# Results — R-MCR-2: Remainder \(k\) Ruled Out as a One-Convention Derivation

### FAIL_DERIVE_K · one-convention Sciama+\(\alpha\) is \(H_0\sim11\) · splice not a theorem · no promote

**CPT Framework — absolute-scale integrity**  
**Date:** 2026-08-16  
**Status:** **BANKED** · `PASS_R_MCR2_FAIL_DERIVE_K`  
**Design:** `a_geom_r_mcr2_remainder_k_design.md`  
**Script:** `scripts/investigations/s3_geometry/r_mcr2_remainder_k_census.py`  
**Data:** `scripts/investigations/s3_geometry/data/r_mcr2_remainder_k_census.json`  
**Parents:** R-MCR · R-XCHECK-D · AST · `s3_sciama_integral.py`

Tags: **[E algebra]** · **[E honesty]** · **[Programme]** · **[F derivation]**

---

## 0. One-line

> **The remainder that would make M13 one-convention is not derivable from a single Mach functional.** Every named Green / raw / Sciama option misses the same-slot targets \(k_{\mathrm{raw}}=\alpha\cdot 3/8\) and \(k_{\mathrm{green}}=3\alpha/(16\pi)\). The unique one-convention \(\alpha\)-extensions of the Sciama book \(\Omega x^2=1\) are **S-Gα** / **S-Rα**, they are consistent with each other (factor \(2\pi\)), and both dual to \(H_0^{(\sqrt{6})}\approx10.98\), not \(68.97\). The published hit is the Green value parked in the raw slot. That splice is now **ruled out as a derivation**. G-H0 stays convention-conditional **[P]**. No promote.

---

## 1. Targets (from R-MCR, not retuned)

\[
A:\quad k_{\mathrm{raw}}=\alpha\cdot\frac38\approx 2.737\times 10^{-3},
\qquad
B:\quad k_{\mathrm{green}}=\frac{3\alpha}{16\pi}=\frac{A}{2\pi}\approx 4.355\times 10^{-4}.
\]

A legal hit must produce \(A\) as a **raw** \(k\) or \(B\) as a **Green** \(k\).

---

## 2. Legal routes — no same-slot hit

| ID | Slot | Produced \(k\) | vs \(A\) | vs \(B\) | Verdict |
|----|------|----------------|:--------:|:--------:|---------|
| S-G | Green | \(3/8\) | | | NO_HIT |
| **S-Gα** | Green | \(\alpha\cdot 3/8\) | **equal** | \(\times 2\pi\) | **WRONG_SLOT** |
| S-R | Raw | \(3\pi/4\) | | | NO_HIT |
| **S-Rα** | Raw | \(\alpha\cdot 3\pi/4\) | \(\times 2\pi\) | | NO_HIT |
| S-1G / S-αG | Green | \(1\), \(\alpha\) | | | NO_HIT |
| S-1R / S-αR | Raw | \(1\), \(\alpha\) | | | NO_HIT |
| S-A / S-Aα | named | \(1/(2\pi)\), \(\alpha/(2\pi)\) | | S-Aα is \(\tfrac83 B\) | NO_HIT |
| S-B / S-Bα | flat | \(1/2\), \(\alpha/2\) | | | NO_HIT |
| S-I / S-Iα | integral | \(1/(16\pi)\), \(\alpha/(16\pi)\) | | S-Iα is \(\tfrac13 B\) | NO_HIT |

**Finding K2-1 [E algebra]:** S-Gα *is* the number \(A\), but \(A\) is the raw-slot target. In the Green slot that number is the G38α book, already scored: \(H_0^{(\sqrt{6})}\approx10.98\). Same-slot hit: **none**.

**Finding K2-2 [E algebra]:** S-Gα and S-Rα are the one-convention pair for \(\Omega x^2=1\) plus \(\alpha\):

\[
k_{\mathrm{raw}}=\alpha\cdot\frac{3\pi}{4}=2\pi\cdot\bigl(\alpha\cdot\tfrac38\bigr)=2\pi\,k_{\mathrm{green}}.
\]

They describe the **same** \(R\) and the **same** invert \(\approx 10.98\). One-convention Sciama+\(\alpha\) cannot produce \(68.97\).

---

## 3. Illegal routes — rejected, not scored as hits

| ID | Move | Why rejected |
|----|------|----------------|
| **X-JUX** | slogan \(k_{\mathrm{raw}}=\alpha\times(3/8)\) | Two slots: Green value into raw equation. This *is* M13, not a derivation. |
| **X-CONV** | \(A/2\pi\Rightarrow B\) | Converts the published splice into the other slot. |
| **X-FAM** | invert \(\Omega_{\mathrm{pack}}(2\pi/\alpha)=1/6\) | Package tautology of \(R_{\mathrm{eff}}\). |
| **X-16** | \(\Omega=1/6\), \(x=\sqrt{6}\), then \(\times\alpha\) | Inserts the dual and the compatibility residual. Recovers S-Gα, which misses. |

**Finding K2-3 [E honesty]:** The only constructions that *equal* \(A\) or \(B\) are X-JUX / X-CONV. Both were illegal *before* numbers. They are the splice, restated.

---

## 4. Near-miss, not adopted

\[
3\cdot\frac{\alpha}{16\pi}=\frac{3\alpha}{16\pi}=B
\]

exactly (census). The \(3\) is the \(3H^2\) inside \(\rho_c\) of the named \(\Omega x^2=1\) book, not a free Ricci license to multiply the Green angular integral. **K-MCR2-1:** do not adopt S-Iα \(\times 3\).

---

## 5. What is ruled out

| Claim | Status |
|-------|--------|
| \(k_{\mathrm{raw}}=\alpha\cdot 3/8\) is a one-functional theorem | **RULED OUT** |
| \(k_{\mathrm{green}}=3\alpha/(16\pi)\) is a one-functional theorem | **RULED OUT** |
| One-convention Sciama+\(\alpha\) yields \(H_0\simeq69\) | **RULED OUT** (\(10.98\)) |
| Published M13 invert as a derivation | **Still not** · remains convention-conditional **[P]** |
| G-H0 / G-EXT observational gates | **Still OPEN** on the published book |
| \(p=2\), mixed-mass, unnamed \(R\) | **Untouched** |

**Finding K2-4:** The coefficient line is closed. Further \(k\)-shopping is not a research step. What remains of M13 is the named package plus the three observational / ontology gates, not a pending coefficient theorem.

---

## 6. Gates

| Gate | Status |
|------|--------|
| PASS_TARGETS_NAMED | **PASS** |
| PASS_ROUTES_FROZEN | **PASS** |
| PASS_S_GA_EQUALS_A_WRONG_SLOT | **PASS** |
| PASS_S_RA_NOT_A | **PASS** |
| PASS_ONE_CONVENTION_NOT_69 | **PASS** (\(10.98\)) |
| PASS_NO_SAME_SLOT_HIT | **PASS** |
| PASS_NEAR_MISS_NOT_ADOPTED | **PASS** |
| PASS_ILLEGAL_REJECTED | **PASS** |
| PASS_NO_H0_USED_TO_BUILD_K | **PASS** |
| PASS_NO_R_ONTOLOGY | **PASS** |
| PASS_NO_PROMOTE | **PASS** |

**Status:** `PASS_R_MCR2_FAIL_DERIVE_K`

---

## 7. Kill board

| ID | Kill |
|----|------|
| **K-MCR2-1** | S-Iα \(\times 3\) sold as a derivation of \(B\) |
| **K-MCR2-2** | X-JUX / X-CONV sold as a theorem |
| **K-MCR2-3** | New \(\Omega\) or \(x=\sqrt{6}\) insert to mint \(k\) |
| **K-MCR2-4** | Name \(R\); mint \(m_e\); retire G-H0 because \(k\) failed |
| **K-MCR-*** | Held |

---

## 8. One-line

> **§R-MCR-2: remainder k FAIL_DERIVE; one-convention Sciama+α is H0~11; published 69 is the splice, not a theorem; gates stay OPEN; no promote.**