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.