# Results — FT-BEND sequel: dropping the graph does not remove the neck

### Parametric 3-volume still has \(\rho_{\min}=\sqrt{mR}\simeq 5\,\mathrm{kpc}\) · \(0.1\,\mathrm{kpc}\) has no 2-sphere · \(50\,\mathrm{kpc}\) is Kepler · leftover CMC is \(10^{-6}\) · 3-Hilbert is the wrong book · \(\nu(y)\) not extracted · prize OPEN

**CPT Framework — FT-BEND / FT-CONST-ROUTE / FT-WE / FT-CONST-PRIZE / Book G**  
**Date:** 2026-09-22  
**Status:** **BANKED** · `PASS_FT_BEND_SEQ_KILL_NECK`  
**Stimulus:** User — continue with the FT-BEND sequel (non-graph 3-surface in leftover \(B^4\), named energy, survive/kill at \(0.1\), \(8.5\), \(50\,\mathrm{kpc}\))  
**Parents:** FT-BEND · FT-CONST-ROUTE · FT-WE · BP-1  
**Script:** `scripts/investigations/s3_geometry/ft_bend_seq.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_bend_seq.json` · `ft_bend_seq.png`

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

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## 0. One-line

> **Kill, as pre-registered.** A graph over leftover \(3\)-space going vertical at \(r_{\mathrm{vert}}=\sqrt{mR}\simeq 5.01\,\mathrm{kpc}\) was a coordinate complaint. The same named energy — \(3\)-volume, vacuum \(H=0\) — as a parametric generating curve \((\rho(s),w(s))\) still has a **neck**: \(\sin\theta=C/\rho^2=1\) at \(\rho_{\min}=r_{\mathrm{vert}}\). There is no \(2\)-sphere smaller than \(5\,\mathrm{kpc}\) for \(M=5\times 10^{10}M_\odot\). Folding onto a second sheet does not restore \(\rho<\rho_{\min}\). S2 sits inside the Sgr A* neck (\(r_{\mathrm{vert}}\simeq 0.045\,\mathrm{kpc}\)). Where the surface exists, the proper-slope readout \(g=a_\star\,\mathrm{d}w/\mathrm{d}s\) is **identically** \(g_N\); the graph-slope readout enhances *toward the neck* (\(\nu=1.066\) at \(8.5\,\mathrm{kpc}\), \(\nu=1.00005\) at \(50\,\mathrm{kpc}\)). Leftover CMC \(H_{\mathrm{sum}}=3/R\) adds \(\rho/R\sim 5\times 10^{-6}\) at \(50\,\mathrm{kpc}\). Induced \(3\)-Hilbert is \(3\)D vacuum GR (flat; no well). Survive (centre Newton **and** \(50\,\mathrm{kpc}\) boosted) **fails**. \(\nu(y)\) not extracted. Fat \(r_y\) zone remains the live constitutive route. No \(\mu\) shop.

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## 1. Pre-registered gates

| Gate | Need | Result |
|------|------|--------|
| \(0.1\,\mathrm{kpc}\) Newton | surface exists, \(0.8<\nu<1.2\) | **NO surface** (\(\sin\theta\simeq 2510\)) |
| \(50\,\mathrm{kpc}\) boosted | \(\nu>2\) | \(\nu_{\mathrm{graph}}=1.00005\), \(\nu_{\mathrm{proper}}=1\) |
| Survive | both | **NO** |
| Kill | centre wrecked **or** outskirts Kepler | **YES** (both) |

MW \(M=5\times 10^{10}M_\odot\), leftover clothes \(R=\sqrt{6}\,c/H_0\), \(a_\star=c^2/R\simeq 2.31\,a_0\).

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## 2. Why “not a graph” is not a new energy

Vacuum \(3\)-volume: \(\rho^2\,\mathrm{d}w/\mathrm{d}s=C=mR\). Then \(\sin\theta=C/\rho^2\) and \(\rho_{\min}=\sqrt{C}\). That \(\rho_{\min}\) is **areal**. A catenoid-like fold still has no smaller \(2\)-sphere.

| \(r\) (kpc) | surface | \(\nu\) (graph \(\mathrm{d}w/\mathrm{d}\rho\)) | \(\nu\) (proper \(\mathrm{d}w/\mathrm{d}s\)) |
|-------------|---------|-----------------------------------------------|-----------------------------------------------|
| \(0.1\) | **no** | — | — |
| \(8.5\) | yes | \(1.066\) | \(1\) |
| \(50\) | yes | \(1.00005\) | \(1\) |

**Finding S1 [E algebra].** Proper slope of the \(H=0\) surface **is** Newton wherever it exists. Graph-slope blow-up is the neck, i.e. the **wrong RAR end** (UV/transition), not the outskirts.

**Finding S2 [E].** Sgr A* \(4.1\times 10^6 M_\odot\): \(r_{\mathrm{vert}}\simeq 0.045\,\mathrm{kpc}\). S2 periapsis \(\sim 120\,\mathrm{AU}\) is inside the neck. The UV gate fails on a named orbit, not only on a \(0.1\,\mathrm{kpc}\) slogan.

Leftover CMC (\(\sin\theta=\rho/R+C/\rho^2\)): neck still \(\simeq 5.01\,\mathrm{kpc}\); at \(50\,\mathrm{kpc}\) the extra \(\rho/R\) is \(\sim 5\times 10^{-6}\) against \(C/\rho^2\simeq 0.01\). Not an IR tail.

Induced \(3\)-Hilbert: \(3\)D vacuum Einstein \(\Rightarrow\) Riemann \(=0\Rightarrow\rho=s\). No dent. Wrong book (Book G is not \(3\)D EH).

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## 3. What this closes vs what stays live

| Branch | Status |
|--------|--------|
| Graph \(u=(R/c^2)\Phi\) | already **wrong end** (FT-BEND) |
| Parametric \(3\)-volume / leftover CMC | **KILL** (this mill) |
| \(3\)-Hilbert of the slice | **wrong book** |
| \(4\)D Gauss / Willmore | still **forbidden / parked** |
| Fat yield zone \(\sim r_y\), ODE in \(y=g/a_0\) | **still LIVE** (FT-CONST-ROUTE B) |
| Analytic local \(W(E)\) | still **CLOSED** |

Do not retry \(3\)-volume in another coordinate. Do not extract a \(\nu(y)\) from the neck.

Figure: `scripts/investigations/s3_geometry/data/ft_bend_seq.png`.

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| Graph dropped | **YES** |
| Neck gone | **NO** (\(\rho_{\min}=5.01\,\mathrm{kpc}\)) |
| Centre Newton | **NO** |
| \(50\,\mathrm{kpc}\) extra \(\Phi\) | **NO** |
| \(\nu(y)\) extracted | **NO** |
| \(\mu(x)\) / SPARC \(I\) / CDM | **NO** |
| Prize closed | **NO** |

**Exit:** `PASS_FT_BEND_SEQ_KILL_NECK`

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## 5. One-line for agents

> **§FT-BEND-SEQ: parametric 3-volume/CMC still necks at \(\sqrt{mR}\); \(0.1\,\mathrm{kpc}\) has no 2-sphere; \(50\,\mathrm{kpc}\) Kepler; \(\nu(y)\) not extracted; fat \(r_y\) zone remains the live constitutive route; prize OPEN.**