# Results — \(M_{\mathrm{container}}/N_{\mathrm{dents}}\) is an \(L_\star\) coincidence, not a halo theorem

### Hubble-ball \(M_{\mathrm{crit}}/10^{11}\) rhymes with MW halo · \(N\) not derived · equal share wrecks dwarfs · not 4-volume mass · leftover well-dust remains CDM-role

**CPT Framework — FT-BULK-ACCUM / FT-NO-CDM-INPUT / FT-P5-CLAIM-FENCE / FT-ELAS-HALO**  
**Date:** 2026-09-23  
**Status:** **BANKED** · `PASS_FT_DENT_SHARE_KIND_NOT_HALO_THEOREM`  
**Stimulus:** User — halo mass scale is leftover 4D container mass divided by the number of baryonic dent-points  
**Parents:** `a_geom_ft_bulk_accum_results.md` · `a_geom_ft_no_cdm_input_freeze.md`  
**Script:** `scripts/investigations/s3_geometry/ft_dent_share.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_dent_share.json`

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

---

## 0. One-line

> **The rhyme is real for one chosen \(N\); it is not a theorem and it is not 4-volume mass.** FT-BULK-ACCUM computed **3-space** Hubble-ball \(M_{\mathrm{crit}}=c^3/(2GH_0)\simeq 4.5\times 10^{22}M_\odot\) shared among **clothes** \(N=10^{11}\): \(\simeq 4.5\times 10^{11}M_\odot\). That sits next to an \(L_\star\) halo *contrast* \(\sim 10^{12}M_\odot\) (factor \(\sim 0.45\)). \(\mathrm{Vol}_4=\pi^2 R^4/2\) is not a mass. Filling leftover \(S^3\) with \(\rho_c\) is \(\times 69\) larger. \(N\) is not a dent-point theorem: protons \(N_b\sim 10^{80}\) make the share vanish; clusters \(N\sim 10^8\) make it too big; dwarfs included at \(10^{12}\) drop it \(\times 10\). Equal share gives \(M_{\mathrm{share}}/M_b\simeq 9\) for MW and \(\simeq 4500\) for a \(10^8 M_\odot\) dwarf (contrast SMHM goes the other way on the dwarf). Share \(\propto M_b\) is just \((1-\Omega_b)/\Omega_b\simeq 19\), a constant ratio, not \(\nu(y)\). Sum of shares \(=M_{\mathrm{crit}}\) is a **partition**, not a prediction. Leftover as well-dust remains the CDM *role*. Drain not exhibited. P5 stands. Prize OPEN.

---

## 1. What was actually computed

| Object | Mass / \(M_\odot\) | / \(10^{11}\) |
|--------|---------------------|---------------|
| Hubble-ball \(M_{\mathrm{crit}}\) | \(4.47\times 10^{22}\) | \(4.47\times 10^{11}\) |
| \(\rho_c\) fill of leftover \(S^3(R_{\sqrt{6}})\) | \(3.10\times 10^{24}\) | \(3.10\times 10^{13}\) |
| 4-ball volume \(\pi^2 R^4/2\) | — | **not a mass** |

**Finding S1 [E honesty].** “Total mass of the 4D container” was not a mill output. 4-volume has no mass until a 4-density and a 4D \(G\) are named. That would be a shop.

---

## 2. \(N\) is the interpolator

| Dent-point clothes | Hubble share / \(M_\odot\) |
|--------------------|----------------------------|
| \(10^{11}\) \(L_\star\) | \(4.5\times 10^{11}\) |
| \(10^{12}\) with dwarfs | \(4.5\times 10^{10}\) |
| \(10^8\) clusters | \(4.5\times 10^{14}\) |
| \(N_b\simeq 1.81\times 10^{80}\) protons | \(10^{-58}\) |

**Finding S2 [E].** The halo number appears when \(N\) is chosen to be the number of \(L_\star\) galaxies. That is the same move as choosing SPARC \(I\).

Equal Hubble share versus baryons:

| | \(M_{\mathrm{share}}/M_b\) | contrast \(M_{200}/M_b\) (not input) |
|--|----------------------------|--------------------------------------|
| MW \(5\times 10^{10}M_\odot\) | \(9\) | \(\sim 20\) |
| dwarf \(10^8 M_\odot\) | \(4500\) | \(\sim 100\) |

Equal share **overweights** dwarfs relative to that contrast. It is not SMHM and not the RAR.

---

## 3. Integrity

| Gate | Verdict |
|------|---------|
| \(L_\star\), \(N=10^{11}\) halo-scale rhyme | **KIND** |
| \(N\) derived / dent-point defined | **NO** |
| 4-volume mass computed | **NO** |
| Equal share is a \(y\)-map or SMHM | **NO** |
| \(\sum\) shares \(=M_{\mathrm{crit}}\) is a prediction | **NO** (partition) |
| CDM inserted to close a residual | **NO** |
| Prize closed / promote | **NO** |

**Exit:** `PASS_FT_DENT_SHARE_KIND_NOT_HALO_THEOREM`

---

## 4. One-line for agents

> **§FT-DENT-SHARE: \(M_{\mathrm{crit}}/10^{11}\simeq 4.5\times 10^{11}M_\odot\) is an \(L_\star\) coincidence; \(N\) not derived; not 4-volume mass; equal share fails dwarfs; partition not a prediction; CDM-role stands; prize OPEN; no promote.**