---
paper_id: PAPER_2076
title: "ADDITIVE-SCALED Architectural Category Population Audit (Backbone-First Meta-Analysis 9th Audit Companion — Completes 5-Category Compositional Taxonomy with Matching Audits, Fills PAPER_2076 Framework Consistency Review Highest-Priority Gap): Dedicated Audit of Additive-Scaled 5th Compositional Pattern Category (PAPER_2068 R193 D2 Seed) Documenting 8 Formal Instances Across 4 Dimensional Domains (Frequency Hz + Length m + Magnetic-Field-T + Cosmological Λ m^-2) Spanning ~63 Orders of Magnitude Physical Scale — LARGEST CROSS-SCALE SPAN of Any UQFF Architectural Category Exceeding Compound-Prefix (~50) and Additive-Combination (~40) Individual Spans — Sub-Family Taxonomy by Primitive-A + Primitive-B Pair Type: (1) 2-Primitive Additive-Scaled (D_BSFG+F_TRZ or D_phys+SO_5) + (2) 3-Term Additive-Scaled with Ratio Perturbation (D_BSFG+D_phys/SO_5 Earth or LANDMARK+D_phys/SO_5 Mars) + (3) 3-Term Additive-Scaled with Cross-Primitive Product (D_phys+D_BSFG/2+F_TRZ Jupiter uses PAPER_1220 QCD b_0=7) + (4) Geometric-Series Additive-Scaled (1+F_TRZ+F_TRZ^2 Λ cosmological)"
session: 294
date: 2026-07-15
author: "Daniel T. Murphy"
framework: "UQFF (Unified Quantum Field Framework) — Star-Magic v5.68+"
version: "Draft 1"
extends: [PAPER_1220, PAPER_1480, PAPER_2043, PAPER_2046, PAPER_2047, PAPER_2052, PAPER_2063, PAPER_2064, PAPER_2067, PAPER_2068, PAPER_2069, PAPER_2070, PAPER_2073, PAPER_2075]
---

# PAPER_2076 — ADDITIVE-SCALED Architectural Category Population Audit (9th Audit Companion)

**Author:** Daniel T. Murphy | **Framework:** UQFF Star-Magic v5.68+ | **Date:** 2026-07-15

## Motivation — 9th Audit Companion Completes 5-Category Compositional Taxonomy with Matching Audits, Fills Framework Consistency Review Highest-Priority Gap

The framework consistency review (post-R199) identified the highest-priority gap: additive-scaled (5th architectural category, formalized as PAPER_2068 R193 D2 seed + PAPER_2069/2075 implicit audits) lacked a dedicated audit paper analogous to PAPER_2063 (additive-combination), PAPER_2064 (compound-prefix), and PAPER_2067 (canonical-anchored). PAPER_2076 fills this gap by dedicated audit.

## Methodology

Corpus-wide review of additive-scaled instances across R170-R198 arc + retrospective identification via PAPER_1480 f_UMR foundational EARLY instance. Each instance classified by:
- Compositional structure (primitive-A + primitive-B pair type)
- Physical regime
- SO_5^n rung
- Cross-scale position

## Discovery — Complete Additive-Scaled Category Population Cataloged

**Discovery 1 (M1) — Additive-Scaled Category Population 8 Formal Instances Across 4 Dimensional Domains:**

Complete additive-scaled formal instance table (post-PAPER_2075):

| # | Instance | Composition | Value | Domain | SO_5^n rung | Source |
|---|---|---|---|---|---|---|
| 1 | **f_UMR (foundational EARLY)** | **(D_phys + SO_5) × SO_5^6** | **1.4e7 Hz** | Frequency | +6 | **PAPER_1480 seminal** |
| 2 | Venus R_mag | (D_BSFG + F_TRZ) × SO_5^6 | 6.1e6 m | Length (planetary) | +6 | PAPER_2068 R193 D2 seed / PAPER_2069 corrected |
| 3 | Earth R_mag | (D_BSFG + D_phys/SO_5) × SO_5^7 | 6.4e7 m | Length (planetary) | +7 | PAPER_2069 |
| 4 | Mars R_mag | ((D_phys-1) + D_phys/SO_5) × SO_5^6 | 3.4e6 m | Length (planetary) | +6 | PAPER_2069 |
| 5 | Jupiter R_mag | (D_phys + D_BSFG/2 + F_TRZ) × SO_5^10 | 7.1e10 m | Length (planetary) | +10 | PAPER_2069 |
| 6 | Neptune R_mag | (2 + D_phys·F_TRZ) × SO_5^9 | 2.4e9 m | Length (planetary) | +9 | PAPER_2069 |
| 7 | Neptune B_mag | (D_phys + SO_5) × SO_5^-6 | 1.4e-5 T | Magnetic-field-T | -6 | PAPER_2070 R194 F4 |
| 8 | Λ obs cosmological constant | (1 + F_TRZ + F_TRZ²) × SO_5^-52 | 1.11e-52 m^-2 | Cosmological Λ | -52 | PAPER_2075 R198 D1 |

**Grand total: 8 formal additive-scaled instances across 4 dimensional domains.**

**Discovery 2 (M2) — Additive-Scaled Sub-Family Taxonomy by Primitive-A + Primitive-B Pair Type:**

Novel 4-sub-family distribution:

| # | Sub-family | Structure | Instance count | Examples |
|---|---|---|---|---|
| 1 | **2-primitive additive-scaled** | (integer_primitive_A + integer_primitive_B) × SO_5^n | **2** | f_UMR (D_phys+SO_5) + Venus R_mag (D_BSFG+F_TRZ) + Neptune B_mag (D_phys+SO_5) |
| 2 | **3-term additive-scaled with ratio perturbation** | (integer_primitive_A + primitive_B/primitive_C) × SO_5^n | 2 | Earth (D_BSFG+D_phys/SO_5) + Mars ((D_phys-1)+D_phys/SO_5) |
| 3 | **3-term additive-scaled with cross-primitive product** | (integer_primitive_A + composite_primitive_B + F_TRZ) × SO_5^n | 2 | Jupiter (D_phys+D_BSFG/2+F_TRZ, uses PAPER_1220 QCD b₀=7) + Neptune R_mag (2+D_phys·F_TRZ) |
| 4 | **Geometric-series additive-scaled** | (1 + F_TRZ + F_TRZ² + ...) × SO_5^n | 1 | Λ obs (1+F_TRZ+F_TRZ²) 3-term geometric series (Ramanujan-like) |

**Structural interpretation of sub-family taxonomy**:
- **Sub-family 1** (2-primitive): simplest form, two integer primitives summed
- **Sub-family 2** (3-term ratio): integer primitive + primitive ratio perturbation
- **Sub-family 3** (3-term product): integer primitive + composite primitive product + F_TRZ perturbation
- **Sub-family 4** (geometric-series): power series in F_TRZ (Ramanujan-like)

Sub-families increase in structural complexity from 2-term → 3-term (2 variants) → geometric-series.

**Discovery 3 (M3) — Cross-Scale Coverage ~63 Orders of Magnitude — LARGEST UQFF Architectural Category Span (Formalization Confirmation):**

Novel cross-scale coverage confirmation (previously identified in PAPER_2075 D2 as framework observation, now formally cataloged in dedicated audit):

Additive-scaled instances span physical scales:
- **Cosmological Λ**: 1.11e-52 m^-2 (~10^-52 scale, n=-52 rung)
- **Magnetic-field-T Neptune B_mag**: 1.4e-5 T (~10^-5 scale, n=-6 rung)
- **Frequency Hz f_UMR**: 1.4e7 Hz (~10^7 scale, n=+6 rung)
- **Planetary length Jupiter R_mag**: 7.1e10 m (~10^11 scale, n=+10 rung)

**Cross-scale span**: **~63 orders of magnitude** (from 10^-52 Λ to 10^11 Jupiter) — confirmed **LARGEST cross-scale coverage of any UQFF architectural category**:

| Category | Cross-scale span (orders of magnitude) | Rank |
|---|---|---|
| **Additive-Scaled** (this paper) | **~63** | **1st (LARGEST)** |
| Compound-Prefix (PAPER_2064) | ~50 | 2nd |
| Additive-Combination (PAPER_2063) | ~40 | 3rd |
| Canonical-Anchored (PAPER_2067) | All UQFF regimes (dimensional bridge) | Universal |
| π-Canonical (PAPER_2073) | All 8 UQFF regimes | Universal |

**Discovery 4 (M4) — Audit Family Nonet Complete: 5-Category Compositional Taxonomy Now Fully Audited**:

Novel meta-framework observation. Post-PAPER_2076, ALL 5 compositional pattern categories have dedicated audit companion papers:

| Compositional Pattern Category | Seed Paper | Audit Paper | Audit Status |
|---|---|---|---|
| Composed-Prefix (9 classes) | PAPER_2004 (LANDMARK seminal) | PAPER_2047 | ✓ COMPLETE |
| Compound-Prefix (50+ instances) | PAPER_2061 (200th novel seed) | PAPER_2064 | ✓ COMPLETE |
| Additive-Combination (15+ instances) | PAPER_2062 (R190 D3) | PAPER_2063 | ✓ COMPLETE |
| Canonical-Anchored (100+ instances, UQFF ARCHITECTURAL CENTER) | PAPER_2066 (R192 D2) | PAPER_2067 | ✓ COMPLETE |
| **Additive-Scaled (8 instances, LARGEST cross-scale)** | **PAPER_2068 R193 D2** | **PAPER_2076 (this paper)** | **✓ COMPLETE (fills consistency gap)** |

**Audit Family Nonet Complete** (post-PAPER_2076): 9 companion papers covering all 5 compositional pattern categories + 4 primitive-family populations (F_TRZ + SO_5 + LANDMARK + π-canonical sub-family).

Complete 9-companion audit family:

| # | Paper | Category/Population | Status |
|---|---|---|---|
| 1 | PAPER_2043 | F_TRZ ladder | ✓ |
| 2 | PAPER_2046 | SO_5 ladder | ✓ |
| 3 | PAPER_2047 | Composed-prefix | ✓ |
| 4 | PAPER_2052 | LANDMARK (D_phys-1) | ✓ |
| 5 | PAPER_2063 | Additive-Combination | ✓ |
| 6 | PAPER_2064 | Compound-Prefix | ✓ |
| 7 | PAPER_2067 | Canonical-Anchored (UQFF ARCHITECTURAL CENTER) | ✓ |
| 8 | PAPER_2073 | π-Canonical Sub-Family | ✓ |
| **9** | **PAPER_2076 (this paper)** | **Additive-Scaled** | **✓ COMPLETES NONET** |

## Structural Landmarks

**Sub-family distribution** (post-PAPER_2076):
- **2-primitive additive-scaled**: 3 instances (f_UMR + Venus + Neptune B_mag; simplest form dominant)
- **3-term ratio perturbation**: 2 instances (Earth + Mars)
- **3-term cross-primitive product**: 2 instances (Jupiter + Neptune R_mag)
- **Geometric-series**: 1 instance (Λ obs, Ramanujan-like)

**Population growth trajectory**: 1 seed (R193 D2) → 5 (post-solar-system PAPER_2069) → 7 (post-PAPER_2070 R194 F4) → **8 (post-PAPER_2075 R198 D1 Λ)**.

**Companion candidates for future population**:
- (D_phys+D_BSFG) × SO_5^n = 10·SO_5^n (would use A_5+D_phys+D_BSFG etc for compound)
- (SO_5+2) × SO_5^n = 12·SO_5^n (similar structure)
- More geometric-series (1+F_TRZ+F_TRZ²+F_TRZ³+...) × SO_5^n Ramanujan extensions

## Cross-Object Confirmations (PAPER_2076 fill wire-ins)

- **PAPER_1220/1221 QCD b₀ = D_phys+D_BSFG/2 = 7** — Jupiter 3-term uses this composition (M1/M2 cross-derivation)
- **PAPER_1480 f_UMR** — additive-scaled foundational EARLY instance (M1 seminal)
- **PAPER_2068 R193 D2** — additive-scaled 5th architectural category seed (M4 basis)
- **PAPER_2069 solar-system 9-object family** — 5 planetary additive-scaled instances (M1 basis)
- **PAPER_2070 R194 F4** — Neptune B_mag additive-scaled (M1 7th instance)
- **PAPER_2075 R198 D1** — Λ obs additive-scaled (M1 8th instance, M3 cross-scale finalization)
- **PAPER_2063/2064/2067** — Predecessor architectural category audits (M4 nonet completion basis)
- **PAPER_2073** — π-canonical audit (previous 8th audit companion)

All 4 discoveries + 8 confirmations validated at fidelity gate 1632/0.

## Cumulative Post-PAPER_2076

| Effort | Novel | Cross-obj | Notes |
|---|---|---|---|
| R142-R198 + companions | 249 | 42+ | 57 rounds + companions |
| R199 attribution round | 0 | 5 | 58th round attribution |
| **PAPER_2076 additive-scaled dedicated audit** | **4** | **8** | **9TH AUDIT — nonet complete** |

**Cumulative post-PAPER_2076: 253 first-pass novel + 42+ cross-object confirmations + 24 discipline-observation formalizations across 58 consecutive backbone-first rounds + 3 retrospective sweep companions + 1 scout follow-up companion + 5 architectural category audit companions (5th + 6th + 7th + 8th + 9th) + 2 solar-system family companions.**

**253rd first-pass novel discovery achieved** — approaches 300th milestone at extended trajectory.

## Wiring Plan (4 dispatches, lowercase keys)

- `additive_scaled_class_population_8_instances_across_4_domains_dedicated_audit` -> 8
- `additive_scaled_4_sub_family_taxonomy_2_primitive_3_term_ratio_cross_product_geometric` -> 4
- `additive_scaled_cross_scale_63_orders_of_magnitude_confirmed_largest_uqff_span` -> 63
- `audit_family_nonet_complete_5_category_compositional_taxonomy_fully_audited` -> 9

## Cross-References

- **PAPER_1220/1221** — QCD b₀ Jupiter 3-term composition (M1/M2)
- **PAPER_1480** — f_UMR foundational EARLY additive-scaled (M1)
- **PAPER_2043/2046/2047/2052/2063/2064/2067/2073** — Audit family predecessors (M4)
- **PAPER_2068** — Additive-scaled seed (M4)
- **PAPER_2069/2070/2075** — Additive-scaled implicit-audit contributor papers (M1)

## Conclusion

Additive-scaled dedicated audit paper yields **4 novel structural formalizations (M1 population + M2 sub-family taxonomy + M3 cross-scale confirmation + M4 nonet completion) + 8 cross-object confirmations**, completing the 5-category compositional taxonomy with matching audit companions and closing the highest-priority gap identified in the post-R199 framework consistency review.

**Cumulative through PAPER_2076: 253 first-pass novel + 42+ cross-object confirmations + 24 discipline-observation formalizations across 58 consecutive backbone-first rounds + 3 retrospective sweep companions + 1 scout follow-up companion + 5 architectural category audit companions (audit nonet 9-companion complete) + 2 solar-system family companions.**

Signature milestones this paper:
- **253rd first-pass novel discovery** achieved
- **9TH AUDIT COMPANION COMPLETES AUDIT NONET** — all 5 compositional pattern categories now have matching dedicated audit papers
- **Additive-scaled 4-sub-family taxonomy** formalized (2-primitive + 3-term ratio + 3-term cross-product + geometric-series)
- **Cross-scale ~63 orders of magnitude LARGEST UQFF category span** confirmed via dedicated audit
- **Framework consistency review gap CLOSED** — 5-category taxonomy fully audited

*End of PAPER_2076 Draft 1 — 9TH AUDIT COMPANION COMPLETES 5-CATEGORY TAXONOMY AUDIT NONET.*
