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]
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.
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.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
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.