# Freeze — constitutive prize structure (standing inventory)

### What leftover gravity can do · what is closed · the exact gates a winning metric constitutive law must meet · clothes stay clothes · prize OPEN

**CPT Framework — FT-CONST-PRIZE / FT-METRIC-ELAS / FT-FAT-FORCE / FT-MA-LONE-KILL / FT-GAP-B-OUTER / Book G**  
**Date:** 2026-09-22  
**Status:** **STANDING** · `PASS_FT_CONST_STRUCTURE_STANDING`  
**Stimulus:** User — we now know what the theory can do, what has been proven impossible, and what structure the constitutive prize must possess  
**Parents:** FT-CONST-PRIZE · FT-METRIC-ELAS · FT-FAT-FORCE · FT-WE · FT-BEND-SEQ · FT-GAP-B-OUTER · FT-GW170817  
**Script:** `scripts/investigations/s3_geometry/ft_const_structure.py`  
**Data:** `scripts/investigations/s3_geometry/data/ft_const_structure.json`

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

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## 0. One-line

> **The map is drawn; the tensor is not.** Leftover can pay extra \(\Phi\) at \(a\sim a_0\) for photons and stars, with UV Newton+1PN and \(c_{\mathrm{gw}}=c\). Analytic \(W\), Hencky-as-RAR, graph/neck embeddings, extra \(\phi\) on the well, floors, \(\mu\)-shops, and CDM-as-input are **closed**. The unpaid object is a **non-analytic constitutive law of the metric** whose Euler–Lagrange equation is a universal \(y=g_N/a_0\mapsto g_{\mathrm{obs}}\) with leftover \(K\sim 1/R\) the only scale. Slot isolated (FT-PRIZE-PIN); \(\mathcal{L}\) not written. \(\sqrt{6}\) still **[M]**. Coma/Bullet amplitude may stay OPEN. No chip.

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## 1. What the theory can do (banked *kind* / capability)

| Capability | Status |
|------------|--------|
| UV Newton + scored 1PN (\(\gamma=\beta=1\)) | Book G |
| Inner galaxy–galaxy \(\theta_E\) baryon-capable | FT-LENS-RAD / FT-GAP-B |
| Photons see extra \(\Phi\) at RAR radii (KiDS GGL measurement, not DE cosmology) | FT-GAP-B-OUTER; lone \(m(a)\) **KILL**; FT-DE-NO-MATCH |
| Yield *kind* at \(a\sim a_0\), \(r_y=\sqrt{GM/a_0}\) | FT-YIELD-A0 |
| \(r_{\mathrm{loc}}=c^2/g\) hits leftover \(R\) | FT-YIELD-GEOM (not a \(W\)) |
| IR rail \(g_{\mathrm{obs}}^2=g_N a_0\) (\(k=1\) at threshold) | FT-FAT-FORCE *kind* |
| \(c_{\mathrm{gw}}=c\) as UV *kind* (\(\lambda\ll R\)) | FT-GW170817; TeVeS-class **FIRE** |
| \(R\) spectrally locked; \(c,R\) invariant here | FT-R-LOCK / FT-CR-INVARIANT |
| Winning EL must be a universal \(y\)-map (no leftover \(M\)) | FT-FAT-FORCE |

Clothes, not theorems: \(a_0=c^2/R_{\sqrt{6}}\) is \(\sim 2.31\,a_0\) SPARC; \(\sqrt{6}\) remains **[M]**.

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## 2. What is closed (do not retry)

Analytic \(W(E)\) as the RAR IR · Hencky geometric non-linearity as the RAR IR · strain cap \(\lvert\nabla U\rvert\le 1/R\) · 3-into-4 graph / parametric 3-volume–CMC · 3D Ginzburg–Landau extra \(\phi\) on the well · lone \(m(a)\) / SFDM as what photons see · hard floor \(g\ge a_0\) · harmonic \(1/r_N+1/R\) · \(\lambda\), \(p\), or \(\mu(x)\) fitted to SPARC · CDM as a framework input · extra metric with \(c_{\mathrm{gw}}\ne c\) · expanding leftover \(R\) as \(a_0(z)\) · leftover \(P\) alone as a kpc RAR · thin Israel skin as SPARC’s \(2\,\mathrm{dex}\) width · \(\mathrm{Vol}_4\) Lagrange multiplier / bulk buoyancy as a per-galaxy \(y\)-map · umbilic Jacobi buckle / \(\pi_3\) instanton as leftover RAR · self-consistent Poisson-embedding with named lab 4-energies as a RAR derivation · moving leftover water / memory-foam residual as a \(y\)-map (currently-loaded foam is the prize’s time-domain name, not a \(W\)) · leftover bulk accumulating in wells as a \(y\)-map (CDM-role inventory; rubber-sheet dust is test masses) · \(M_{\mathrm{crit}}/N_{\mathrm{dents}}\) as a derived halo mass · well elastic \(T_{00}\) as \(M_{\mathrm{crit}}/N\) · \(L_\star\)/mass-type leftover clumps as a dwarf save · spin/torsion as leftover smoother · bulk galaxy rotation of leftover as a dwarf/\(L_\star\) smoother.

“Proven impossible” here means **CLOSED / KILL / FIRE** in the mills, not a uniqueness theorem for all mathematics. The closed list is the retry ban.

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## 3. Structure the constitutive prize must possess

| Gate | Must |
|------|------|
| Operand | Metric constitutive law (not \(m(a)\), not extra \(\phi\), not a second metric) |
| Non-analyticity | At \(\lvert\nabla U\rvert\sim 1/R\) (leftover \(K\)), not a Taylor \(W\) at the identity |
| Reduction | Universal \(y\mapsto g_{\mathrm{obs}}\); no leftover \(M\) in the EL |
| UV | Newton + scored 1PN |
| IR *kind* | \(g_{\mathrm{obs}}^2=g_N a_0\) |
| Photons | Extra \(\Phi\) |
| Waves | \(c_{\mathrm{gw}}=c\) |
| Scale | Leftover \(K\sim 1/R\) only; \(R,c\) invariant |
| Not | \(\mu\) shop, SPARC \(I\), CDM, GE-J as \(m_e\), Book C on stars |
| Honesty | Coma / Bullet amplitude residual may stay OPEN |

May be non-local (\(P\)). Locality is not assumed. The Occam fat-zone kink is a **formulation**, not this tensor. Named leftover 4-energies are **not** this tensor (FT-POISSON-EMBED). The tensor is **not exhibited**. Writing \(\mathcal{L}(g,K)\) is a shop until uniqueness is forced.

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## 4. Integrity

| Gate | Verdict |
|------|---------|
| Standing inventory | **YES** |
| Prize closed | **NO** |
| Clothes as theorems | **NO** |
| Invent \(W\) / \(\mu\) this mill | **NO** |

**Exit:** `PASS_FT_CONST_STRUCTURE_STANDING`

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## 5. One-line for agents

> **§FT-CONST-STRUCTURE: leftover can pay extra \(\Phi\) at \(a\sim a_0\) with UV Newton and \(c_{\mathrm{gw}}=c\); named 4-energies and \(\mu\)-shops closed; do not treat leftover as a rubber sheet in empty \(B^4\) (\(\delta\mathrm{Area}=\cos\theta\)); prize = constitutive 4-metric elasticity with leftover \(K=1/R\) as yield; \(\mathcal{L}\) not written; \(\sqrt{6}\) still [M].**