# Results — leftover clumping by mass type / \(L_\star\) dents does not save dwarfs

### Lowest-\(g\) needs the most extra \(\Phi\) · stars-only misses gas-rich BTFR · equal \(L_\star\) share overweights dwarfs · \(M_b^\alpha\) is a shop · DF2 vs dSph not a mass-type split · CDM-role stands

**CPT Framework — FT-DENT-SHARE / FT-A0-KIND / FT-DF2-CLAMP / FT-NO-CDM-INPUT / FT-P5-CLAIM-FENCE**  
**Date:** 2026-09-23  
**Status:** **BANKED** · `PASS_FT_DENT_CLUMP_WRONG_END_NOT_DWARF_SAVE`  
**Stimulus:** User — \(N=10^{11}\) \(L_\star\) might mean localised clumping depending on massive mass type, and might explain dwarfs  
**Parents:** `a_geom_ft_dent_share_results.md` · `a_geom_ft_a0_kind_results.md` · `a_geom_ft_df2_clamp_results.md`  
**Script:** `scripts/investigations/s3_geometry/ft_dent_clump.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_dent_clump.json`

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

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## 0. One-line

> **Localisation is the right *kind* of leftover contrast; restricting dents to massive types is the wrong RAR end, and it does not explain dwarfs.** The RAR’s loudest extra \(\Phi\) is in the faintest, lowest-\(g\) systems (\(dSph\) at \(\sim 0.027\,a_0\) wants \(\nu\simeq 6\), not Kepler). If only \(L_\star\) / compact / “massive” objects nucleate leftover clumps, isolated dwarfs are starved — Newton, which they are not. If every dwarf still gets the \(L_\star\) packet \(4.5\times 10^{11}M_\odot\), \(M_{\mathrm{share}}/M_b\simeq 4500\) versus a few hundred even in SMHM contrast. A power \(M_{\mathrm{share}}\propto M_b^{0.61}\) that hits both ends is an interpolator, not a dent-point theorem. BTFR uses stars **plus gas**; a mass-type that is stellar \(T(2,3)\) packing misses gas-rich dwarfs. DF2 and a dSph are the same mass class (\(2\times 10^8\) vs \(10^8\)–\(10^6\)); mass type does not split Newton vs boost. Host leftover as environment is Sciama/EFE, parked. Localised leftover contrast around baryons is the CDM *role*. No CDM input. Prize OPEN.

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## 1. Three ways to read “\(N=10^{11}\) is localisation”

| Reading | Dwarfs | End |
|---------|--------|-----|
| **A.** Only \(L_\star\) nucleate leftover clumps | Isolated dwarfs have no packet | Newton. **Wrong end** (dSphs are deep-MOND). Satellites might sit in a host well — that is a halo, CDM-role, and field dwarfs remain. |
| **B.** Every baryonic well gets the same packet | Dwarfs get \(4.5\times 10^{11}M_\odot\) | \(M/M_b\simeq 4500\). **Too much**, not a \(y\)-map. |
| **C.** Packet depends on mass type / \(M_b^\alpha\) | Fit \(\alpha\simeq 0.61\) through MW share and a \(10^{10}M_\odot\) dwarf contrast | **Shop** (SMHM in leftover clothes). Not derived from leftover \(K\). |

**Finding C1 [E algebra].** RAR polarity: extra \(\Phi\) *grows* as \(g_N\) falls. Massive/compact systems are the UV (Newton). Dwarfs and LSB disks are the IR. A nucleation threshold on “massive type” fires on the UV side.

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## 2. Mass type is not a free handle

**Stars vs gas.** SPARC/BTFR \(M_b=M_\star+M_{\mathrm{gas}}\). Gas-dominated dwarfs lie on the same relation. Leftover that clumps only on stellar \(T(2,3)\) packing would track \(M_\star\), not \(M_b\). Forbidden as a RAR save.

**Book G vs Book C.** Book G is all mass. Diffuse HI is still mass. Book C is not a galactic leftover mill.

**DF2 vs dSph.** DF2 baryons \(\simeq 2\times 10^8 M_\odot\), \(a_{\mathrm{int}}\simeq 0.048\,a_0\) (would want \(\nu\simeq 4.6\) if isolated leftover). Observed \(\sigma\simeq\sigma_N\). A dSph at \(0.027\,a_0\) wants \(\nu\simeq 6\) and *has* extra \(\Phi\). Same mass-type bin; opposite kinematics. Occupancy slogan for DF2 stays fair (FT-DF2-CLAMP); mass-type clumping does not split them. Host-field clamp does not fire at 80–150 kpc. Sciama stays parked.

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## 3. Integrity

| Gate | Verdict |
|------|---------|
| Localised leftover contrast *kind* | **YES** (any \(P\rho\); also CDM-role) |
| \(L_\star\)-only dents pay isolated dSph | **NO** |
| Massive-type clump has RAR polarity | **NO** (wrong end) |
| Stars-only tracks BTFR | **NO** |
| Equal \(L_\star\) share explains dwarfs | **NO** (\(\times 4500\)) |
| \(M_b^\alpha\) derived | **NO** (shop) |
| DF2 vs dSph split by mass type | **NO** |
| Sciama / EFE unparked · CDM inserted · promote | **NO** |
| Prize closed | **NO** |

**Exit:** `PASS_FT_DENT_CLUMP_WRONG_END_NOT_DWARF_SAVE`

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## 4. One-line for agents

> **§FT-DENT-CLUMP: \(L_\star\)/massive-type leftover clumps are the wrong RAR end for dwarfs; equal packet overweights; \(M_b^\alpha\) is a shop; gas-rich BTFR tracks all baryons; DF2 vs dSph not a mass-type split; CDM-role stands; prize OPEN; no promote.**