Research Library · Source Document

a_geom_ft_bend_seq_results.md

Download raw
92 lines·~673 words·a_geom_ft_bend_seq_results.md

This is a raw source document. Integrity tags [E] [M] [O] [F] are coloured for readability. Content is as written in the research file.

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]

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.

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 \(R=\sqrt{6}\,c/H_0\), \(a_\star=c^2/R\simeq 2.31\,a_0\).

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).

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.

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

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.
← Back to Research LibraryCPT Physics Home