# 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]**

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## 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.

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## 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 \(y\).

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## 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.

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## 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.

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## 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`

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## 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.**