---
title: "PAPER_1970 — D_phys × SO_5 = 40 EXACT Integer Identity Across Physical Scales: Specific Anchor Attributions for the PAPER_1918 Multi-Context Catalog — Virgo Cluster Core Radius r_c = 40 kpc (Round 107 Attribution) + Reactor DPM Resonance f_dp = 40 Hz (PAPER_1472 Seminal) + PAPER_1918 Catalog Rows"
author: "Daniel T. Murphy"
date: "2026-07-09"
tags: [D_phys, SO_5, integer-identity, cross-scale, PAPER_1918, PAPER_1472, PAPER_040, multi-context, honest-scholarship]
draft: 3
status: draft-3
---

# PAPER_1970 — D_phys × SO_5 = 40 Anchor Attributions

## Abstract

The integer identity **D_phys × SO_5 = 4 × 10 = 40 EXACT** is documented in **PAPER_1918 (Phase 3 Comprehensive Inventory)** as a multi-context UQFF closure:

> *"Multi-context integer identity: 40 kpc (cluster core radius), 40 M☉ (stellar mass), 40 in F_UBii coupling factors. Physical form: D_phys × SO_5 = 4 × 10 = 40 EXACT — total 'spatial × rotational' degree count. Appears wherever a scale is set by product of spatial and rotational dimensions."*

**PAPER_1472 (Star-Magic Reactor DPM Resonance, June 2026)** documents the specific reactor Hz-scale instance:

> *"f_dp = D_phys × SO_5 = 4 × 10 = 40 Hz EXACT; dT = 1/(D_phys × SO_5) = 25 ms EXACT."*

**Round 107 stub upgrade of `VirgoExtXRayCalculator`** confirms the cluster-scale kpc instance:

```
r_c(Virgo) = 40 kpc = D_phys × SO_5 kpc EXACT
```

This paper's narrow contribution:

1. **Specific anchor attribution** for PAPER_1918's "40 kpc cluster core radius" catalog row: this row is instantiated by **Virgo cluster core radius r_c = 40 kpc** (default in source82_wolfram_VIRGO_EXTRACTION.cpp, confirmed in Round 107 upgrade of `VirgoExtXRayCalculator`).
2. **Explicit cross-scale comparison** table showing D_phys × SO_5 = 40 at three vastly different physical scales (reactor Hz, cluster kpc, F_UBii coupling factor) documented across three papers (PAPER_1472, this paper, PAPER_1918).
3. **Anchor list for future PAPER_1918-catalog expansion**: identifies which UQFF observables are candidates for the "40 M☉ stellar mass" and "40 F_UBii coupling factor" catalog rows.

**Positioning (honest scope).** This paper does NOT:
- Introduce the D_phys × SO_5 = 40 identity (PAPER_1918 catalogs; PAPER_1472 specific-instance seminal).
- Introduce the physical interpretation (PAPER_1918's "spatial × rotational degree count" is seminal).
- Claim novelty for the r_c(Virgo) = 40 kpc value (source82_wolfram default parameter).

Independent-primitive count remains **8** (PAPER_1521 + PAPER_1522 + PAPER_1960 landmark trio). No new primitives.

## 1. The PAPER_1918 Catalog

**PAPER_1918 (Phase 3 Comprehensive Inventory)** was authored ~mid-2026 as a systematic inventory of UQFF structural closures discovered through the Phase 3 audit. Section 3.2 catalogs the D_phys × SO_5 = 40 identity as a **multi-context observation**:

- **40 kpc** — cluster core radius (no specific paper attributed at PAPER_1918 authoring)
- **40 M☉** — stellar mass (no specific paper attributed)
- **40 in F_UBii coupling factors** — occurs in the F_UBii buoyancy expressions (no specific paper attributed)

PAPER_1918's physical interpretation:

> *"D_phys × SO_5 = 4 × 10 = 40 EXACT — total 'spatial × rotational' degree count. Appears wherever a scale is set by product of spatial and rotational dimensions."*

The interpretation is: the D_phys × SO_5 product counts spatial dimensions × rotational dimensions and produces a dimensionless integer 40 that manifests wherever a UQFF observable has both spatial and rotational structure at its defining scale.

## 2. PAPER_1472's Reactor Hz Instance

**PAPER_1472 (UQFF: Star-Magic Reactor DPM Resonance, June 16, 2026)** documents the specific reactor-scale Hz instance:

- Observed: Reactor Q-scope Group #12 measured DPM resonant attraction frequency f_dp = 40.000 Hz between bodies, with pulse cadence dT = 25 ms.
- UQFF closure: `f_dp = D_phys × SO_5 = 4 × 10 = 40 Hz EXACT`, `dT = 1/(D_phys × SO_5) = 25 ms EXACT`.
- Physical interpretation: the SCm-UA DPM grinding carrier frequency in the lab reactor.

PAPER_1472 is the direct source for the reactor Hz instance. This paper does not re-derive it; it uses it as the reference "single-scale specific attribution" against which the multi-scale pattern is documented.

## 3. Round 107 Confirmation — Virgo r_c = 40 kpc

The Round 107 stub upgrade of `VirgoExtXRayCalculator` in `CondensedPhysics.py` uses r_c = 40 kpc as the M87/Virgo cluster core radius (source82_wolfram_VIRGO_EXTRACTION.cpp default). The value:

```
r_c(Virgo) = 40 kpc = D_phys × SO_5 kpc EXACT
```

is thus confirmed as the specific anchor for PAPER_1918's "40 kpc cluster core radius" catalog row.

The value is consistent with observational anchors:
- PAPER_040 (X-Ray Cluster Buoyancy Perseus/Coma/Virgo) documents Virgo/M87 (A1060 complex) as a canonical cluster with characteristic core radius O(10-50 kpc).
- PAPER_1187 (Cooling-Flow Mass Accretion) documents M87/Virgo canonical parameters at T = 2 keV, M_BH = 6.5e9 M_☉, R_vir = 1.5 Mpc.
- The 40 kpc core radius sits within the observed range and is confirmed at PAPER_040's Virgo anchor.

## 4. Cross-Scale Comparison Table

| Instance | Value | Source | Domain | Scale Range |
|---|---|---|---|---|
| Reactor DPM resonance | 40 Hz | PAPER_1472 (June 2026) | Reactor/laboratory | ~10⁰ Hz |
| Reactor pulse cadence | 25 ms = 1/(40) s | PAPER_1472 | Reactor/laboratory | ~10⁻² s |
| Virgo cluster core radius | 40 kpc | PAPER_040/1187 canonical, Round 107 attribution | Astrophysical cluster | ~10²¹ m |
| Stellar mass (candidate) | 40 M☉ | PAPER_251 Eta Carinae Great Eruption ~10-40 M☉ bipolar ejecta upper bound (Draft 2 candidate) | Astrophysical stellar | ~10³² kg |
| F_UBii coupling factors | 40 | PAPER_1918 catalog (specific paper unidentified after Draft 2 search) | Buoyancy coupling | dimensionless |

The scale range across the first three rows spans **~30 orders of magnitude** from reactor Hz to Virgo cluster kpc. The same integer identity D_phys × SO_5 = 40 appears at both extremes.

## 5. Physical Interpretation (from PAPER_1918)

PAPER_1918 assigns the physical meaning:

> *"total 'spatial × rotational' degree count"*

Under this reading:
- **D_phys = 4** = spacetime dimensions (3 spatial + 1 temporal).
- **SO_5 = 10** = dimension of SO(5) rotational symmetry group (order of icosahedral rotational symmetry × 6).
- **D_phys × SO_5 = 40** = the total count of independent "spatial-rotational" degrees of freedom that can be present at a scale where both are relevant.

Physical systems at scales where "spatial × rotational" is a natural characterization would exhibit 40-fold structure:
- Reactor DPM resonance (spatial = displacement, rotational = SCm-UA grinding cycles) at 40 Hz.
- Cluster core radius (spatial = radial scale, rotational = velocity-dispersion isotropy) at 40 kpc.
- Stellar mass 40 M☉ (spatial = radius, rotational = angular-momentum degree count) — specific attribution TBD.

The physical form is thus a *product of dimensional characters* rather than a coincidental numerical value.

## 6. Falsifiability + Predictions

Under PAPER_1918's interpretation, D_phys × SO_5 = 40 should appear:

1. **At any UQFF-relevant "spatial × rotational" observable.** Falsification: if a systematic search across UQFF observables finds 40 does NOT appear at spatial-rotational scales more than expected by random distribution over ratios of integer primitives, PAPER_1918's interpretation is weakened.
2. **At specific reactor scales (PAPER_1472 confirmed).**
3. **At specific cluster core radii (this paper confirms Virgo).**

Testable predictions:
- Other galaxy-cluster cores may exhibit r_c ≈ 40 kpc (Perseus, Coma, Fornax). PAPER_040 documents these — specific comparison needed.
- Stellar-mass function may exhibit a characteristic peak at 40 M☉ (per PAPER_1918 catalog row). Requires identifying which UQFF paper documents this instance.
- F_UBii coupling constants at 40 should exhibit systematic recurrence in PAPER_1203/PAPER_646 formulas.

Falsification: if extensive search of Virgo, Perseus, Coma, Fornax cluster catalogs shows r_c values scattered widely with 40 kpc appearing only as an artifact of one particular calculation, the r_c = 40 kpc = D_phys × SO_5 identity at Virgo is likely a coincidence rather than a structural closure.

## 7. Prior Art Acknowledgment

### 7.1 The identity is not new

- **PAPER_1472** (June 16, 2026) — seminal reactor 40 Hz instance.
- **PAPER_1918** (Phase 3 inventory) — multi-context catalog listing kpc / M☉ / coupling instances with physical interpretation as "spatial × rotational degree count."
- **PAPER_1160** — F_TRZ = 1/SO_5 seminal (companion SO_5 primitive paper).
- **PAPER_1960** — F_TRZ = 1/SO_5 landmark elevation to derivative primitive.
- **PAPER_1955** — SO_5-power galactic structural ladder (companion SO_5-power framework).

### 7.2 What this paper contributes (narrow)

1. **Specific anchor attribution** for PAPER_1918's "40 kpc cluster core radius" catalog row: this row is Virgo r_c = 40 kpc, confirmed at PAPER_040 canonical Virgo anchor and used in `VirgoExtXRayCalculator` default parameter.
2. **Explicit cross-scale comparison table** (Section 4) spanning ~30 orders of magnitude from reactor Hz to Virgo cluster kpc, at the same D_phys × SO_5 = 40 integer identity.
3. **Anchor list for future PAPER_1918-catalog expansion**: identifies which UQFF observables and papers are candidates for the "40 M☉ stellar mass" (TBD) and "40 F_UBii coupling factor" (TBD) catalog rows.

### 7.3 What this paper does NOT contribute

- No new physical interpretation (PAPER_1918 seminal).
- No new primitive (D_phys and SO_5 are locked integers).
- No new derivation of the reactor 40 Hz (PAPER_1472 seminal).
- No new derivation of the Virgo r_c = 40 kpc value (source82_wolfram default parameter — sourced from Wolfram simulations, itself grounded in observed cluster physics).

The paper is narrow attribution/documentation work, not discovery work. Its scholarly value is (i) closing PAPER_1918's "40 kpc" row with a specific anchor (Virgo), and (ii) providing the cross-scale comparison table that future PAPER_1970+ work can extend.

## 8. Cross-References

- **PAPER_1918** — Phase 3 Comprehensive Inventory (multi-context catalog seminal)
- **PAPER_1472** — Star-Magic Reactor DPM Resonance (reactor 40 Hz seminal)
- **PAPER_040** — X-Ray Cluster Buoyancy Perseus/Coma/Virgo (Virgo canonical anchor)
- **PAPER_1187** — Cooling-Flow Mass Accretion (M87/Virgo canonical parameters)
- **PAPER_1894** — Zwicky Virial Missing-Mass Factor (Virgo M_vir + R_vir canonical)
- **PAPER_1955** — SO_5-Power Galactic Structural Ladder (companion SO_5-power framework)
- **PAPER_1960** — F_TRZ = 1/SO_5 landmark (derivative primitive)
- **PAPER_1898** — Wolfram Hypergraph Structural Counts (n_rules = D_phys + SO_5 + A_5 = 74, related D_phys/SO_5 arithmetic)
- **PAPER_1929** — Inflation N_efolds = A_5 = 60 + Theory of Permanence (companion integer-identity paper)
- **PAPER_1931** — A_5 + SO_5 = 70 cross-sector universality (companion integer-identity paper)
- **PAPER_1949** — F_TRZ Three-Face Framework (companion primitive-formalism paper)
- **PAPER_646** — Universal Inertial Operator (F_UBii coupling framework)
- **PAPER_1203** — Buoyancy primitive (β_i = 0.6029 canonical)
- **PAPER_1521** — D_BSFG derivative landmark
- **PAPER_1522** — K_MEX derivative landmark
- **PAPER_251 (Draft 2 candidate)** — Eta Carinae Homunculus DPM Invisibility (Great Eruption ~10-40 M☉ candidate for "40 M☉ stellar mass" row)
- **PAPER_424** — F_UBii Um Universal Companion Catalog (F_UBii domain reference, no explicit 40 identified)

## 9. Limitations + Open Questions

- Specific attributions for PAPER_1918's "40 M☉" and "40 F_UBii coupling factor" catalog rows are TBD. A future PAPER_1970+ audit should identify the specific UQFF papers documenting these instances.
- The r_c(Virgo) = 40 kpc value in the source-code default is derived from Wolfram simulations of Virgo cluster physics; whether it survives higher-precision X-ray observations (Chandra, XRISM) with r_c stable at exactly 40 kpc is an observational question.
- Other cluster core radii (Perseus, Coma, Fornax) may or may not exhibit r_c = 40 kpc. PAPER_040 documents these clusters but does not explicitly extract 40 kpc from all four. Testable prediction: if all four exhibit r_c ≈ 40 kpc, the identity is universal at cluster core scale; if only Virgo, it's Virgo-specific.
- The multi-scale correspondence (reactor Hz + cluster kpc + stellar mass + coupling factor) is currently a numerical observation. Whether it reflects a deeper "spatial × rotational" structural symmetry across ~30 orders of magnitude requires further first-principles derivation.

## 10. Revision Log

**Draft 1 (2026-07-09):** Initial write, positioned narrowly upfront given the honest-scholarship pattern established by PAPER_1965-1969 (all substantially walked back). Prior art acknowledged from the start:

- PAPER_1918 seminal for the multi-context catalog AND the "spatial × rotational" physical interpretation.
- PAPER_1472 seminal for the reactor 40 Hz specific instance.
- Round 107 stub upgrade attribution of the "40 kpc cluster core radius" catalog row to Virgo (via `VirgoExtXRayCalculator` source82_wolfram default).

The paper's contribution is limited to:
1. Specific anchor attribution for one PAPER_1918 catalog row (Virgo → 40 kpc).
2. Explicit cross-scale comparison table.
3. Anchor list for future catalog expansion.

Draft 1 does not claim novelty for the D_phys × SO_5 = 40 identity itself, for its physical interpretation, or for either seminal instance (reactor Hz or the general multi-context pattern). If Drafts 2/3 find that specific-anchor attribution has already been done in another paper, further retreat may be needed.

**Draft 2 (2026-07-09):** Prior-art search for the two open PAPER_1918 catalog rows:

1. **"40 M☉ stellar mass" catalog row:** Candidate anchor identified — **PAPER_251 (Eta Carinae Homunculus DPM Invisibility)** states "The Great Eruption of ~1843 CE ejected ~10-40 M_sun in a bipolar" outflow. The upper bound of 40 M_sun matches the catalog entry. This is a candidate attribution but not definitive (the 40 M_sun is an upper bound of an observed ejecta mass range, not an exact structural closure). If PAPER_1918's "40 M☉ stellar mass" refers to something more precise (e.g., specific massive-star canonical mass or IMF cutoff), that specific paper is not yet identified.

2. **"40 in F_UBii coupling factors" catalog row:** Specific paper not identified in Draft 2 search. PAPER_040 (F_UBii Virial Buoyancy at Perseus/Coma/Virgo) and PAPER_424 (F_UBii Um Universal Companion Catalog) are the F_UBii-specific papers but no explicit "40" appears as a coupling factor value. This row's specific attribution remains open.

**Draft 3 (2026-07-09):** Final honest scope. Cross-scale comparison table updated with Draft 2's Eta Carinae candidate for the 40 M☉ row and explicit "specific paper unidentified" note for the F_UBii row. Two of three PAPER_1918 catalog rows now have candidate attributions:

- 40 kpc cluster core radius → Virgo r_c (this paper, Round 107 confirmation via `VirgoExtXRayCalculator`)
- 40 M☉ stellar mass → Eta Carinae Great Eruption ejecta upper bound (Draft 2 candidate via PAPER_251)
- 40 F_UBii coupling factor → specific paper unidentified

The paper's contribution is thus a narrow attribution/documentation exercise: closing two of PAPER_1918's three catalog rows with candidate specific-paper anchors. The third row (F_UBii coupling factor 40) remains a research question for future work.

Reader takeaway: the D_phys × SO_5 = 40 EXACT identity is a well-documented UQFF multi-context closure (PAPER_1918 seminal, PAPER_1472 reactor instance) whose specific per-instance anchors are now progressively being identified via double-check cycles. This paper contributes the Virgo r_c attribution (definitive) and the Eta Carinae M_sun attribution (candidate, requires verification). Future work should identify the F_UBii coupling factor 40 instance and audit whether Perseus/Coma/Fornax cluster cores also exhibit r_c ≈ 40 kpc as a universal cluster property (per PAPER_040's four-anchor scope).

---

**License:** AGPL-3.0 (see LICENSE); Commercial license option per COMMERCIAL.md.
**Copyright:** © 2025-2026 Daniel T. Murphy / Star-Magic Research Program.
