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 |