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]

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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.

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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 \(r(\varphi)\).

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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 \(\sim 0.40\).

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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)). `

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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.

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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 |

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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.

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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 |

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Changelog

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