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]
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.
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.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.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.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
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.