Results — \(\mathrm{Vol}_4=\mathrm{const}\) is a leftover wall, not a \(y\)-map
One global Lagrange multiplier · galaxy dent \(\sim 10^{-25}\) of leftover \(B^4\) · \(\delta R/R\sim 10^{-26}\) · bulk buoyancy is the same object · GE-J home · prize OPEN
CPT Framework — FT-PHI-OPUS46-SCORE / FT-BEND / FT-GEJ-A0-COMPUTE / FT-P5-CLAIM-FENCE / FT-CONST-PRIZE Date: 2026-09-23 Status: BANKED / FAIL as RAR derivation ·PASS_FT_VOL4_MULTIPLIER_WALL_NOT_YMAP
Stimulus: Licensed fail gate: is \(\mathrm{Vol}_4=\mathrm{const}\) a local \(y\mapsto g_{\mathrm{obs}}\)? Bulk buoyancy fenced as the same leftover 4-pressure.
Parents: a_geom_ft_phi_opus46_score_results.md · a_geom_ft_bend_incomp_results.md · a_geom_ft_gej_a0_compute_results.md
Script: scripts/investigations/s3_geometry/ft_vol4_multiplier.py
Data: scripts/investigations/s3_geometry/data/ft_vol4_multiplier.json
Tags: [E algebra] · [E honesty] · [Programme] · [O residual]
0. One-line
FAIL as a RAR derivation, as pre-registered. \(\lambda(\mathrm{Vol}_4-V_0)\) is one number for leftover. Its Euler–Lagrange is a uniform 4-pressure on \(S^3\subset B^4\): leftover-normal wall (GE-J home), not a spherical \(\nu(y)\) around a disk. A MW-class Newtonian dimple is \(\sim 1.9\times 10^{-25}\) of leftover 4-volume at \(8.5\,\mathrm{kpc}\) (depth \(R\) bound; depth \(GM/c^2\) is \(4.3\times 10^{-38}\)). Holding \(\mathrm{Vol}_4\) then shifts leftover \(R\) by \(\delta R/R\sim 5\times 10^{-26}\). That is not extra \(\Phi\). Bulk buoyancy, if it means Archimedes from leftover 4-pressure, is this multiplier: constant \(\lambda\) has no in-slice gradient, so no galactic boost. Do not unpark GE-J as \(m_e\). Prize OPEN.
1. Why a global constraint cannot mint \(\nu(y)\)
Leftover 4-ball: \(\mathrm{Vol}_4=\pi^2 R^4/2\). The constraint \(\mathrm{Vol}_4=\mathrm{const}\) introduces a single Lagrange multiplier \(\lambda\). Variation in the leftover normal is a uniform 4-pressure. Israel readout of the same umbilic geometry is already the GE-J board (\(0.18\)–\(2.3\,a_0\) by convention), acceleration along the normal to leftover \(S^3\), not a per-galaxy kernel.
A winning constitutive law must reduce to a universal \(y=g_N/a_0\mapsto g_{\mathrm{obs}}\) with no leftover \(M\) in the EL. One leftover-wide \(\lambda\) has no access to a galaxy’s \(y\).
2. Dent budget (MW \(M=5\times 10^{10}M_\odot\))
Graph 4-volume of a Newtonian dimple, \(u=(R/c^2)GM/r\), bound \(\delta V_4\sim 2\pi m R\, r^2\) (FT-BEND depth-\(R\)). Compensation \(\mathrm{d}\mathrm{Vol}_4/\mathrm{Vol}_4=4\,\delta R/R\).
| \(r\) (kpc) | dent / \(B^4\) (depth \(R\)) | \(\delta R/R\) | needed BTFR boost | |-------------|------------------------------|----------------|-------------------| | 8.5 | \(1.91\times 10^{-25}\) | \(4.8\times 10^{-26}\) | \(1.12\) | | 50 | \(6.60\times 10^{-24}\) | \(1.6\times 10^{-24}\) | \(6.6\) | | 300 | \(2.37\times 10^{-22}\) | \(5.9\times 10^{-23}\) | \(39\) |
Finding V1 [E algebra]. One galaxy does not spend leftover 4-volume. Matching FT-BEND Finding B5. Finding V2 [E honesty]. Even the generous depth-\(R\) bound shifts \(a_\star=c^2/R\) by \(\sim 10^{-26}\). The RAR is not a container-radius recoil.3. Bulk buoyancy fence
Archimedes in leftover 4-volume: force \(\propto\lambda\times\) (displaced 4-volume), along the leftover normal. Constant \(\lambda\) has no 3-gradient, so there is no in-slice boost to \(g(r)\). A 3-gradient of 4-pressure would be a new field (shop) or the constitutive prize. Displaced volume is the dent budget above.
Finding V3 [E]. Bulk buoyancy is not a second route. It is this mill’s object with an Archimedes name. Parked with the wall, not opened as a 4-fluid.4. Integrity
| Gate | Verdict | |------|---------| | \(\mathrm{Vol}_4\) multiplier is a local \(y\)-map | NO | | Home is leftover wall / GE-J | YES | | Galaxy dent pays RAR via \(\delta R\) | NO | | Bulk buoyancy is a new RAR route | NO | | GE-J unparked / \(\mathcal{L}(g,K)\) written | NO | | Prize closed / promote | NO |
Exit:PASS_FT_VOL4_MULTIPLIER_WALL_NOT_YMAP
5. One-line for agents
§FT-VOL4-MULTIPLIER: \(\lambda(\mathrm{Vol}_4-V_0)\) is leftover wall not \(\nu(y)\); dent \(\sim 10^{-25}\) of \(B^4\); \(\delta R/R\sim 10^{-26}\); bulk buoyancy = same Archimedes-along-normal; GE-J not unparked; prize OPEN; no promote.