# S7-P13 — Effective Length Packaging for \(K/L\)

### Half double-loop = geometric fibre · Fourier deficit not a length bug in \(\omega\)

**CPT Framework — S⁷ generation track**  
**Date:** 2026-08-09  
**Status:** **S7-P13 BANKED** · length dictionary **[E]** · 2×KL_eff package **[M]** · first-principles \(K/L\) **OPEN**  
**Script:** `scripts/investigations/s7_geometry/s7_p13_effective_length_KL.py`  
**Data:** `scripts/investigations/s7_geometry/data/s7_p13_effective_length_KL.json`  
**Stimulus:** User — effective length ≠ double loop; half double-loop  
**Priors:** P3 · P4-C · P5 · P6 · P9 · P11

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

---

## 0. User question

> Have we the effective length correct? It would not be the double loop as it is only the effective length. Potentially half the double loop length.

---

## 1. Length dictionary **[E]**

| Symbol | Definition | Period / \(\omega\) |
|--------|------------|----------------------|
| \(L_{\mathrm{geom}}\) | \(2\pi R\) single Hopf fibre | \(\omega_{\mathrm{geom}}=c/R\) |
| \(L_{\mathrm{double}}\) | \(4\pi R\) spinor / MSCT 4π return | \(\omega_{\mathrm{double}}=c/(2R)\) |
| \(L_{\mathrm{eff}}\) | \(L_{\mathrm{double}}/2 = L_{\mathrm{geom}}\) | same as geom |

**Finding L0 [E]:** “Half the double loop” **is** the geometric fibre length. That is the right *dictionary* for dynamical path length.

---

## 2. What the banked code already uses **[E]**

| Construction | Packaging |
|--------------|-----------|
| P4/P5 \(\Delta\omega\propto\|1/\cos\alpha-1/\sin\alpha\|\) | \(\omega_{\mathrm{geom}}\) already |
| P6/P9/P11 Fourier \(E=\sec\alpha=\sqrt{2}\,r(\varphi)\) | mass–latitude shape; no \(L_{\mathrm{double}}\) in \(\omega\) |

**Finding L1 [E honesty]:** The \(\sim 0.20\) Fourier \(K/L_{\mathrm{eff}}\) is **not** caused by mistakenly using double-loop frequency. The banked \(\omega\) is already single-loop / half-double-loop.

---

## 3. Baseline numbers

| Quantity | Value |
|----------|------:|
| \(K/L_{\mathrm{obs}}\) (Foot) | 0.395952 |
| \(K/L_{\mathrm{eff}}(\sec\alpha)\) | 0.203604 |
| obs/eff | 1.944715 |
| \(2\times K/L_{\mathrm{eff}}\) | 0.407209 |
| rel err of \(2\times\mathrm{eff}\) vs obs | 2.84% |

### Fourier survey (φ ∈ [0, 2π))

| Model | \(K/L_{{\mathrm{{eff}}}}\) | obs/eff | err of 2×eff vs obs |
|-------|---------------------------:|--------:|--------------------:|
| sec (tension / path energy) | 0.203604 | 0.5142 | 2.84% |
| sec + u (λ=1, L_eff) | 0.203618 | 0.5142 | 2.85% |
| u_geom only (L_eff) | 0.795821 | 2.0099 | 301.98% |
| u only (ω_double lf=1/2) | 0.795821 | 2.0099 | 301.98% |
| W_full=1−u (L_eff) | 0.795821 | 2.0099 | 301.98% |
| W_P4half=1−u/2 | 0.795821 | 2.0099 | 301.98% |
| √2 r (mass map) | 0.203604 | 0.5142 | 2.84% |

---

## 4. Double-cover Fourier trap **[E]**

On spinor parameter \(\theta\in[0,4\pi)\), \(\varphi=\theta/2\). Foot potential is \(-K\cos(3\varphi)+L\cos\varphi\). Integer Fourier \(\cos(n\theta)\) on the double period **mis-labels** harmonics. Correct extraction uses \(\varphi\)-modes on a \(2\pi\) circle — i.e. **half the double-loop parameter** as the angle — recovering the L2 baseline.

---

## 5. Uniform \(\omega\to\omega/2\) does not fix the ratio **[E]**

Setting all frequencies to double-loop (\(\Delta\omega\to\Delta\omega/2\)) scales \(u\\propto(\\Delta\omega)^2\) by \(1/4\). Fourier \(c_3/c_1\) of a pure shape is scale-invariant — **cannot** turn 0.20 into 0.40 by itself.

---

## 6. Packaging reading of the user claim **[M]**

If the Foot coefficient \(L\) were *identified* with a geometric length that was wrongly taken as \(L_{{\mathrm{{double}}}}\), then correcting to \(L_{{\mathrm{{eff}}}}=L_{{\mathrm{{geom}}}}=L_{{\mathrm{{double}}}}/2\) **doubles** \(K/L\).

Applied to the banked Fourier ratio: \(2\times K/L_{\mathrm{eff}}(\sec)= 0.407209\) vs obs \(0.395952\) (rel err **2.84%**).

| Claim | Tag |
|-------|-----|
| Numerical near-match of 2×Fourier_eff to obs | **[E num]** |
| This is forced by S⁷ fibre geometry alone | **[M package]** — needs freeze |
| Already true because code used L_double in ω | **[F]** — code uses geom |

**Honesty:** the coincidence is real; elevating it to a derivation requires a named principle that Foot \(L\) tracks double-loop length while \(K\) does not (or an equivalent half-period identification). That principle is **not** yet banked as **[E]**.

---

## 7. Do not merge factor-of-two stories

| Story | Content | Status |
|-------|---------|--------|
| P4-C | Full Weber \(W=1-u\) vs velocity-only \(1-u/2\) | **[E]** constraint |
| Length | \(L_{{\mathrm{{double}}}}\) vs \(L_{{\mathrm{{eff}}}}=L_{{\mathrm{{geom}}}}\) | dictionary **[E]**; K/L fix **[M]** |
| P9/P11 | Two-channel energy doubles Fourier ratio | **CLOSED F** |
| Coincidence | obs ≈ 2 × Fourier_eff(sec) | **[E]** num |

---

## 8. Integrity board

| Claim | Tag |
|-------|-----|
| L_eff = L_double/2 = L_geom (dictionary) | [E] |
| Banked Δω uses ω_geom = c/R already | [E] |
| Fourier K/L_eff~0.20 is mass-map shape | [E] |
| obs/eff ≈ 1.94 near factor 2 | [E coincidence] |
| 2×KL_eff(sec) ≈ KL_obs within ~3% | [E num] err 2.84% |
| 2×KL_eff as theorem from L_double→L_eff | [M package / not yet E] |
| Uniform ω→ω/2 fixes Fourier ratio | [F] |
| Integer Fourier on 4π period = K/L | [F] |
| Two-channel energy doubles K/L_eff | [CLOSED F] P11 |
| P4-C W half vs full = same as length 2× | [F merge] |
| K/L_obs from masses depends on loop length | [F] — pure Foot algebra |
| First-principles K/L closed | [O still] |

---

## 9. Verdict

1. **Dictionary [E]:** Effective length = half the double-loop = geometric fibre loop. Correct.
2. **Code [E]:** Banked \(\omega\) already uses that effective length.
3. **Fourier gap [E honesty]:** \(K/L_{{\mathrm{{eff}}}}\sim 0.20\) is the mass-map shape, not a double-loop frequency bug.
4. **Package [M]:** \(2\times K/L_{\mathrm{eff}}\approx K/L_{\mathrm{obs}}\) (err 2.84%) is a freeze *candidate* if Foot \(L\) is double-loop-misidentified; not a theorem yet.
5. **Residual [O]:** first-principles geometric \(K/L\) remains open.

> **One-liner:** Half double-loop **is** the right effective length and is already what \(\omega\propto 1/R\) uses; the factor-of-two \(K/L\) coincidence is real but is not fixed by flipping fibre frequency to double-loop — it needs a named Foot-\(L\) packaging (or new functional), still **OPEN** as theorem.

---

## 10. Do not

- Claim code used double-loop \(\omega\) (it does not)
- Sell 2×Fourier_eff as theorem without packaging freeze
- Merge with P4-C Weber half-u or P11 two-channel doubling
- Change Foot \(K/L_{{\mathrm{{obs}}}}\) with loop length (mass algebra)

## 11. Related bank

| Doc | Role |
|------|------|
| `a_geom_s7_p9_two_component_weber_KL_results.md` | 1.94× coincidence; two-comp FAIL |
| `a_geom_s7_p11_joint_fourier_KL_doubling_results.md` | joint Fourier doubling CLOSED F |
| `s7_p4_constraint_weber_acceleration.py` | distinct Weber factor-of-2 |
| `a_geom_s3_natural_spin_structure_bridge.md` | double cover / 4π monodromy |

---

## Follow-on freeze

Named package: **`a_geom_s7_p_L_effective_length_package_card.md`** (\(\mathcal{P}_L\) v1).  
Residual map: **`a_geom_s7_delta_KL_residual_status.md`**.

## Changelog

| Date | Change |
|------|--------|
| 2026-08-09 | S7-P13: L_eff dictionary; 2×KL_eff package; K/L still OPEN |
| 2026-08-09 | Link \(\mathcal{P}_L\) freeze + residual status one-pager |