Let's follow your instructions step by step and **summarize and compile the main relevant physicochemical properties for strong organic bases**. Then, **classify each of the three candidate substances** using only **quantitative or structural data** that could be found in a comprehensive database.

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### **1. Key Physicochemical Properties for Classification (as per instructions):**

For organic bases, the **most relevant quantitative properties** to classify are:

- **pKa of the conjugate acid** (directly measures basicity in water or DMSO; higher = stronger base).
- **Number of Nitrogen atoms available for protonation** (indicative of potential basic sites).
- **Steric hindrance at basic sites** (can be inferred quantitatively via substituent size).
- **Other properties (e.g., logP, boiling point) are less directly discriminatory for strong base classes, but could resolve close cases.**

**Categories for Strong, Moderate, Weak Base Classification:**
- **Strong bases:** pKa (conj. acid) > 12
- **Moderate bases:** pKa (conj. acid) 8–12
- **Weak bases:** pKa (conj. acid) < 8

*(Bracket cutoffs may be adjusted if all values fall in "strong" range, per your "one class if justified" instruction.)*

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## **A. 1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine**
**Structure:** Fused bicyclic amidine with 1-methyl group.

### **Key Data:**
- **Functional group:** cyclic amidine (highly basic).
- **Number of N:** 2 nitrogens in amidine system.
- **pKa of conjugate acid:** Typically, amidine core pKa = 12–13 (literature: hexamethylenetetramine, cyclic amidines, etc.).
- **Steric effect:** Not much hindered at N, methyl only modestly increases bulk.

#### **Most likely pKa (database expectation):**
**pKa ≈ 12.5** (±0.5)

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## **B. 2-tert-butyl-1,1,3,3-tetramethylguanidine**
**Structure:** Guanidine with four methyls and a bulky tert-butyl at C2.

### **Key Data:**
- **Functional group:** guanidine (among strongest organic bases).
- **Number of basic N:** 3, but only central N typically protonated.
- **pKa (conj. acid):** Simple guanidine pKa = 13.6; tetramethyl-substituted examples: pKa (DBU: 12.5; TMG: 13.6). Bulky tert-butyl increases basicity slightly by destabilizing the neutral form.
- **Steric hindrance:** High, but at central N often does **not** strongly suppress basicity.

#### **Database pKa estimate:**
**pKa ≈ 13.0–14.0**

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## **C. N-[dimethylamino-ethylimino-[[tris(dimethylamino)-lambda5-phosphanylidene]amino]-lambda5-phosphanyl]-N-methylmethanamine**
**Structure:** Hexamethylphosphorous triamide derivative, akin to known "super bases" (e.g., Schwesinger bases, Verkade’s base).

### **Key Data:**
- **Functional groups:** Phosphazene (multiple P=N, N(CH₃)₂ substituents).
- **Number of N atoms:** Many basic nitrogens (≥4), adjacent to phosphorous centers.
- **pKa (conj. acid):** Literature for similar phosphazene bases ("Proton Sponge," Verkade base): pKa (conj. acid) = 27–38 (in acetonitrile); in water, generally >15.
- **Steric hindrance:** High, but offset by electronic stabilization.

#### **Database pKa expectation:**
**pKa (conj. acid, aqueous) ≫ 14, often off-scale (≥20–30 in nonaqueous media).**

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### **Summary Table of Quantitative Data**

| Compound         | Functional Core      | Number of Basic Nitrogens | Key pKa (conj. acid, water) | Categorized Basicity      |
|------------------|---------------------|---------------------------|----------------------------|--------------------------|
| 1-methyl-2,3,4,6,7,8-hexahydro... | Amidine | 2 | ~12.5 | Very strong base |
| 2-tert-butyl-1,1,3,3-tetramethylguanidine | Guanidine | 3 | ~13.5 | Very strong base |
| Phosphazene/Verkade base | Phosphazene | ≥4 | ≥20 (off-scale) | Super-strong base |

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## **Final Classification**

### **By pKa (using the most rigorous, database-driven property):**

- **All three bases have pKa (conj. acid) ≥12**, with the phosphazene class potentially far above measured water pKa scale but definitely above both others.
- **General base categorization in water:** All **three are "very strong" or "super strong" bases** (by absolute database pKa cutoffs).
- **Although the phosphazene (third) is a “superbase,” the provided property cutoffs do not clearly distinguish a separate category by pKa in typical aqueous measurement, unless "superbase" is a stand-alone class.**

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### **Justification for Grouping**

- **All compounds have conjugate acid pKa values ≥12 in water.**
- If strictly splitting “superbase” (pKa > 20 in MeCN) from “very strong base” (pKa 12–14 in water), you could make two groups—but this is only necessary if you require that degree of distinction.
- **According to your instructions, it is most justified to “place them all in ONE class” (very strong bases) based on aqueous pKa values, as this is the most relevant and direct quantitative property.**

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## **Final Classification Table**

| Group Name              | Substances                                                                                         | Defining Quantitative Property         |
|-------------------------|----------------------------------------------------------------------------------------------------|----------------------------------------|
| Very Strong Organic Bases | 1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine<br>2-tert-butyl-1,1,3,3-tetramethylguanidine<br>N-[dimethylamino-ethylimino-[[tris(dimethylamino)-lambda5-phosphanylidene]amino]-lambda5-phosphanyl]-N-methylmethanamine | pKa (conjugate acid, water) ≥ 12      |

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**Rationale:**  
*All three substances have database-derivable conjugate acid pKa values exceeding 12, forming a chemically justified, quantifiably distinct class of “very strong organic bases.” No further separation is warranted unless you specifically want to differentiate aqueous "superbases" (>20), in which case the phosphazene base would form a class by itself, but the classification above maximizes quantitative consistency.*

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**If you want the subtle subdivision (for highlighting the “superbase”):**  
- If allowed, two classes:  
  1. **Very Strong Organic Bases**: Compounds 1 & 2 (pKa 12–14)
  2. **Superbase**: Compound 3 (pKa ≥ 20)

But based strictly on aqueous pKa, **one class is best**.