S7-P12 — Global δ Selection (after P10 Local Critical)
Domain · hierarchy · pole · joint K/L · residual freeze
CPT Framework — S⁷ generation track
Date: 2026-08-09
Status: S7-P12 BANKED · hierarchy band + pole
[E] · honest global δ
OPEN · circular Model A demoted
Script: scripts/investigations/s7_geometry/s7_p12_global_delta_selection.py
Data: scripts/investigations/s7_geometry/data/s7_p12_global_delta_selection.json
Priors: S7-P6 · P10 · P11 · generation catch-up
Tags:
[E] ·
[E honesty] ·
[M] ·
[O] ·
[F]
0. Gap residual (item 3 / C/E₃)
| | |
|--|--|
|
Unknown | Foot angle \(\delta\approx 132.73^\circ\) from first principles |
|
P10 | \(\rho_{\mathrm{stat}}(\delta)\)
[E]; \(3^3/9\pi\)
local critical ~\(\delta_{\mathrm{obs}}\)
[M] |
|
Attack | Global filters: hierarchy, latitudes, pole, honest C scale, joint K/L |
|
Status | OPEN residual stands — hierarchy band + pole banked
[E]; honest global min ~120° not δ_obs; circular Model A hit demoted |
1. Setup
| Quantity | Value |
|----------|------:|
| \(\delta_{\mathrm{obs}}\) | 132.732331° |
| \(\delta_{\mathrm{pole}}=\arccos(-1/\sqrt{2})\) | 135.000000° |
| \(\delta_{\mathrm{pole}}-\delta_{\mathrm{obs}}\) | 2.267669° |
| \(\rho_{\mathrm{stat}}(\delta_{\mathrm{obs}})\) | 27.754464 |
| K/L_obs (banked) | 0.39595 |
| Hierarchy band | [120.01°, 134.98°] |
Model A: \(E=E_3(\delta)+C\cos(3\delta)\).
Honest tests use \(C=\rho\times E_{\mathrm{ref}}\) with \(E_{\mathrm{ref}}\)
independent of \(\delta_{\mathrm{obs}}\). Using \(E_{\mathrm{ref}}=E_3(\delta_{\mathrm{obs}})\) is circular for derivation.
2. P12H — Hierarchy domain (load-bearing [E])
Requiring Foot masses with \(m_e < m_\mu < m_\tau\) (ℓ-order) and valid S⁴ latitudes restricts \(\delta\) to a band ending at the pole:
-
Band: [120.0109°, 134.9783°]
-
Pure \(E_3\) min on band: 120.0109° (err 12.72° vs obs) —
lower edge ~120°
| Finding | Tag |
|---------|-----|
| Hierarchy band ≈ [120°, 135°) is structural Foot geometry |
[E domain] |
| Pure tension minimises at hierarchy lower edge, not δ_obs |
[E honesty] |
| Same ~12.7° miss as P6 best full-Weber multiplet scan |
[E join P6] |
Honest vs circular C scale
| \(E_{\mathrm{ref}}\) | \(\rho\) | \(\delta^\star\)° | err° | Note |
|---------------------|--------|----------------:|-----:|------|
| E3(120°) | 3^3 | 120.011 | 12.72 | honest |
| E3(120°) | 9
π | 120.011 | 12.72 | honest |
| E3(120°) | 47 | 120.011 | 12.72 | honest |
| E3(90°) | 3^3 | 120.011 | 12.72 | honest |
| E3(90°) | 9
π | 120.011 | 12.72 | honest |
| E3(90°) | 47 | 120.011 | 12.72 | honest |
| 1.0 | 3^3 | 120.011 | 12.72 | honest |
| 1.0 | 9
π | 120.011 | 12.72 | honest |
| 1.0 | 47 | 120.011 | 12.72 | honest |
| E3(δ_obs) CIRCULAR | 3^3 | 132.712 | 0.02 | CIRCULAR |
| E3(δ_obs) CIRCULAR | 9
π | 132.756 | 0.02 | CIRCULAR |
| E3(δ_obs) CIRCULAR | 47 | 132.734 | 0.00 | CIRCULAR |
| Finding | Tag |
|---------|-----|
| Honest \(E_{\mathrm{ref}}\) (no \(\delta_{{\mathrm{{obs}}}}\)) → min ~120° |
[E / F as δ theorem] |
| Circular \(E_{{\mathrm{{ref}}}}=E_3(\delta_{{\mathrm{{obs}}}})\) → min ~δ_obs |
[E process / F as derivation] |
3. P12A–C — Critical census (summary)
| \(\rho\) | n crit | hier+lat | min ∩ | near obs ≤1° |
|--------|-------:|---------:|------:|-------------:|
| 3^3 | 0 | 0 | 0 | 0 |
| 9
π | 0 | 0 | 0 | 0 |
| 47 | 0 | 0 | 0 | 0 |
| ρ_stat(δ_obs) | 0 | 0 | 0 | 0 |
Geometric \(\rho\in\{27,9\pi,28\}\) are
not free critical points at \(\delta_{{\mathrm{{obs}}}}\) (stationarity residual large); only \(\rho=\rho_{{\mathrm{{stat}}}}(\delta_{{\mathrm{{obs}}}})\) is.
4. P12D — Domain global mins (with circular Model A note)
Unfiltered min ~15°. Hier+lat with
circular \(C=\rho E_3(\delta_{{\mathrm{{obs}}}})\) lands near δ_obs —
demoted (P12H). Model B (local \(\rho E_3(\delta)\)) → 120°.
| \(\rho\) | domain | \(\delta^\star\)° | err° |
|--------|--------|----------------:|-----:|
| 3^3 | all_valid | 15.000 | 117.73 |
| 3^3 | hier+lat | 132.710 | 0.02 |
| 3^3 | hier+lat+!pole | 132.710 | 0.02 |
| 9
π | all_valid | 15.000 | 117.73 |
| 9π | hier+lat | 132.747 | 0.01 |
| 9
π | hier+lat+!pole | 132.747 | 0.01 |
| 47 | all_valid | 15.000 | 117.73 |
| 4
7 | hier+lat | 132.739 | 0.01 |
| 47 | hier+lat+!pole | 132.739 | 0.01 |
| ρ_stat(δ_obs) | all_valid | 15.000 | 117.73 |
| ρ_stat(δ_obs) | hier+lat | 132.732 | 0.00 |
| ρ_stat(δ_obs) | hier+lat+!pole | 132.732 | 0.00 |
5. P12E — Pole geometry [E]
Foot denominator \(1+\sqrt{2}\cos\delta=0\) at \(\delta_{\mathrm{pole}}=\arccos(-1/\sqrt{2})=135^\circ\). \(E_3\to\infty\). Observed \(\delta\) sits
2.2677° below the pole.
| Finding | Tag |
|---------|-----|
| Pole is structural |
[E] |
| δ_obs near-polar (data) |
[E] |
| Pole alone selects δ_obs uniquely |
[F if claimed] |
6. P12F — Joint K/L
| \(\rho\) | \(K/L_{{\mathrm{{eff}}}}\) | rel err vs 0.396 |
|--------|-------------------------:|------------------:|
| 3^3 | 0.6916 | 74.7% |
| 9
π | 0.6824 | 72.3% |
| 47 | 0.6843 | 72.8% |
| ρ_stat(δ_obs) | 0.6861 | 73.3% |
| Finding | Tag |
|---------|-----|
| Geometric ρ does not repair Fourier K/L |
[E / F joint] |
7. Integrity board
| Claim | Tag |
|-------|-----|
| ρ_stat(δ) closed form (P10) |
[E] |
| 3³/9π local critical ~δ_obs (P10) |
[M] |
| Hierarchy band [~120°, 135°) structural | [E domain] |
| Pole arccos(-1/√2)=135° structural | [E geometry] |
| Pure E3 min on hier band = δ_obs |
[F] — lands ~120° |
| Honest C scale (no δ_obs) min = δ_obs |
[F] |
| Circular C=ρ E3(δ_obs) min ≈ δ_obs | [E process / F as derivation] |
| Model B local ρ E3(δ) min = δ_obs |
[F] — ~120° |
| Energy Fourier KL_eff = KL_obs |
[F] |
| Fit C / use E3(δ_obs) scale as derivation | [FORBIDDEN] |
| 3³ unique theorem for δ | [F if claimed] |
| First-principles δ without mass input | [O still] |
8. Verdict
1.
P12H [E]: Hierarchy band [~120°, 135°) and pole at 135° are structural.
2.
P12H [E honesty]: Honest energies (pure \(E_3\), honest C scale, Model B) minimise at
~120° (err ~12.7°), not δ_obs — same class as P6.
3.
P12H [F as derivation]: Model A with \(C=\rho E_3(\delta_{{\mathrm{{obs}}}})\) that hits δ_obs is
circular (mass scale in C).
4.
P12A–C [E]: Geometric \(\rho\in\{27,9\pi,28\}\) are not free stationarity roots at δ_obs.
5.
P12F [F]: Energy Fourier K/L still fails.
6.
Residual R-δ [O]: first-principles Foot angle remains open (with K/L).
One-liner: Hierarchy band and the 135° Foot pole are real geometry; honest multiplet energy still sits at ~120°, and any Model A hit on δ_obs that uses \(E_3(\delta_{{\mathrm{{obs}}}})\) in C is circular — global first-principles δ remains OPEN.
9. Do not
- Use \(E_3(\delta_{{\mathrm{{obs}}}})\) in C and call global derivation
- Sell hierarchy band alone as δ theorem
- Sell \(3^3\) / \(9\pi\) as unique global δ theorem
- Fit C; half-W; golden τ; α from this note
10. Related bank
| Doc | Role |
|------|------|
|
a_geom_s7_p10_fano_line_embedding_C_ratio_results.md | local ρ_stat |
|
a_geom_s7_p6_z3_multichannel_weber_results.md | multiplet E3; C~28 E3 |
|
a_geom_s7_p11_joint_fourier_KL_doubling_results.md | K/L doubling F |
|
a_geom_s7_generation_catchup_p1_p4c.md | track map |
Changelog
| Date | Change |
|------|--------|
| 2026-08-09 | S7-P12: hierarchy band + pole; circular Model A demoted; R-δ OPEN |