# S7-P11 — Joint Fourier \((q_1,q_2)\) ≟ Double \(K/L_{\mathrm{eff}}\) (Gap Fill)

### Factor-of-two coincidence real · two-channel sum does **not** double the ratio

**CPT Framework — S⁷ generation track**  
**Date:** 2026-08-08  
**Status:** **S7-P11 BANKED** · doubling hypothesis **CLOSED F** · \(K/L\) still **OPEN**  
**Script:** `scripts/investigations/s7_geometry/s7_p11_joint_fourier_KL_doubling.py`  
**Data:** `scripts/investigations/s7_geometry/data/s7_p11_joint_fourier_KL_doubling.json`  
**Priors:** S7-P9 (same hypothesis family) · P6 · P7 · P3

Tags: **[E]** · **[E algebra]** · **[E honesty]** · **[M]** · **[O]** · **[F]** · **[CLOSED F]**

---

## 0. Gap statement

| | |
|--|--|
| **P3** | \(K/L_{\mathrm{obs}}=\sin\delta/(3\sin 3\delta)\approx 0.396\) exact Z₃ algebra from masses |
| **P6** | Single-channel Fourier on Foot circle \(K/L_{\mathrm{eff}}\approx 0.206\) |
| **Coincidence** | \(0.396/0.206\approx 1.92\approx 2\) |
| **Hypothesis** | Joint \((q_1,q_2)\) Hopf Weber Fourier sum doubles \(K/L_{\mathrm{eff}}\to\approx 0.40\) |
| **Status** | Hypothesis **CLOSED F** as geometric derivation; coincidence **[E]**; \(K/L\) first-principles **OPEN** |

**Join to P9:** P9 already failed sec+csc / 2×E / double Weber. This gap-fill makes the **joint Fourier sum** construction explicit (linearity) and packages the inventory attack.

---

## 1. Baseline **[E]**

| Quantity | Value |
|----------|------:|
| \(K/L_{\mathrm{obs}}\) (Foot / P3) | 0.395952 |
| \(K/L_{\mathrm{eff}}\) single \(E=\sec\alpha\) | 0.203604 |
| \(K/L_{\mathrm{eff}}\) single \(\sec+u\) | 0.203618 |
| obs / single | **1.9447** |

**Analytic:** on the Foot map, \(\cos\alpha=1/(\sqrt{{2}} r)\Rightarrow \sec\alpha=\sqrt{{2}}\,r(\varphi)\). Single-channel Fourier is essentially the Fourier of the mass map \(r(\varphi)\).

---

## 2. Joint Fourier constructions

| Model | \(K/L_{{\mathrm{{eff}}}}\) | × single | err vs obs | Doubles? | Near obs? |
|-------|---------------------------:|---------:|-----------:|:--------:|:---------:|
| J0 single sec | 0.203604 | 1.0000 | 48.6% | no | no |
| J1 2×sec | 0.203604 | 1.0000 | 48.6% | no | no |
| J2 sec+csc | 0.203601 | 1.0000 | 48.6% | no | no |
| J2 sec+csc+u | 0.203614 | 1.0000 | 48.6% | no | no |
| J3 joint spectral sum | 0.203601 | 1.0000 | 48.6% | no | no |
| J4 cos²sec+sin²csc | 0.801588 | 3.9370 | 102.4% | no | no |
| J4 |cos|sec+|sin|csc | — | — | — | no | no |
| J5 (√ω1+√ω2)² | 0.209491 | 1.0289 | 47.1% | no | no |
| J5 (ω1−ω2)² | -0.144667 | -0.7105 | 136.5% | no | no |
| J5 u_P5 only | 0.795821 | 3.9087 | 101.0% | no | no |

### Linearity **[E algebra]**

```text
Fourier(E1 + E2) = Fourier(E1) + Fourier(E2)
⇒ K/L from joint spectral sum ≡ K/L from Fourier(sec+csc)
linearity match confirmed: True
```

| Channel | c1 | c3 | K/L |
|---------|----:|----:|----:|
| q1 = secα | 1.4183e+03 | -2.8878e+02 | 0.203604 |
| q2 = cscα | -1.0422e-02 | 7.2927e-03 | 0.699751 |
| joint sum | 1.4183e+03 | -2.8877e+02 | 0.203601 |

**Finding:** q2 = csc contributes a different spectrum, but the **joint** \(c_3/c_1\) stays \(\sim 0.20\), not \(\sim 0.40\).

---

## 3. Why the factor-of-two does not become a theorem

```text
Foot circle φ
  r(φ) from mass map (P3)
  sec α = √2 r(φ)     →  Fourier → K/L_eff ≈ 0.20
  2 × same shape      →  ratio unchanged  [scale kill]
  sec + csc           →  still ≈ 0.20     [joint sum]
  spectral sum q1+q2  →  ≡ Fourier(sec+csc)
  pure u_P5(α(φ))     →  ≈ 0.80 ≈ 2×obs   [wrong target]

Observed K/L is a *Foot potential* coefficient, not automatically
the Fourier of path energy E(α(φ)).
```

---

## 4. Mixture scan (diagnostic only)

- Best \(\sec+\lambda\csc\): \(\lambda=-2.000\), \(K/L=0.2036\), err=48.6%
- Best \(\sec+\mu u\): \(\mu=5.000\), \(K/L=0.2037\), err=48.6%

**[FORBIDDEN as derivation]** — tuning \(\lambda,\mu\) to hit 0.396 is a fit, not S⁷ geometry.

---

## 5. Integrity board

| Claim | Tag |
|-------|-----|
| Single K/L_eff ~0.204 on Foot (P6/P9/P11) | [E] |
| obs/single ≈ 1.94 near factor 2 | [E coincidence] |
| 2×E doubles K/L_eff | [CLOSED F] scale invariance |
| Joint E=sec+csc doubles K/L_eff | [CLOSED F] |
| Joint Fourier sum of q1,q2 spectra doubles K/L_eff | [CLOSED F] |
| Measure-weighted joint doubles K/L_eff | [CLOSED F] |
| Pure u_P5 Fourier ≈ 0.80 ≈ 2×obs | [E curiosity] |
| Fit λ,μ to force K/L_obs | [FORBIDDEN as E] |
| Foot K/L from masses is Z3 algebra (P3) | [E algebra] |
| First-principles geometric K/L | [O] remains |
| Gap hypothesis 'two-channel doubles 0.206→0.40' | CLOSED F as derivation; coincidence stands |

---

## 6. Verdict

1. **Coincidence [E]:** \(K/L_{{\mathrm{{obs}}}}/K/L_{{\mathrm{{eff}}}}^{{\mathrm{{single}}}}\approx 1.94\approx 2\).
2. **Joint two-channel Fourier [CLOSED F as doubling theorem]:** sec+csc, spectral sum, measure weights do **not** produce \(\approx 0.40\).
3. **Scale kill [E]:** \(2\times E\) cannot change \(c_3/c_1\).
4. **Curiosity [E]:** pure \(u_{{\mathrm{{P5}}}}\) Fourier \(\approx 0.80\approx 2\times K/L_{{\mathrm{{obs}}}}\) — different functional, not a pass.
5. **Residual [O]:** first-principles geometric \(K/L\) (same as P6–P10).

> **Gap fill one-liner:** Combining both S⁷ Hopf channels does **not** double Fourier \(K/L_{{\mathrm{{eff}}}}\) from \(\sim 0.20\) to \(\sim 0.40\); the factor-of-two coincidence survives, the two-component doubling theorem does not.

---

## 7. Do not redo

- Sell obs≈2×single as a two-channel theorem
- Claim 2×E doubles a Fourier ratio
- Fit λ,μ and call it geometric derivation
- Reopen P9 family without a new functional principle

## 8. Related bank

| Doc | Role |
|-----|------|
| `a_geom_s7_p9_two_component_weber_KL_results.md` | Same hypothesis family FAIL |
| `a_geom_s7_p6_z3_multichannel_weber_results.md` | Single Fourier ~0.21 |
| `a_geom_s7_p7_fano_channel_weights_results.md` | q1,q2 Hopf law |
| `a_geom_s7_p10_fano_line_embedding_C_ratio_results.md` | C/E3 / δ residual |

---

## Changelog

| Date | Change |
|------|--------|
| 2026-08-08 | P11 gap fill: joint Fourier sum explicit; doubling hypothesis CLOSED F |