\(\mathcal{P}_L\) — 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]
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 mayread* 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.
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 \(K/L\) |
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.
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 |
Changelog
| Date | Change | |------|--------| | 2026-08-09 | \(\mathcal{P}_L\) v1 freeze from P13 |
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.