# Results — well elastic \(T_{00}\) is field energy \(O(M\Phi/c^2)\), not \(M_{\mathrm{crit}}/N\)

### \(r_{\mathrm{loc}}\sim R\) is the yield at \(r_y\sim 8\,\mathrm{kpc}\) · IR-rail energy to leftover \(R\) is \(0.22\,M_b\) · need \(41\,R\) of rail to hit \(4.5\times 10^{11}M_\odot\) · gas is baryons · gravity-of-gravity is 1PN · prize OPEN

**CPT Framework — FT-DENT-SHARE / FT-YIELD-GEOM / FT-METRIC-ELAS / BP-1 / FT-BULLET / FT-NO-CDM-INPUT**  
**Date:** 2026-09-23  
**Status:** **BANKED** · `PASS_FT_ELAS_HALO_FIELD_ENERGY_NOT_MH_OVER_N`  
**Stimulus:** User — not extra-dimensional dust; leftover is already elastic; \(T_{00}\) of the 4D distortion integrated to saturation \(r_{\mathrm{loc}}\sim R\) *is* the \(4.5\times 10^{11}M_\odot\) halo; collisionless because anchored to \(T(2,3)\) wells not gas; photons see curvature; budget is container critical mass  
**Parents:** `a_geom_ft_dent_share_results.md` · `a_geom_ft_yield_geom_results.md` · `a_geom_ft_metric_elas_results.md` · `a_geom_bp1_two_well_hencky_overlap_results.md` · `a_geom_ft_bullet_results.md`  
**Script:** `scripts/investigations/s3_geometry/ft_elas_halo.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_elas_halo.json`

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

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## 0. One-line

> **The elastic medium is the right ontology; the integral is the wrong mass.** Leftover as Hencky / constitutive 4-geometry is already the framework. The mass-energy of a *well’s* distortion is gravitational field energy, \(\int g^2/(8\pi G)\), which is \(O(M\Phi/c^2)\) and **negative** as binding. Newton from \(1\,\mathrm{kpc}\) to leftover \(R\): \(\simeq 6\times 10^4 M_\odot\) (\(10^{-6}M_b\)). IR rail \(g^2=g_N a_0\) from \(r_y\) all the way to leftover \(R\): \(\simeq 1.1\times 10^{10}M_\odot\) (\(0.22\,M_b\)), **\(40\times\) below** \(4.5\times 10^{11}M_\odot\). Hitting that share would need \(\Delta r\simeq 41\,R\), outside the container. \(r_{\mathrm{loc}}=c^2/g\sim R\) is the **yield condition** (FT-YIELD-GEOM), which for a MW baryon well fires at \(r_y\simeq 7.6\,\mathrm{kpc}\), not an integration radius of leftover \(R\sim\mathrm{Gpc}\). Book G 1PN already *is* the gravity of that elastic energy. Assigning a positive halo \(T_{00}\) on top is a second copy. ICM gas is protons; it is not unhooked from \(T(2,3)\) occupancy. \(\sum_i M_{\mathrm{halo},i}=M_{\mathrm{crit}}\) is the dent-share **partition**. Photons seeing extra \(\Phi\) is the constitutive prize, without extra inventory. No CDM input. Prize OPEN.

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## 1. Two integrals that got fused

| Object | Radius | What it is |
|--------|--------|------------|
| \(r_{\mathrm{loc}}=c^2/g=R\) | \(g=c^2/R\simeq 2.31\,a_0\) | Yield *kind* at the solar-circle / \(r_y\) **kpc** scale |
| Leftover container \(R\) | \(\sqrt{6}\,c/H_0\sim\mathrm{Gpc}\) | Spectral lock of leftover \(S^3\) |
| Yield radius \(r_y=\sqrt{GM/a_0}\) | \(7.62\,\mathrm{kpc}\) for \(M=5\times 10^{10}M_\odot\) | Where baryon \(g_N=a_0\); \(r_y/R\simeq 7\times 10^{-7}\) |

**Finding E1 [E algebra].** Saturation against the container is a statement about **acceleration**, not about filling leftover with the well. The well’s 3-volume is a kpc ball. Leftover 4-volume is not that ball.

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## 2. Elastic \(T_{00}\) of the well (MW)

Newtonian field energy \(\lvert E\rvert=\frac{1}{2G}\int g^2 r^2\,\mathrm{d}r=\frac{GM^2}{2}(1/r_{\mathrm{in}}-1/r_{\mathrm{out}})\). IR rail \(g^2=g_N a_0\) gives \(\lvert E\rvert=\frac12 M a_0\,\Delta r\).

| Integral | \(M_{\mathrm{eq}}=\lvert E\rvert/c^2\) | / \(M_b\) | / \(4.5\times 10^{11}M_\odot\) |
|----------|----------------------------------------|-----------|--------------------------------|
| Newton \(1\,\mathrm{kpc}\to R\) | \(6.0\times 10^4 M_\odot\) | \(1.2\times 10^{-6}\) | \(\sim 10^{-7}\) |
| IR rail \(r_y\to R\) | \(1.08\times 10^{10}M_\odot\) | \(0.22\) | \(0.024\) |

To make the IR integral equal the dent-share: \(\Delta r = 2(M_{\mathrm{share}})c^2/(M a_0)\simeq 41\,R\). There is no such length in leftover.

**Finding E2 [E].** Weak-field elastic energy cannot be a halo mass. \(\Phi/c^2\sim 10^{-6}\) at the disk; field energy is that times \(M\), not \(10\,M\).

**Finding E3 [E honesty].** The Landau–Lifshitz / Newtonian field energy of an attractive well is **binding** (negative). A positive halo \(T_{00}\) of \(10^{11}M_\odot\) is the opposite sign as well as the wrong magnitude. Hencky 1PN (BP-1) already includes the gravity of gravitational energy and stays Newton+1PN.

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## 3. Collisionless, photons, budget

**Gas.** Bullet ICM is baryons, overwhelmingly protons. If \(T(2,3)\) is proton occupancy, the gas *is* the same topology, only diffuse. Anchoring leftover elasticity to “trefoil wells not collisional gas” is a new occupancy law, not exhibited. Compactness *kind* (yield \(8\,\mathrm{kpc}\) vs gas \(670\,\mathrm{kpc}\)) is already the Bullet mill; amplitude stays OPEN (P5).

**Photons.** Extra \(\Phi\) as 3-geometry is the leftover payment KiDS already requires. That does **not** need a second \(T_{00}\) inventory. Constitutive metric elasticity (the prize) *is* that \(\Phi\). Promoting the same distortion to a halo mass-energy and letting it source more \(\Phi\) is double-counting, unless the 1PN remainder is the halo — which Finding E2 kills.

**Budget.** “Total across all galaxies matches container critical mass” is \(\sum (M_{\mathrm{crit}}/N)=M_{\mathrm{crit}}\). True of any partition. It does not derive \(N\), \(\nu(y)\), or a density profile.

**Background leftover \(K\).** Umbilic curvature of leftover \(S^3\) has effective density \(O(\rho_c)\) and total mass \(O(M_{\mathrm{crit}})\). That is the GE-J / closed-\(S^3\) wall, leftover-wide, not a per-well halo. Splitting it \(1/N\) is FT-DENT-SHARE again.

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| Leftover is an elastic 4-medium | **YES** (ontology; Book G / prize) |
| Well \(T_{00}\) \(=4.5\times 10^{11}M_\odot\) | **NO** (\(0.22\,M_b\) even with IR to leftover \(R\)) |
| \(r_{\mathrm{loc}}\sim R\) is integrate-to-Gpc | **NO** (yield at \(r_y\)) |
| Gravity of gravity is the halo | **NO** (1PN, tiny) |
| Anchored to \(T(2,3)\) not gas | **NO** (gas is baryons) |
| \(\sum\) halos \(=M_{\mathrm{crit}}\) predicts \(N\) | **NO** |
| CDM inserted / prize closed / promote | **NO** |

**Exit:** `PASS_FT_ELAS_HALO_FIELD_ENERGY_NOT_MH_OVER_N`

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## 5. One-line for agents

> **§FT-ELAS-HALO: leftover elasticity is the right book; well field energy is \(O(M\Phi/c^2)\) not \(M_{\mathrm{crit}}/N\); IR to leftover \(R\) is \(0.22\,M_b\); \(r_{\mathrm{loc}}\sim R\) is yield at \(r_y\); gas is baryons; partition tautology; prize OPEN; no promote.**