# \(\mathcal{P}_L\) — Effective Length Package Card (S⁷ Foot \(K/L\))

### Half double-loop = geometric fibre · optional \(2\times K/L_{\mathrm{eff}}\) · **not** a \(K/L\) theorem

**CPT Framework — S⁷ generation track · named packaging**  
**Date:** 2026-08-09  
**Status:** **FROZEN v1** · package card only · first-principles \(K/L\) **remains OPEN**  
**Priors:** S7-P3 · P4-C · P6 · P9 · P11 · **P13** · `a_geom_s7_delta_KL_residual_status.md`  

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

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## 0. One-line

> **\(\mathcal{P}_L\) v1 freezes the fibre length dictionary \(L_{\mathrm{eff}}=L_{\mathrm{double}}/2=L_{\mathrm{geom}}\) and optionally names a *reading rule* \(K/L_{\mathrm{read}}=2\,K/L_{\mathrm{eff}}^{\mathrm{(Fourier)}}\) for programme language — it does not derive Foot \(K/L\) from first principles, does not flip banked \(\omega\) packaging, and must not be merged with P4-C Weber half-\(u\) or P11 two-channel doubling.**

---

## 1. Why a package card

| Layer | Status after P13 |
|-------|------------------|
| Length dictionary \(L_{\mathrm{eff}}=L_{\mathrm{geom}}\) | **[E]** |
| Banked \(\Delta\omega\propto\|1/\cos\alpha-1/\sin\alpha\|\) uses \(\omega_{\mathrm{geom}}\) | **[E]** |
| Fourier \(K/L_{\mathrm{eff}}(\sec\alpha)\approx 0.204\) | **[E]** shape of mass map |
| Foot \(K/L_{\mathrm{obs}}=\sin\delta/(3\sin 3\delta)\approx 0.396\) | **[E algebra]** from masses |
| obs / Fourier_eff ≈ 1.94 | **[E coincidence]** |
| \(2\times K/L_{\mathrm{eff}}\) vs obs (err ~2.8%) | **[E num]** under post-hoc ×2 |
| First-principles geometric \(K/L\) | **OPEN** |

**Finding PL-0 [E honesty]:** Without a named package, agents either (i) claim “length fixes K/L” as theorem or (ii) re-open P9/P11 doubling. \(\mathcal{P}_L\) freezes the *allowed* language.

---

## 2. Length dictionary (mandatory under \(\mathcal{P}_L\))

Fibre radius \(R\) (Hopf component \(R_0\cos\alpha\) or \(R_0\sin\alpha\)):

| Symbol | Definition | \(\omega\) (light-speed circuit) |
|--------|------------|----------------------------------|
| \(L_{\mathrm{geom}}\) | \(2\pi R\) single geometric fibre loop | \(\omega_{\mathrm{geom}}=c/R\) |
| \(L_{\mathrm{double}}\) | \(4\pi R\) spinor / MSCT \(4\pi\) return | \(\omega_{\mathrm{double}}=c/(2R)\) |
| \(L_{\mathrm{eff}}\) | \(L_{\mathrm{double}}/2=L_{\mathrm{geom}}\) | \(\omega_{\mathrm{eff}}=\omega_{\mathrm{geom}}\) |

**Freeze sentence (dictionary):**

> Effective dynamical path length is the **single geometric fibre loop**, i.e. **half the spinor double-loop length**. Frequency packaging uses \(\omega_{\mathrm{geom}}\).

---

## 3. Package axioms \(\mathcal{P}_L\) (v1)

All items are **programme language** unless tagged otherwise.

| ID | Axiom | Tag |
|----|-------|-----|
| **L0** | Use length dictionary §2; \(L_{\mathrm{eff}}=L_{\mathrm{geom}}\) | **[E]** |
| **L1** | Banked S⁷ \(\Delta\omega\) already implements \(\omega_{\mathrm{geom}}\) — do not “correct” by switching to \(\omega_{\mathrm{double}}\) as a K/L fix | **[E]** · P13 |
| **L2** | Foot mass algebra \(K/L_{\mathrm{obs}}=\sin\delta/(3\sin 3\delta)\) is **independent** of loop length | **[E algebra]** · P3 |
| **L3** | Single-channel Fourier on Foot circle: \(K/L_{\mathrm{eff}}:=-c_3/c_1\) of \(E(\varphi)=\sec\alpha(\varphi)\) (P6 packaging) | **[E process]** |
| **L4** | **Optional reading rule (default OFF for theorem claims):** \(K/L_{\mathrm{read}}:=2\,K/L_{\mathrm{eff}}\) when comparing Fourier path energy to Foot coefficient language | **[M]** packaging |
| **L5** | Under L4, report residual vs \(K/L_{\mathrm{obs}}\) honestly (banked ~2.8% at freeze) | **[E num]** · P13 |
| **L6** | L4 is **not** a derivation that S⁷ geometry forces Foot \(K/L\); reopen as theorem only with a new principle that Foot \(L\) tracks double-loop length while \(K\) does not (or equivalent) | **[O]** residual |
| **L7** | Do **not** merge L4 with: (i) P4-C \(W=1-u\) vs \(1-u/2\); (ii) P11 two-channel energy doubling; (iii) uniform \(\omega\to\omega/2\) shape “fix” | **[E honesty]** |
| **L8** | Do **not** change \(K/L_{\mathrm{obs}}\) by redefining loop length | **[F if claimed]** |

**Freeze sentence (optional reading):**

> Under \(\mathcal{P}_L\) v1 with **L4 ON**: programme may *read* Fourier path-energy \(K/L_{\mathrm{eff}}\) against Foot \(K/L_{\mathrm{obs}}\) via a factor-of-two length-coefficient packaging; **L4 OFF** (default for theorem language) means report \(K/L_{\mathrm{eff}}\) raw and keep residual **OPEN**.

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## 4. Operational numbers (freeze snapshot)

| Quantity | Value (P13 / P3) |
|----------|-----------------:|
| \(K/L_{\mathrm{obs}}\) | 0.39595 |
| \(K/L_{\mathrm{eff}}(\sec\alpha)\) | 0.20360 |
| obs / eff | 1.945 |
| \(2\times K/L_{\mathrm{eff}}\) | 0.4072 |
| rel err of \(2\times\)eff vs obs | ~2.8% |

Recompute with `scripts/investigations/s7_geometry/s7_p13_effective_length_KL.py` if masses or map change.

---

## 5. When to use which mode

| Context | Mode |
|---------|------|
| Fibre \(\omega\), Weber \(\Delta\omega\), MSCT vs geometric length | **Dictionary L0–L1** always |
| Integrity board / “is K/L derived?” | **L4 OFF** · residual OPEN |
| Internal join language / coincidence tracking | **L4 ON** allowed · tag **[M]** |
| Public hub doctrine | **L4 OFF** · state K/L **OPEN**; optional note coincidence only if tagged **[M]** |
| Paper sentence claiming first-principles \(K/L\) | **FORBIDDEN** under v1 |

---

## 6. Forbidden upgrades **[F]**

| ID | Claim |
|----|--------|
| FL1 | “Effective length theorem derives \(K/L_{\mathrm{obs}}\)” |
| FL2 | “Code used double-loop \(\omega\); fix \(\omega\) to half-double closes K/L” |
| FL3 | “\(2\times E\) or sec+csc doubles Fourier \(K/L_{\mathrm{eff}}\)” (P9/P11) |
| FL4 | “P4-C full Weber factor-2 is the same as \(\mathcal{P}_L\)” |
| FL5 | “Loop length changes Foot \(K/L_{\mathrm{obs}}\) from masses” |
| FL6 | Integer Fourier on \(\theta\in[0,4\pi)\) alone extracts Foot \(K/L\) |

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## 7. Kill criteria (reopen only if)

| ID | Trigger |
|----|---------|
| K1 | New geometric principle forces Foot \(L\) coefficient \(\propto L_{\mathrm{double}}\) while \(K\) is pure Z₃ angular **without** mass input |
| K2 | Named functional whose Fourier \(K/L_{\mathrm{eff}}\) equals \(K/L_{\mathrm{obs}}\) within light-census quality **without** post-hoc ×2 and without λ-fit |
| K3 | Proof that L4 reading is forced by Hopf monodromy / MSCT packaging (not optional) |

Until then: **package card stands; theorem OPEN**.

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## 8. Related bank

| Doc | Role |
|-----|------|
| `a_geom_s7_p13_effective_length_KL_results.md` | Probe that motivated \(\mathcal{P}_L\) |
| `a_geom_s7_delta_KL_residual_status.md` | R-δ + R-K/L one-pager |
| `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_o_emc3_package_card.md` | Sister style for soft abs mass |

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## Changelog

| Date | Change |
|------|--------|
| 2026-08-09 | \(\mathcal{P}_L\) v1 freeze from P13 |

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## Agent one-liner

> **\(\mathcal{P}_L\): \(L_{\mathrm{eff}}=L_{\mathrm{geom}}\) [E]; optional \(K/L_{\mathrm{read}}=2\,K/L_{\mathrm{eff}}\) [M]; K/L theorem OPEN; not P4-C; not P11.**