# 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]**

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## 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.

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## 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.

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## 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 |

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## 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 |

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## 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** |

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## 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 |

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## 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 |

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## 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.

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## 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`  

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## 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 + \(\mathcal{P}_L\) |

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## 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.**