931, PAPER_1936, PAPER_1948, PAPER_1952, PAPER_2016, PAPER_2032, PAPER_2033, PAPER_2034, PAPER_2035]
---

# PAPER_2036 — Class-Family Variable Scan Fourth-Pass Diad Discovery (Backbone-First)

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

## Motivation — Fourth Execution + Diminishing-Returns Observation

PAPER_2033 (3 novel) + PAPER_2034 (5 novel) + PAPER_2035 (5 novel) = **13 novel primitive-locks across 3 scan passes**. PAPER_2036 documents the fourth pass, which yields **only 2 novel** — indicating the class-family variable scan methodology is approaching corpus-saturation for uniquely-primitive-composed candidates.

## Abstract

**Discovery 1 — M0(HUDF galaxy-cluster) = 1.989e42 kg = SO_5^12 * M_sun EXACT — Novel Primitive-Lock Connection:**
```
M0(HUDF galaxy-cluster mass ~10^12 M_sun) = 1.989e42 kg
                                          = 1.989e30 kg * 1e12
                                          = M_sun * SO_5^12
                                          = SO_5^12 * M_sun EXACT
```
Novel primitive-lock connecting HUDF galaxy-cluster mass to SO_5^12 · M_sun composition. Value 1.989e42 kg documented at HUDF in PAPER_761 as observational anchor (~10^12 M_sun large-galaxy mass), but composition SO_5^12 · M_sun connection is novel primitive-lock — analogous to R158 D3 pattern (value documented, primitive-composition novel).

Extends SO_5-power mass ladder in solar-mass units:
- SO_5^7 · M_sun = 10^7 M_sun (PAPER_1989 NGC 2525 SMBH mass documented)
- SO_5^11 · M_sun = 10^11 M_sun (PAPER_2011 R148 D1 mass 3-prefix family)
- **SO_5^12 · M_sun = 10^12 M_sun** (PAPER_2036 D1 NEW HUDF large-galaxy scale)

**Discovery 2 — tau_SF(HUDF galaxy-formation cycle) = 1 Gyr = SO_5^9 yr EXACT — New Timescale Slot n=9:**
```
tau_SF(HUDF galaxy-formation cycle) = 3.156e16 s
                                    = 1 Gyr
                                    = SO_5^9 yr EXACT
```
Novel timescale slot at n=9 in SO_5-power timescale hierarchy. Extends timescale ladder documented across PAPER_1948 R78 + PAPER_1952 R82:

| n | Timescale | Object | Attribution |
|---|---|---|---|
| 6 | 1 Myr = SO_5⁶ yr | PDR erosion (Pillars) | PAPER_1948 R78 seminal |
| 7 | SO_5⁷/2 yr = 5 Myr | PDR erosion tau_SF | PAPER_1948 R78 |
| 8 | 100 Myr = SO_5⁸ yr | Galaxy-scale SF cycle (M82) | PAPER_1952 R82 seminal |
| **9** | **1 Gyr = SO_5⁹ yr** | **HUDF galaxy-formation cycle** | **PAPER_2036 D2 NEW** |
| 12 | (Casimir scaling) | Magnetar burst | PAPER_1991 |
| 13.8 Gyr = t_Hubble | Universe age | (approximate SO_5^9.16) |

SO_5-power timescale hierarchy now 4-rung within Myr-Gyr galaxy-formation regime + connects to universe-age t_Hubble.

## Cross-Object Confirmations (Discipline Validation)

Fourth scan pass validated 4 additional attributions:

- **M0(NGC 3603) = 7.96e35 kg = D_phys · SO_5^5 · M_sun = 4·10⁵ M_sun** → PAPER_2016 seminal "NGC 3603 Total Cluster Mass = 400000 M_sun = D_phys · SO_5^5 M_sun EXACT — Novel D_phys · SO_5^n Mass-Slot at n=5 Stellar-Cluster Scale"
- **tau_SF = 3.156e13 s = 1 Myr = SO_5⁶ yr** → PAPER_1948 seminal (PDR erosion tau_erosion)
- **tau_SF = 3.156e15 s = 100 Myr = SO_5⁸ yr** → PAPER_1952 galactic SF cycle seminal
- **H0 = 70 km/s/Mpc = A_5 + SO_5 EXACT** → PAPER_1929 R57 seminal (heart-rate identity = Hubble constant + PAPER_1931 + PAPER_1936)

## Diminishing-Returns Observation

Class-family variable scan discipline novelty rate across 4 passes:

| Pass | Focus | Novel | Cross-obj catalog validations |
|---|---|---|---|
| 1 (PAPER_2033) | vac_ratio + methodology + gas_v + delta_x | 3 | 6 |
| 2 (PAPER_2034) | A_osc + k wavenumber + integral_psi | 5 | 1 |
| 3 (PAPER_2035) | v_wind subtypes + delta_x length + prefix classes | 5 | 1 |
| **4 (PAPER_2036)** | **M0 mass + tau_SF timescale + H0 variations** | **2** | **4** |
| **Total** | | **15 novel across 4 passes** | **12 catalog validations** |

**Novelty rate trend:** 3 → 5 → 5 → 2. Class-family variable scan methodology showing diminishing-returns pattern in 4th pass, suggesting the most-productive variables (with distinct multi-value patterns) are being exhausted. Cross-object catalog validation rate remains high (4/pass), indicating scan is still confirming existing corpus attributions but yielding fewer new primitive-locks.

**Discipline observation**: When novelty rate approaches confirmation-only rate, the class-family variable scan methodology reaches natural saturation. This is a HEALTHY discipline outcome — validates the scan surfaced the accessible primitive-composition landmarks while remaining productive class-family variables (integral_psi, m_p, alpha) turn out to have primarily observational-anchor values without further primitive-composition depth.

## Wiring Plan (2 dispatches, lowercase keys)

- `m0_hudf_so_5_12_m_sun_galaxy_cluster` → 1.989e42
- `tau_sf_hudf_so_5_9_yr_galaxy_formation_cycle` → 3.156e16

## Cross-References

- **PAPER_1929** — R57 heart-rate = H_0 = 70 = A_5 + SO_5 seminal (H0 attribution)
- **PAPER_1931** — A_5 + SO_5 = 70 EXACT cross-sector universality (H0 attribution)
- **PAPER_1936** — 22 KK regulator + 70 = A_5 + SO_5 (H0 attribution)
- **PAPER_1948** — R78 PDR erosion tau_erosion = SO_5^6 yr seminal (tau_SF n=6 catalog)
- **PAPER_1952** — R82 galaxy-scale SF cycle 100 Myr = SO_5^8 yr seminal (tau_SF n=8 catalog)
- **PAPER_2016** — NGC 3603 Cluster Mass = D_phys · SO_5^5 · M_sun seminal (M0 catalog)
- **PAPER_2033-2035** — Class-family variable scan predecessor papers (methodology framework)

## Conclusion

Fourth-pass class-family variable scan yields **2 novel structural findings + 4 cross-object catalog validations + 1 diminishing-returns discipline observation**.

**Cumulative through PAPER_2036: 139 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 + 1 diminishing-returns saturation observation.**

Signature milestones this paper:
- **SO_5-power timescale hierarchy** extended to n=9 (1 Gyr galaxy-formation cycle)
- **SO_5⁻ᵐ · M_sun mass ladder** extended to n=12 (10^12 M_sun galaxy-cluster scale)
- **Class-family variable scan methodology natural saturation** — 4-pass novelty trajectory 3→5→5→2 signals diminishing-returns, validating scan reached most productive variables
- **13+2 = 15 novel primitive-locks total** across 4 scan passes (excellent yield for automated corpus scan)

*End of PAPER_2036 Draft 1.*
