# Results — P6: leftover \(S^3\) has no named 3-wave vertex; downconversion PARK

### Linear \(\Delta_{S^3}\) · SO(4) triangle \(\neq\) a coupling · GeV mode \(\ell\sim 10^{42}\) · \(\sim 140\) binary splits if a vertex existed · \(\ell=1\) is \(10^{-33}\,\mathrm{eV}\)

**CPT Framework — FT-COHERENCE-FALSIFY-PLAN / FT-S3-SPECTRUM / FT-T23-COOL-NO-EXP / FT-T23-SINK-RANK**  
**Date:** 2026-09-23  
**Status:** **PARKED** · `PASS_FT_P6_DOWNCONVERT_PARK_NO_VERTEX`  
**Stimulus:** User — proceed with P6  
**Parents:** `a_geom_ft_coherence_falsify_plan.md` · `a_geom_ft_s3_spectrum_results.md` · `a_geom_ft_t23_cool_no_exp_results.md`  
**Script:** `scripts/investigations/s3_geometry/ft_p6_downconvert.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_p6_downconvert.json`

Tags: **[E algebra]** · **[E honesty]** · **[Programme]** · **[O residual]**

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## 0. One-line

> **PARK.** Named leftover geometry is the linear Laplace–Beltrami operator: free waves, **no cubic vertex**. SO(4) Clebsch–Gordan for \((\ell/2,\ell/2)\) would only tell *which* modes could couple **if** a vertex existed; it does not mint one. A \(1\,\mathrm{GeV}\) quantum on leftover \(S^3\) sits at \(\ell\sim 1.64\times 10^{42}\). Direct dump to \(\ell=1\) is not a 3-wave (\(10\not\to 1+1\)). A binary shower would need \(\sim 140\) splittings **and** a coupling. \(\ell=1\) is \(\simeq 1.1\times 10^{-33}\,\mathrm{eV}\), not a one-step GeV sink. A cubic is either the constitutive prize (P5) or Planck-suppressed GR — neither is this mill. Post-window sinks lose downconversion as a named process.

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## 1. Linear vs selection rules

| Object | Vertex? |
|--------|---------|
| \(\partial_t^2\Phi = c^2\Delta_{S^3}\Phi\) | **NO** |
| \(\int Y_{\ell_1}Y_{\ell_2}Y_{\ell_3}\) on \(S^3\) | Selection only; \(\lvert\ell_1-\ell_2\rvert\le\ell_3\le\ell_1+\ell_2\) |
| Sample \(\ell=10\to 1+1\) | SO(4) **false** |
| Sample equal split \(10\to 5+5\) | SO(4) true; **energy** \(\sqrt{\ell(\ell+2)}\) **false** at these \(\ell\) |

**Finding V1 [E algebra].** Free leftover harmonics do not downconvert. **Finding V2.** Even with a vertex, \(\omega=c\sqrt{\ell(\ell+2)}/R\) is not harmonic in \(\ell\) at low \(\ell\); kinematics are not a free-for-all cascade.

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## 2. Integrity

| Gate | Verdict |
|------|---------|
| Named 3-wave vertex | **NO** |
| Downconversion as post-window sink | **PARK** |
| \(\ell=1\) GeV dump | **NO** |
| Invented cubic / \(\mu\) shop | **NO** |
| P7 opened | **NO** (plan: only if still standing; P6 was the mill asked) |

**Exit:** `PASS_FT_P6_DOWNCONVERT_PARK_NO_VERTEX`

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## 3. One-line for agents

> **§FT-P6-DOWNCONVERT: leftover \(\Delta_{S^3}\) is linear — no 3-wave vertex; downconversion PARK; \(\ell_{\mathrm{GeV}}\sim 10^{42}\); \(\ell=1\) not a GeV sink.**