# SDRPC-PRIME-001 Validation Phase I — Round 06
## Operator-Relative Prime Core + Morphism Classification

**日期：** 2026-09-23  
**Finite domain：** `n ≤ 120,000`

**作者：** Neo.K  
**機構：** EveMissLab／一言諾科技有限公司  
**版本：** 原稿未標註（本次整理版）  

---

## 1. 本輪核心

Round 05 抽出 prime 的 canonical operator-relative core：

```text
multiplicative irreducibility
relative to:
  multiplication + unit + factorization relation
```

Round 06 直接分類：不同 morphism 到底保不保這個 core。

## 2. Morphism Catalog

| Morphism | Class | Prime-core status | Counterexample / note |
|---|---|---|---|
| ISO_ODD_CODE | isomorphism_with_transported_operation | preserved | Irreducibility preservation follows from isomorphism; finite check is only a sanity control. |
| EMBED_Odd | submonoid_embedding | preserved_on_tested_elements | divisor-closed on checked range |
| EMBED_C6 | submonoid_embedding | preserved_on_tested_elements | divisor-closed on checked range |
| EMBED_C30 | submonoid_embedding | preserved_on_tested_elements | divisor-closed on checked range |
| EMBED_C210 | submonoid_embedding | preserved_on_tested_elements | divisor-closed on checked range |
| EMBED_R1mod6 | submonoid_embedding | not_preserved | {"n": 25, "prime_in_Nplus": false, "irreducible_in_submonoid": true, "factorization": {"5": 2}} |
| EMBED_Squares | submonoid_embedding | not_preserved | {"n": 4, "prime_in_Nplus": false, "irreducible_in_submonoid": true, "factorization": {"2": 2}} |
| QUOT_MOD_6 | quotient_homomorphism | not_well_defined_on_quotient_classes | {"residue": 2, "prime": 2, "composite": 8} |
| QUOT_MOD_30 | quotient_homomorphism | not_well_defined_on_quotient_classes | {"residue": 2, "prime": 2, "composite": 32} |
| QUOT_MOD_210 | quotient_homomorphism | not_well_defined_on_quotient_classes | {"residue": 2, "prime": 2, "composite": 212} |
| HOM_V2 | noninjective_monoid_homomorphism | not_preserved | {"v2": 1, "prime": 2, "composite": 6} |
| HOM_BIG_OMEGA | noninjective_monoid_homomorphism | task_sufficient_but_factorization_dependent | For n>1, Ω(n)=1 iff n is prime; however Ω computation encodes full factorization multiplicity. |
| LOSSY_RANKMOD100_C210 | lossy_projection | not_preserved | {"rank_mod_100": 1, "prime_value": 11, "composite_value": 2197} |

## 3. Class A：Isomorphism

`ISO_ODD_CODE` 把 `f(n)=2n+1` 當 carrier encoding，並把 multiplication 一起 transported。

這種情況 prime core 完整保存。

SDRPC 對它的定位不是新 theorem，而是 canonical preservation rule：

```text
structure-preserving isomorphism -> irreducibility preserved
```

## 4. Class B：Submonoid Embedding

`Odd / C6 / C30 / C210` 在本輪都沒有 prime-vs-domain-irreducible disagreement。

共同特徵不是『看起來像質數候選』，而是：

```text
multiplicative closure + divisor closure + same unit
```

這三個條件使 ambient factorization 不會因 domain restriction 而被刪掉。

相反，`R1mod6` 與 `Squares` 都不是 divisor-closed，因此會把 ambient composites 變成 domain irreducibles。

這一類差異已經不是 representation artifact，而是 operator-domain semantics 的真差異。

## 5. Class C：Quotient Homomorphism

`mod 6 / 30 / 210` 都是 many-to-one quotient maps。

它們可以保留 residue necessary conditions，但 primality 不能下降成 residue-class predicate：同一 residue fiber 中可同時有 prime 與 composite。

所以：

```text
quotient may preserve constraints
without preserving prime ontology
```

## 6. Class D：Noninjective Homomorphism

`v2(n)` 滿足：

```text
v2(ab)=v2(a)+v2(b)
```

是真正的 monoid homomorphism，但它會把大量 prime/composite 壓到同一值，所以不保 primality。

這證明：

```text
homomorphism alone is too weak for prime-core preservation
```

需要更強的 faithful / isomorphic 條件，或至少 task-specific fiber condition。

## 7. 特別控制：Ω(n)

`Ω(n)`（含重數的質因子個數）同樣滿足加法型 homomorphism：

```text
Ω(ab)=Ω(a)+Ω(b)
```

而對 `n>1`：

```text
Ω(n)=1 iff n is prime
```

所以它雖然高度 noninjective，卻對 primality 是 task-sufficient。

但這不是新 primality compression：計算 Ω 一般需要取得完整 factorization multiplicity。

它非常適合作為 SDRPC 的控制案例：

```text
task sufficiency != computational novelty
```

## 8. Class E：Lossy Projection

把 C210 survivor 映射成 rank mod 100，完全不保 original multiplication。

prime/composite 立即混在同一 projected fiber。

因此 lossy coordinate projection 只能作 observation，不能自動承載 prime core。

## 9. 目前的 Morphism Preservation Ladder

本輪可以先整理成：

```text
Isomorphism / faithful transport
    -> preserves prime core

Divisor-closed submonoid embedding
    -> preserves prime core on retained elements

General submonoid embedding
    -> may create new irreducibles

Quotient / noninjective homomorphism
    -> generally does not preserve prime core
       unless a task-specific fiber theorem is available

Lossy non-homomorphic projection
    -> no preservation guarantee
```

## 10. 算子本體論與 SDRPC 的合流

到 Round 06，SDRPC 的 structural relativity 已經可以重寫成：

```text
Carrier symbols are secondary.
Operator/relational structure determines the predicate semantics.
Morphisms determine which predicates survive transport.
Probes observe only selected images of that structure.
```

也就是：

```text
Domain -> Operator Ontology -> Morphism -> Invariant -> Probe
```

而不是先從 digit / residue pattern 開始。

## 11. 這對 prime 問題最大的修正

現在可以很明確地分三層：

### Ontology layer
`multiplication + unit + factorization relation`。

### Morphism layer
哪些 domain transforms 保留這些結構？

### Probe layer
在不完整保留 ontology 時，哪些 images 仍保留 task-relevant information？

因此 mod 6 的位置被徹底固定：它屬於 Probe / quotient layer，不是 Ontology layer。

## 12. Round 06 的 theorem-candidate / control 結果

本輪整理出五條：

1. isomorphism preservation；
2. divisor-closed embedding preservation；
3. non-divisor-closed embedding may create irreducibles；
4. quotient/homomorphic image need not preserve prime core；
5. noninjective map can still be task-sufficient if its fibers respect the task label。

其中第 1 條是標準代數事實；其餘在 SDRPC 中的價值主要是分類與 proof-engineering，不應冒充全新代數定理。

## 13. 下一輪

Round 07 應該開始真正碰『問題的問題』：

```text
Prime Ontology Compression vs Probe Compression
```

把所有目前方法分成：

- ontology-preserving compression；
- task-preserving compression；
- quotient necessary-condition compression；
- computationally expensive exact compression；

然後比較 proof burden。

核心問題會是：

> 有沒有比完整 factor structure 更小、又比 modulus/wheel 更強的 canonical task state？

這才真正開始逼近 `MinDep(Q_prime)` 的有意義版本。

## 14. Round 06 結論

```text
prime core preservation is primarily a morphism problem over operator structure;
probe patterns are downstream observations of that structure.
```

這一輪把『算子本體論 × 序列域相對論 × 探針演算』第一次正式合成同一張 preservation map。

---

```text
Round 01: COMPLETE
Round 02: COMPLETE
Round 03: COMPLETE
Round 04: COMPLETE
Round 05: COMPLETE
Round 06: COMPLETE
Next: Round 07 — Prime Ontology Compression vs Probe Compression
```