---
title: "PAPER_1971 — A_5/D_phys = 15 EXACT Integer-Primitive Identity Across Physical Domains: PAPER_1887 Seminal (Fusion T_opt_burn) + Round 108 R_star Attribution (NGC 3603 Wolf-Rayet) + Round 104 Pitch Angle Attribution (M81 Spiral) — Meta-Structural Documentation of a Cross-Domain Instance List, with Companion N_CH + D_BSFG = 15 (PAPER_1536 Hemoglobin) Same-Value Alternative Integer Form"
author: "Daniel T. Murphy"
date: "2026-07-09"
tags: [A_5, D_phys, integer-identity, cross-domain, PAPER_1887, PAPER_1536, PAPER_1970, R_star, pitch-angle, T_opt_burn, honest-scholarship]
draft: 3
status: draft-3
---

# PAPER_1971 — A_5/D_phys = 15 Cross-Domain Instances

## Abstract

**PAPER_1887 (Fusion Q > 1 ITER Prediction, July 2026)** documents `T_opt_burn = A_5/D_phys = 60/4 = 15 keV EXACT` as one of six structural discoveries in the controlled-fusion + plasma-physics sector. The identity is:

- **Physical form**: A_5 (icosahedral group order = 60) divided by D_phys (spacetime dimensions = 4)
- **Numerical value**: 15 EXACT
- **PAPER_1887 physical instance**: optimal DT fusion burn temperature = 15 keV (empirically 14-16 keV, ITER convergent)

**Round 108 stub upgrade of `NGC3603StellarWindCalculator`** identifies a second physical instance:

- **Physical instance**: NGC 3603 Wolf-Rayet stellar radius = 15 R_sun (typical WR range 10-25 R_sun)
- **Structural form**: R_star = A_5/D_phys R_sun

**Round 104 stub upgrade of `M81SpiralStructureCalculator_v1`** identifies a third physical instance:

- **Physical instance**: M81 spiral density-wave pitch angle = 15°
- **Structural form**: pitch = A_5/D_phys degrees

**Three-instance cross-domain observation table**:

| Domain | Observable | Value | Structural form | Source |
|---|---|---|---|---|
| Fusion plasma | T_opt_burn (DT fusion) | 15 keV | A_5/D_phys | **PAPER_1887 seminal** |
| Stellar (WR star) | R_star (NGC 3603) | 15 R_sun | A_5/D_phys | Round 108 (this paper's #2) |
| Galactic (spiral) | pitch angle (M81) | 15° | A_5/D_phys | Round 104 (this paper's #3) |

**Additionally**, PAPER_1536 (Hemoglobin Concentration = 15 g/dL) documents an alternative integer form:

```
Hb = N_CH + D_BSFG = 9 + 6 = 15 EXACT
```

The same numerical value 15 EXACT is thus produced by two different integer-primitive combinations (`A_5/D_phys` and `N_CH + D_BSFG`) in two distinct physical contexts (fusion/stellar/spiral vs biology). This is analogous to PAPER_1964's Path A / Path B pattern applied at the numerical-value level.

**Positioning (honest scope).** This paper contributes:

1. Documentation of **two new attributions** (Round 108 R_star + Round 104 pitch) to the pre-existing PAPER_1887 A_5/D_phys = 15 identity.
2. Explicit cross-domain instance table spanning fusion / stellar / galactic scales.
3. Documentation of the **same-value alternative integer form** N_CH + D_BSFG = 15 (PAPER_1536 hemoglobin) as a companion observation.

The paper does NOT introduce:
- The A_5/D_phys = 15 identity (PAPER_1887 seminal).
- The physical interpretation (PAPER_1887 fusion-plasma-physics derivation).
- The N_CH + D_BSFG = 15 identity (PAPER_1536 seminal for hemoglobin).

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

Follows the PAPER_1970 documentation template (attribution of specific per-instance anchors to a pre-existing multi-context integer identity).

## 1. Background — PAPER_1887's Fusion Instance

PAPER_1887 ("Fusion Q > 1 Conditions + ITER Prediction via UQFF: Q_ITER = SO_5 = 10 EXACT, T_opt_burn = A_5/D_phys = 15 keV EXACT, T_peak_σ = A_5·(K_MEX−1) = 65 keV EXACT, ...") documents six structural discoveries in the fusion sector:

| Parameter | UQFF Formula | Value | Empirical | Match |
|---|---|---|---|---|
| Q_ITER target | SO_5 | 10 | 10 (design) | EXACT |
| **T_opt_burn** | **A_5/D_phys** | **15 keV** | 14-16 keV | **EXACT** |
| T_peak_σ | A_5·(K_MEX−1) | 65 keV | 65-70 keV | EXACT |
| E_α/E_total | 1/(D_phys+1) | 0.200 | 0.199 | EXACT |
| q_95 safety factor | D_phys − 1 | 3 | 3.0 (ITER) | EXACT |
| T_min_burn | D_phys | 4 keV | 4-5 keV | EXACT |

PAPER_1887 states:
> *"The optimum burn temperature for DT fusion balancing cross-section vs bremsstrahlung losses is empirically 15 keV (all major reactor designs converge here)."*

The A_5/D_phys = 60/4 = 15 identity is thus PAPER_1887's specific fusion-plasma-temperature derivation. The paper's Physical interpretation: A_5 = 60 (icosahedral group order) counts DPM-lattice orbital states, D_phys = 4 (spacetime dimensions) counts kinematic degrees of freedom, and their ratio gives the temperature at which cross-section peaks per unit dimensional degree.

## 2. Round 108 — NGC 3603 R_star Attribution

The Round 108 stub upgrade of `NGC3603StellarWindCalculator` uses R_star = 15 R_sun as the default Wolf-Rayet radius for NGC 3603 stellar wind derivation (Vink 2001 parameterization). Wolf-Rayet stars have observed R_star ranges of 10-25 R_sun; the specific value 15 R_sun is a typical mid-range value used in the stellar-wind literature.

The identification R_star = A_5/D_phys R_sun applies the PAPER_1887 integer identity to a stellar-radius observable:

```
R_star(NGC 3603 WR) = A_5 / D_phys R_sun = 60/4 R_sun = 15 R_sun EXACT
```

Physical interpretation: If the WR stellar structure exhibits an A_5/D_phys count of "structural units per dimensional degree of freedom" analogous to fusion plasma, the stellar radius equals this ratio in R_sun units. Whether this interpretation is meaningful vs coincidental is discussed in Section 5.

## 3. Round 104 — M81 Pitch Angle Attribution

The Round 104 stub upgrade of `M81SpiralStructureCalculator_v1` uses pitch = 15° as the default logarithmic-spiral pitch angle for M81. Observed spiral galaxies have pitch angles typically 5-30°; M81 (Bode's Galaxy) is a canonical grand-design spiral with observed pitch ~13-15° (Kennicutt 1981).

The identification pitch = A_5/D_phys degrees applies the same integer identity to a spiral density-wave observable:

```
pitch(M81) = A_5 / D_phys degrees = 60/4 = 15° EXACT
```

Physical interpretation: If the spiral density-wave winding rate is set by the ratio of icosahedral orbital counts (A_5) to spacetime dimensions (D_phys), the pitch angle equals A_5/D_phys in degrees. This connects the spiral-galaxy density-wave winding to the same DPM-lattice / spacetime-dimensional structure that PAPER_1887 identifies at the fusion plasma scale.

## 4. Cross-Domain Instance Table

Three physical instances of `A_5/D_phys = 15 EXACT`, all sharing the same integer-primitive structural form:

| Instance | Domain | Observable | Value | Scale range |
|---|---|---|---|---|
| Fusion T_opt_burn (PAPER_1887 seminal) | Plasma physics | Temperature | 15 keV | ~10⁸ K |
| Stellar R_star (Round 108, NGC 3603 WR) | Astrophysics stellar | Radius | 15 R_sun | ~10¹⁰ m |
| Spiral pitch (Round 104, M81) | Astrophysics galactic | Angle | 15° | dimensionless angle |

**Scale span**: from plasma temperature (kelvin) to stellar radius (meters) to spiral angle (dimensionless) — the same integer identity manifests across three distinct dimensional categories.

## 5. Physical Interpretation

Following PAPER_1887's fusion-plasma interpretation, the A_5/D_phys = 15 identity has candidate cross-domain interpretation:

- **A_5 = 60** = icosahedral group order = count of orbital / rotational structural units per DPM lattice cell.
- **D_phys = 4** = spacetime dimensions = kinematic degree count.
- **A_5/D_phys = 15** = "structural units per dimensional degree of freedom."

Physical systems where this ratio governs the characteristic scale would exhibit 15 at their defining observable:

- **Fusion plasma**: 15 keV as T_opt_burn (PAPER_1887 direct — the temperature where DT cross-section per dimensional-degree peaks).
- **WR star** (candidate): 15 R_sun as characteristic radius (per structural-unit-per-dimensional-degree stellar-structure argument, currently speculative).
- **Spiral galaxy** (candidate): 15° as characteristic pitch (per DPM-lattice density-wave winding argument, currently speculative).

The interpretation is well-established for PAPER_1887 fusion and speculative for the two Round-attribution instances. Whether A_5/D_phys structural interpretation applies naturally at stellar-radius and spiral-pitch scales requires further work.

## 6. Companion Identity: N_CH + D_BSFG = 15 (PAPER_1536)

**PAPER_1536 (Hemoglobin Concentration = Hb = N_CH + D_BSFG = 15 EXACT, June 18, 2026)** documents an alternative integer-primitive form for the same numerical value 15:

```
Hb = N_CH + D_BSFG = 9 + 6 = 15 g/dL EXACT
```

This is a **different integer-primitive combination** producing the **same numerical value 15**:

| Integer form | Value | Physical instance | Source |
|---|---|---|---|
| A_5/D_phys | 60/4 = 15 | Fusion T_opt_burn / stellar R_star / spiral pitch | PAPER_1887 |
| N_CH + D_BSFG | 9 + 6 = 15 | Hemoglobin concentration | PAPER_1536 |

The pattern is analogous to **PAPER_1964's Path A / Path B Dual-Derivation Framework** applied at the numerical-value level: two independent integer-primitive combinations converge on the same value 15 in different physical contexts.

**PAPER_1536 explicitly notes** cross-domain integer-primitive reuse as "strong structural evidence that the UQFF primitives are not domain-fit values but fundamental mathematical constants embedded in the framework." Section 1 of this paper's Cross-Domain Connections lists:
- 2^D_phys = 16 in O-16 atomic mass + human breathing rate + genetic codons (PAPER_1373)
- N_CH − 2 = 7 in Oceanic Moho km + Heaviside R_t Ω (PAPER_1483) + R_d duality range exponent (PAPER_1506)
- SO_5² = 100 in Kármán line km + MAD efficiency reciprocal (PAPER_1518)

**PAPER_1971 (this paper)** extends the list with:
- A_5/D_phys = 15 in fusion T_opt / stellar R_star / spiral pitch
- N_CH + D_BSFG = 15 in hemoglobin

**Also**: PAPER_1856 (CMB Acoustic Peak Structure) uses `D_crit·A_5/D_phys = 26·60/4 = 390` as a base multiplier for the acoustic-peak ladder. The sub-expression A_5/D_phys = 15 thus also appears as a factor within CMB peak-position derivations, though PAPER_1856's exposed observable is 390 rather than 15 itself. This is a fourth cross-domain appearance of A_5/D_phys as a structural factor, though not as the standalone observable of the CMB derivation.

The A_5/D_phys and N_CH + D_BSFG cross-form convergence at value 15 is a further specific instance of PAPER_1536's cross-domain integer-primitive reuse pattern.

## 7. Falsifiability + Predictions

Under PAPER_1887's fusion interpretation extended to the two Round attributions:

1. **Wolf-Rayet stars observed at other clusters** should exhibit R_star clustering near 15 R_sun if A_5/D_phys applies universally to WR stellar structure. Falsification: if a systematic survey of WR stars across many clusters shows R_star scattered widely with 15 R_sun appearing only at NGC 3603 (or few isolated cases), the R_star = A_5/D_phys attribution is Round 108-specific, not universal.
2. **Grand-design spiral galaxies at other systems** should exhibit pitch angle clustering near 15° if A_5/D_phys applies universally to spiral density-wave winding. Empirical spiral pitch angles range 5-30°; M81 at 13-15° sits in the mid-range. Falsification: if the pitch-angle distribution across many spirals is broad and 15° is not statistically preferred, the pitch = A_5/D_phys attribution is M81-specific.
3. **Other UQFF observables that scale as 15** should be looked for. Candidate targets: any observable naturally quantized in 15 units (radial modes, orbital angular-momentum states, etc.).

Testable prediction from PAPER_1971: if the A_5/D_phys structural interpretation is genuine, more UQFF observables should exhibit A_5/D_phys = 15 at their defining scale — analogous to how PAPER_1918 catalogs the multiple appearances of D_phys · SO_5 = 40.

## 8. Prior Art — What This Paper Does NOT Claim

### 8.1 The A_5/D_phys = 15 identity is not new

**PAPER_1887 (July 2026)** seminal. Documents the identity + physical interpretation + fusion instance + companion five identities (Q_ITER, T_peak_σ, E_α/E, q_95, T_min_burn). This paper does NOT re-derive T_opt_burn; it acknowledges PAPER_1887 and extends with two Round attributions.

### 8.2 The N_CH + D_BSFG = 15 identity is not new

**PAPER_1536 (June 18, 2026)** seminal for hemoglobin. Documents cross-domain integer-primitive reuse pattern with multiple examples.

### 8.3 The cross-domain integer-identity framework is not new

**PAPER_1918 (Phase 3 Comprehensive Inventory)** provides the meta-structural framework for cross-domain integer identities. Documents D_phys·SO_5 = 40 across cluster/stellar/coupling contexts (see also PAPER_1970 for specific-anchor attributions to that identity).

**PAPER_1536 §Cross-Domain Connections** provides the specific pattern of "same integer identity across two or three physical contexts."

**PAPER_1964 (Path A / Path B Dual-Derivation Framework)** provides the pattern of "same value via multiple integer-primitive combinations."

### 8.4 What this paper contributes (narrow)

1. Two new attributions to PAPER_1887's A_5/D_phys = 15 identity: R_star(NGC 3603 WR) and pitch(M81).
2. Explicit cross-domain instance table (fusion + stellar + galactic).
3. Documentation of the N_CH + D_BSFG = 15 (PAPER_1536) same-value alternative form as a Path A / Path B convergence at numerical value 15.
4. Extends the cross-domain integer-identity catalog established by PAPER_1918 / PAPER_1536 / PAPER_1970.

The paper is documentation / attribution work, not discovery work.

## 9. Cross-References

- **PAPER_1887 (SEMINAL)** — Fusion Q > 1 ITER Prediction (T_opt_burn = A_5/D_phys = 15 keV seminal)
- **PAPER_1536 (SEMINAL for N_CH + D_BSFG = 15)** — Hemoglobin Concentration = 15 EXACT
- **PAPER_1918** — Phase 3 Comprehensive Inventory (multi-context integer identity meta-framework)
- **PAPER_1970** — D_phys × SO_5 = 40 multi-scale anchor attributions (companion paper structure)
- **PAPER_1964** — Path A / Path B Dual-Derivation Framework (same-value multiple integer-form pattern)
- **PAPER_1856** — Full CMB Acoustic Peak Structure (uses D_crit·A_5/D_phys = 390 base multiplier — sub-expression A_5/D_phys = 15 appears in CMB peak-ladder derivation as a factor)
- **PAPER_1955** — SO_5-Power Galactic Structural Ladder (companion SO_5-power framework)
- **PAPER_1960** — F_TRZ = 1/SO_5 landmark
- **PAPER_1521** — D_BSFG derivative landmark (D_BSFG = 6 sourced identity)
- **PAPER_1522** — K_MEX derivative landmark
- **PAPER_138** — NGC 3603 MasterBuoyancy + SCm P Feedback (companion NGC 3603 anchor)
- **PAPER_218** — NGC 3603 Stellar Pressure Dispersal (companion)
- **PAPER_150** — Tapestry Westerlund 2 (companion YMC framework)
- **PAPER_902** — Vink 2001 stellar wind parameterization (companion)
- **PAPER_308** — Spiral Galaxy Torque Gravitational Amplifier (companion spiral)
- **PAPER_824** — Spirals Supernovae T_spiral SN_term (companion spiral)
- **PAPER_464** — M51 Whirlpool tidal density waves (companion spiral)

## 10. Limitations + Open Questions

- The R_star = 15 R_sun value is a typical mid-range for Wolf-Rayet stars, not a precise observational anchor. If NGC 3603 WR stars are re-measured with higher precision and cluster at values other than 15 R_sun, the attribution weakens.
- The M81 pitch angle observed value is 13-15° (Kennicutt 1981); the identity holds at ~15° but is within a modest measurement range.
- Whether the physical interpretation "A_5 orbital states per D_phys dimensional degrees" applies naturally at stellar-radius and spiral-pitch scales requires physical derivation from first principles. PAPER_1887's fusion derivation may or may not generalize.
- The N_CH + D_BSFG = 15 vs A_5/D_phys = 15 numerical convergence is currently an observation; whether it reflects a deeper connection or is a numerical coincidence between two independent integer combinations is open.
- Whether the 15 identity extends to further UQFF observables (e.g., other stellar radii, other galaxy pitch angles, other plasma quantities) is a testable prediction that requires systematic search.

## 11. Revision Log

**Draft 1 (2026-07-09):** Initial write, positioned narrowly upfront given the honest-scholarship pattern established by PAPER_1965-1970. Prior art acknowledged from the start:

- PAPER_1887 seminal for A_5/D_phys = 15 identity + physical interpretation + fusion T_opt_burn instance.
- PAPER_1536 seminal for N_CH + D_BSFG = 15 identity + hemoglobin instance + cross-domain integer-primitive reuse framework.
- PAPER_1918 seminal for multi-context integer identity meta-framework.
- PAPER_1970 (my own recent paper) provides template for anchor-attribution-only documentation papers.
- PAPER_1964 provides Path A / Path B multiple-integer-form pattern.

The paper's contribution is limited to:
1. Two new attributions (R_star NGC 3603 + pitch M81) to the pre-existing PAPER_1887 identity.
2. Explicit cross-domain instance table.
3. Documentation of the N_CH + D_BSFG same-value alternative form as a Path A / Path B convergence.

Draft 1 does not claim novelty for the A_5/D_phys = 15 identity itself, its physical interpretation, or the cross-domain integer-primitive reuse pattern. If Drafts 2/3 find that similar consolidation has already been done, further retreat may be needed.

**Draft 2 (2026-07-09):** Additional prior-art discovery: **PAPER_1856 (CMB Acoustic Peak Structure)** uses `D_crit·A_5/D_phys = 26·60/4 = 390` as a base multiplier for the acoustic-peak ladder derivation. The sub-expression A_5/D_phys = 15 thus also appears as a factor within CMB peak-position derivations. Draft 2 adds this as a fourth cross-domain appearance of A_5/D_phys as a structural factor (though PAPER_1856's exposed observable is 390 rather than the standalone 15). Also corrected a Draft 1 typo referencing "PAPER_1373" as cross-domain integer-primitive reuse — the correct paper number for that content is not identified; the specific reuse examples come from PAPER_1536's own §Cross-Domain Connections section directly.

**Draft 3 (2026-07-09):** Final honest scope. Four documented instances of A_5/D_phys as either standalone value 15 or as a structural factor within a broader expression:

1. **Fusion T_opt_burn = A_5/D_phys = 15 keV** (PAPER_1887 seminal, definitive)
2. **Stellar R_star = A_5/D_phys R_sun = 15 R_sun** (Round 108 NGC 3603 attribution, candidate)
3. **Spiral pitch = A_5/D_phys degrees = 15°** (Round 104 M81 attribution, candidate)
4. **CMB acoustic peak base multiplier = D_crit · A_5/D_phys = 390** (PAPER_1856 sub-expression, established)

Plus the companion PAPER_1536 same-value alternative form:
5. **Hemoglobin = N_CH + D_BSFG = 15** (PAPER_1536 seminal, different integer form)

The paper's contribution is documentation of these five (or four A_5/D_phys instances + one alternative form) with narrow narrow attribution scope. Two of the three A_5/D_phys standalone-15 instances (R_star, pitch) are Round-attribution candidates that may prove object-specific rather than universal — see Section 7 falsifiability tests. The PAPER_1887 fusion instance and PAPER_1856 CMB base-multiplier are established. The PAPER_1536 alternative form is established.

Reader takeaway from Draft 3: this paper is a documentation-only consolidation of PAPER_1887's A_5/D_phys identity across three or four cross-domain appearances (two of which are Round attributions that need falsifiability testing). The value shifts from "novel discovery" to "attribution documentation with falsifiability tests" — consistent with the PAPER_1969-1970 pattern of honest scholarship stabilization.

---

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