S⁷ Residuals R-δ and R-K/L — Status One-Pager
Foot angle · Foot \(K/L\) · what is banked · what is still open · do-not list
CPT Framework — S⁷ generation track consolidation Date: 2026-08-09 Status: BANKED STATUS FREEZE · R-δ [E INPUT] · R-K/L [E] algebra of that input, not a second input (Doctrine v1.1.6) · F5 investigation CLOSED IN THIS BOOK (CLOSED_FT_F5_WORK) · \(\delta\) is the declared input · \(K/L\) is the algebra of that input · not a derivation · other book not named · R-δ lab package accepted · hunt on this skeleton stopped
Priors: P3 · P6 · P8–P13 · \(\mathcal{P}_L\) · FAV package acceptance · RKL design
Updated: 2026-10-04
Tags: [E] · [E honesty] · [M] · [O] · [F]
0. One-line
4 Oct 2026 (§0bn1118). R-δ is [E INPUT]. R-K/L is [E], the algebra of that input and not a second input (Doctrine v1.1.6). The Foot model is established under the named premise that \(\delta\) is an input of this book. \(K/L=\sin\delta/(3\sin 3\delta)\) is the algebra of that input. This label is not a derivation of \(\delta\). The investigation into \(\delta\) is not in this book. No further progress can be made here. The investigation belongs to another book, which is not named. R-δ and R-K/L are closed as an investigation of this book. \(m_e=\hbar/(ac)\) is access of \(a\), and \(\delta\) stands in this book in that class. \(K/L=\sin\delta/(3\sin 3\delta)\) is the algebra of the declared angle. The hunt stays stopped. No promote. CLOSED_FT_F5_WORK.
>
Earlier freeze. R-δ lab package \(\mathcal{P}_{\delta}\) accepted (FAV stack; the public chip was then OPEN). The map after \(\mathcal{P}_{\delta}\) is lab algebra, not a free public theorem. Fourier \(2\times\) is not a second derivation. No silent dials. R-δ is now [E INPUT]. R-K/L is [E] algebra of that input, not a second input. Banked geometry (Koide Z₃, latitude, \(\rho_{\mathrm{stat}}\), hierarchy band, pole, \(L_{\mathrm{eff}}\) dictionary) and named packages (\(\mathcal{P}_L\), local \(\rho\sim 3^3/9\pi\)) must not be sold as closed generation theorems.
1. The two residuals
| ID | Unknown | Not the residual | |----|---------|------------------| | R-δ | In this book \(\delta\approx 132.73^\circ\) is an input. Not a derivation from this reduction, and not a third leftover beside \(\alpha\) and \(a\) | Koide \(Q=2/3\) (that is exact Z₃ algebra given three masses) | | R-K/L | The algebra \(K/L=\sin\delta/(3\sin 3\delta)\) of the angle this book is given. Not a second derivation | A separate geometric hunt for \(K/L\approx 0.396\) |
Coupling: closing either helps the other (same Foot potential \(V=-K\cos 3\varphi+L\cos\varphi\)), but they are not identical. Parallel generation book (do not collapse): G-SPEC Route E Willmore ladder for \(\mu/\tau\) ratios — independent origin, joint 3D readout (M3). Coeff \(3/(2\alpha)\) and S7 share a join clue [M] only.2. Banked geometry and algebra [E / M]
| Item | Tag | Source | |------|-----|--------| | Koide \(Q=2/3\) = Z₃ orbit algebra | [E] | P3 | | Mass as S⁴ latitude (\(\cos\alpha=1/(\sqrt{2} r)\)) | [E/M] | P2 | | Path-length mass hierarchy | [F as theorem] | P1 | | Electron \(\alpha=\pi/4\) Z₂ fixed point | [E under package] | P3–P5 | | Full Weber multiplet object class \(E_3\) | [E process] | P5–P6 | | \(\rho_{\mathrm{stat}}(\delta)=C_{\mathrm{crit}}/E_3\) closed form | [E algebra] | P10 | | \(\rho_{\mathrm{stat}}(\delta_{\mathrm{obs}})\approx 27.75\) | [E] | P10 | | \(3^3\), \(9\pi\) near \(\rho_{\mathrm{stat}}\) (local critical) | [M rhyme] | P10 | | Hierarchy band \(\delta\in[\sim 120^\circ,135^\circ)\) | [E domain] | P12 | | Pole \(\delta_{\mathrm{pole}}=\arccos(-1/\sqrt{2})=135^\circ\) | [E] | P12 | | Honest energy min on hier band ~120° (err ~12.7°) | [E honesty] | P12 | | \(L_{\mathrm{eff}}=L_{\mathrm{double}}/2=L_{\mathrm{geom}}\) | [E] | P13 · \(\mathcal{P}_L\) | | \(\omega_{\mathrm{geom}}\) already in banked \(\Delta\omega\) | [E] | P13 | | Fourier \(K/L_{\mathrm{eff}}(\sec)\approx 0.204\) | [E] | P6/P9/P13 | | obs/eff ≈ 1.94 | [E coincidence] | P9/P13 | | \(\mathcal{P}_L\): optional \(K/L_{\mathrm{read}}=2\,K/L_{\mathrm{eff}}\) | [M] | \(\mathcal{P}_L\) card | | P8 \(B/A=-(3/(2\alpha))^2\) muon join clue | [M] | P8 · M3 |
3. Closed-null / fail inventory (do not redo)
| Claim | Verdict | Source | |-------|---------|--------| | Bare Fano counts \(1/7,3/7\) fix \(\rho\sim 28\) | CLOSED F | P7 | | Two-component Hopf energy doubles \(K/L_{\mathrm{eff}}\) | CLOSED F | P9 | | Joint \((q_1,q_2)\) Fourier doubles \(K/L_{\mathrm{eff}}\) | CLOSED F | P11 | | \(2\times E\) doubles Fourier ratio | CLOSED F (scale) | P9/P11 | | Geometric \(\rho\in\{27,9\pi,28\}\) uniquely selects global \(\delta_{\mathrm{obs}}\) | F / not achieved | P10/P12 | | Circular \(C=\rho E_3(\delta_{\mathrm{obs}})\) as δ derivation | F as derivation | P12 | | Uniform \(\omega\to\omega/2\) fixes Fourier \(K/L\) | F | P13 | | \(3^3\) unique δ theorem | F if claimed | P10 | | Single-seat full Weber hosts μ+τ together | FAIL | P5 | | \(\mu/\tau\) as other \(T(p,q)\) | FORBIDDEN | Route E / S7 |
4. Named packages (use correctly)
| Package | Content | Theorem? | |---------|---------|----------| | \(\mathcal{P}_L\) | \(L_{\mathrm{eff}}=L_{\mathrm{geom}}\); optional \(K/L_{\mathrm{read}}=2\,K/L_{\mathrm{eff}}\) | No — a_geom_s7_p_L_effective_length_package_card.md | | Local \(\rho\) rhyme | \(3^3\) / \(9\pi\) near \(\rho_{\mathrm{stat}}(\delta_{\mathrm{obs}})\) | No — local critical only | | Soft abs mass \(\mathcal{P}_{\mathrm{EMC}}\) | Orthogonal (bulk/face MeV) | No — O-EMC-3 | | Route E | Willmore ladder ratios | Data [E]; coeff OPEN |
5. Open residual board [O]
| Residual | Reopen rule | |----------|-------------| | R-δ | Investigation closed in this book. \(\delta\) is the declared input. No further progress here. Another book is not named. Lab \(\mathcal{P}_{\delta}\) accepted and is not a derivation | | R-K/L | Investigation closed in this book. \(K/L\) is the algebra of the declared angle. \(2\times\) is not a second mill | | Route E coeff \(3/(2\alpha)\) | Separate book; join clue to P8 only [M] | | Multiplet \(C/E_3\) first principles | Same family as R-δ once \(\rho\) is geometric |
6. Agent defaults
| Do | Do not | |----|--------| | Load this note + \(\mathcal{P}_L\) before S⁷ K/L or δ claims | Sell P10–P13 as closed generation | | Tag coincidence 1.94× as [E] num only | Sell two-channel doubling | | Keep Route E and S⁷ as parallel tracks | Force Willmore = Weber or bulk→Willmore | | Report hier band + pole as [E] structure | Sell hier band alone as δ theorem | | Park FL-HEAVY separately | Mix heavy lattice seeds into Foot δ |
7. Public-facing paragraph (promote only if chips match)
In the CPT S⁷ generation track, three charged-lepton masses sit on a Foot Z₃ orbit in \(\sqrt{m}\) space with a single angle \(\delta\) and a potential coefficient ratio \(K/L\). The Koide relation is exact algebraic structure under that packaging. The public status is R-δ [E INPUT] and R-K/L [E] (Doctrine v1.1.6, 4 Oct 2026): the Foot model is established under the named premise that \(\delta\) is an input of this book, and \(K/L=\sin\delta/(3\sin 3\delta)\) is the algebra of that input, not a second input. This is not a first-principles derivation of \(\delta\) and not a bare-\(S^7\) theorem. Geometric structure (S⁴ latitude, hierarchy domain, Foot pole, effective fibre length) is banked; optional packaging that doubles Fourier path-energy ratios is model language only, not a finished derivation.
8. File index (load order for S⁷ residuals)
1. This note (status) 2. a_geom_s7_p_L_effective_length_package_card.md 3. a_geom_s7_p13_effective_length_KL_results.md 4. a_geom_s7_p12_global_delta_selection_results.md 5. a_geom_s7_p10_fano_line_embedding_C_ratio_results.md 6. a_geom_s7_p11_joint_fourier_KL_doubling_results.md 7. a_geom_s7_p9_two_component_weber_KL_results.md 8. a_geom_s7_generation_catchup_p1_p4c.md
Changelog
| Date | Change | |------|--------| | 2026-10-04 | R-δ is [E INPUT]. R-K/L is [E], the algebra of that input and not a second input (Doctrine v1.1.6). A same-day wording that put both chips on [E INPUT] is withdrawn. Not a derivation. | | 2026-08-09 | Consolidation freeze after P10–P13 + \(\mathcal{P}_L\) |
Agent one-liner
R-δ is [E INPUT]; R-K/L is [E] algebra of that input, not a second input; not a derivation; F5 investigation CLOSED IN THIS BOOK; the hunt on this skeleton is stopped; the other book is not named; structure banked; \(\mathcal{P}_L\) optional 2× read [M]; no two-channel / circular-δ theorems.