# Results — Poisson-embedding with named lab energies FAILS the RAR gates

### Linear \(P\rho\) is Newton at kpc · conformal backreaction \(O(\Phi/c^2)\sim 10^{-8}\) · graph already necks · 3-Hilbert / Hencky / \(\mathrm{Vol}_4\) do not survive · remainder is constitutive · prize OPEN

**CPT Framework — FT-PHI-OPUS46-SCORE / FT-S3-PHI / FT-BEND-SEQ / FT-WE / BP-1 / FT-VOL4-MULTIPLIER / FT-JACOBI-UMBILIC / FT-METRIC-ELAS**  
**Date:** 2026-09-23  
**Status:** **BANKED / BRANCH CLOSED as derivation** · `PASS_FT_POISSON_EMBED_NAMED_ENERGIES_FAIL`  
**Stimulus:** Licensed fail gate: self-consistent \(\Delta_g\Phi=4\pi G\,P\rho\) on a 3-geometry deformed only by a **named** leftover 4-energy.  
**Parents:** `a_geom_ft_phi_opus46_score_results.md` · `a_geom_ft_s3_phi_results.md` · `a_geom_ft_bend_seq_results.md` · `a_geom_ft_vol4_multiplier_results.md` · `a_geom_ft_jacobi_umbilic_results.md`  
**Script:** `scripts/investigations/s3_geometry/ft_poisson_embed.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_poisson_embed.json`

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

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## 0. One-line

> **FAIL, as pre-registered. Close this branch as a derivation.** Coupled Poisson on a deformed leftover 3-geometry, using only 4-energies already in the lab, does not pay UV Newton **and** a \(50\,\mathrm{kpc}\) boost. Linear \(P\rho\) on round \(S^3\) is Coulomb at \(r\ll R\). Conformal / lapse backreaction is \(O(\Phi/c^2)\simeq 5\times 10^{-8}\) at \(50\,\mathrm{kpc}\) against a needed BTFR factor \(\simeq 6.6\). Graph 3-volume / CMC still necks at \(\sqrt{mR}\simeq 5.01\,\mathrm{kpc}\) (no \(0.1\,\mathrm{kpc}\) 2-sphere). Induced 3-Hilbert is flat vacuum. Hencky is Newton+1PN. \(\mathrm{Vol}_4\) is a leftover wall. The only non-perturbative slope \(g/a_\star=O(1)\) is the killed neck, and the Jacobi mill found no extra buckling branch. Compactness plus embedding does **not** non-linearize Poisson enough. Remainder is a constitutive map of the metric (standing prize). No \(\mathcal{L}(g,K)\) invented. P5 stands. No promote.

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## 1. Pre-registered gates

Survive if \(0.1\,\mathrm{kpc}\) is Newton **and** \(50\,\mathrm{kpc}\) is boosted. Kill if the centre wrecks **or** the outskirts stay Kepler, if the interpolator depends on leftover \(M\), or if the only new scale is \(\ell=1\simeq 4a_0\). Do not retry 3-volume/CMC in another coordinate. Do not close Coma/Bullet.

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## 2. Expansion parameters (MW \(M=5\times 10^{10}M_\odot\))

| \(r\) (kpc) | \(\Phi/c^2\) | \(g_N/a_\star\) | graph exists | \(O((r/R)^2)\) | needed BTFR |
|-------------|--------------|-----------------|--------------|----------------|-------------|
| 0.1 | \(2.4\times 10^{-5}\) | \(2.5\times 10^3\) | **no** | \(9\times 10^{-17}\) | \(1\) |
| 8.5 | \(2.8\times 10^{-7}\) | \(0.35\) | yes | \(6.6\times 10^{-13}\) | \(1.12\) |
| 50 | \(4.8\times 10^{-8}\) | \(0.010\) | yes | \(2.3\times 10^{-11}\) | \(6.6\) |
| 300 | \(8.0\times 10^{-9}\) | \(2.8\times 10^{-4}\) | yes | \(8.2\times 10^{-10}\) | \(39\) |

**Finding E1 [E algebra].** \(\Phi/c^2\) and \((r/R)^2\) are small data on every RAR radius. Iterating \(\Phi_{n+1}=\mathrm{Green}(g[\Phi_n])\,P\rho\) around Newton has no runaway: the correction is \(O(\Phi/c^2)\) times Newton.

**Finding E2 [E].** The one \(O(1)\) parameter is the graph slope \(g/a_\star\). That is FT-BEND’s conversion \(u=(R/c^2)\Phi\). It is \(O(1)\) only where the graph necks. Self-consistency in *that* variable is the killed embedding, not a new Poisson.

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## 3. Named 4-energies (no new \(W\))

| Energy | \(0.1\,\mathrm{kpc}\) Newton | \(50\,\mathrm{kpc}\) boosted | Note |
|--------|:---------------------------:|:----------------------------:|------|
| Linear \(P\rho\) on round \(S^3\) | yes | **no** | Coulomb; compactness \(10^{-12}\) |
| Conformal / lapse \(\Delta_g\) | yes | **no** | \(O(\Phi/c^2)\sim 10^{-8}\) |
| Graph 3-volume / leftover CMC | **no** | **no** | BEND-SEQ neck |
| Induced 3-Hilbert | **no** | **no** | 3D vacuum GR, no well |
| Hencky 4-metric | yes | **no** | BP-1 Newton+1PN |
| \(\mathrm{Vol}_4=\mathrm{const}\) | yes | **no** | wall, not \(y\)-map |

None survive. Umbilic Jacobi (companion mill) supplies no extra non-perturbative branch.

**Finding E3 [E honesty].** This closes “compactness + embedding non-linearizes Poisson” for **named lab energies**. It is not a uniqueness theorem against every conceivable 4-energy. Writing a new \(\mathcal{L}(g,K)\) to survive is the \(\mu\) shop in embedding clothes, and is refused. The standing remainder is constitutive non-analyticity of the metric (FT-METRIC-ELAS), unpaid.

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| Named-energy Poisson-embed pays RAR | **NO** |
| Conformal bootstrap | **NO** |
| 3-volume/CMC retried as a new coordinate | **NO** |
| Branch closed as a derivation | **YES** |
| Remainder constitutive | **YES** |
| P5 cluster claimed | **NO** |
| \(\mathcal{L}(g,K)\) written / prize closed / promote | **NO** |

**Exit:** `PASS_FT_POISSON_EMBED_NAMED_ENERGIES_FAIL`

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## 5. One-line for agents

> **§FT-POISSON-EMBED: self-consistent \(P\rho\) Poisson with named lab 4-energies FAILS (Newton or neck); conformal \(O(\Phi/c^2)\sim 10^{-8}\); compactness plus embedding does not mint the RAR; remainder constitutive; prize OPEN; no promote.**