# Results — T-SH Kerr Envelope + RT Residual

### GR Kerr silhouette vs spin · literature RT floor · NZ GRMHD/spin **OPEN** · no false GRMHD · no EHT discovery · no promote

**CPT Framework — spin-2 decisive tests · T-SH-K**  
**Date:** 2026-08-14  
**Status:** **BANKED** · `PASS_T_SH_KERR_RT_ENVELOPE`  
**Design:** `a_geom_spin2_t_sh_kerr_rt_envelope_design.md`  
**Parent:** T-SH forecast + plasma honesty · S6 board  
**Census:** `scripts/investigations/s3_geometry/spin2_t_sh_kerr_rt_envelope_census.py` · `data/spin2_t_sh_kerr_rt_envelope_census.json`  

Tags: **[E honesty]** · **[P prediction]** · **[Programme]**

---

## 0. One-line

> **T-SH-K delivers what can be done without inventing physics: GR Kerr geometric diameter envelope \(b_{\mathrm{eff}}\in[4.48,5.20]\,M\) (half-range ~6.9% of Schwarzschild; max shrink ~13.8% at high spin); literature RT bias floor ~4–8%; static NZ \(b_c=2e\) sits **above the entire Kerr band** at fixed \(M/D\); full NZ GRMHD and spinning NZ metric remain **OPEN** (no in-repo GRMHD, no free NZ Kerr); EHT still does **not** select NZ once RT floors are included; ngEHT remains a TARGET not auto-3σ; no promote.**

---

## 1. Integrity (held)

| Rule | Held |
|------|:----:|
| No false claim of NZ GRMHD in this repo | **Yes** |
| No invented spinning NZ metric | **Yes** · status **OPEN** |
| Kerr envelope is GR geometry only | **Yes** |
| RT envelope is literature honesty, not a new simulation | **Yes** |
| No EHT discovery claim | **Yes** |
| No promote | **Yes** |

---

## 2. What was requested vs what was delivered

| Requested | Delivered | Not delivered |
|-----------|-----------|---------------|
| Kerr envelope | **GR** critical-curve diameter vs \(a,i\) | — |
| Full NZ GRMHD ray-trace | Prerequisites + **OPEN** chip | Fluid + RT + NZ metric imaging |
| Combined honesty | Updated \(n_\sigma\) (one- and two-sided) | NZ-side spinning \(b_c(a)\) |

**Repo fact:** no BHAC / ipole / RAPTOR / EHT imaging stack; no banked spinning compression metric.

---

## 3. GR Kerr geometric envelope (computed)

Spherical photon orbits (Teo / Bardeen, \(M=1\)); celestial critical curve; mean diameter proxy \(D=(D_h+D_v)/2\), \(b_{\mathrm{eff}}=D/2\).

| Check | Result |
|-------|--------|
| \(a=0\) | \(b_{\mathrm{eff}}=3\sqrt{3}\) **PASS** |
| Grid | \(a\in\{0\ldots 0.998\}\), \(i\in\{1^\circ,17^\circ,60^\circ,90^\circ\}\) · 36 points |

| Quantity | Value |
|----------|------:|
| \(b_{\mathrm{eff}}\) min / max | **4.479** / **5.196** |
| Half-range \(\sigma_b\) | **0.359** (~**6.9%** of \(3\sqrt{3}\)) |
| Max \|dev\| from Schwarzschild | **0.717** (~**13.8%**) |

**Structural fact on this grid:** Kerr spin **shrinks** the geometric diameter relative to Schwarzschild (\(b_{\mathrm{eff}}(a)\le b_{\mathrm{eff}}(0)\)). The upper edge of the Kerr family at fixed \(M/D\) is the **non-spinning** GR value.

### Absolute µas (banked \(\theta_g\))

| Source | \(\sigma_{\mathrm{Kerr}}\) half-range | max shrink vs \(a=0\) |
|--------|--------------------------------------:|----------------------:|
| Sgr A* | **~3.61 µas** | **~7.22 µas** |
| M87* | **~2.74 µas** | **~5.48 µas** |

---

## 4. Literature RT / plasma envelope (not a simulation)

| Floor | Fractional | Sgr A* (µas) | M87* (µas) |
|-------|------------:|-------------:|-----------:|
| Typical | 4% of \(\theta_{\mathrm{GR}}\) | **~2.09** | **~1.59** |
| Ceiling | 8% of \(\theta_{\mathrm{GR}}\) | **~4.18** | **~3.18** |

These are **honesty ceilings** aligned with EHT model-library practice (ring vs pure geometry can differ by O(few–10%)). They are **not** an in-repo GRMHD run.

---

## 5. Static NZ vs GR Kerr family

| Object | \(b_c\) / \(b_{\mathrm{eff}}\) |
|--------|-------------------------------:|
| GR Kerr band | **[4.48, 5.20]** |
| NZ static | **\(2e \approx 5.437\)** |

**At fixed \(M/D\):** static NZ lies **above the entire GR Kerr geometric band**. Spin does **not** let GR “reach up” to NZ; it moves GR **down**. Spin is therefore **not** an escape that makes NZ look like a high-spin Kerr silhouette at the same mass scale.

| Source | \(\theta_{\mathrm{NZ}}-\theta_{\mathrm{GR}}(a=0)\) | Kerr can close that gap? |
|--------|--------------------------------------------------:|:------------------------:|
| Sgr A* | **+2.42 µas** | **No** (spin widens the gap) |
| M87* | **+1.84 µas** | **No** |

---

## 6. Discrimination (updated honesty)

### 6.1 Two-sided Kerr half-range (pessimistic if misread)

If one naively puts \(\sigma_{\mathrm{Kerr}}\) two-sided into the NZ−GR error budget, ngEHT \(n_\sigma\) collapses to \(\sim 0.5\)–\(0.6\). That is **misleading** for an “NZ above GR” test, because Kerr variation is **one-sided downward**.

### 6.2 One-sided (correct for geometric family test) + RT

Use gap = NZ − GR(\(a=0\)) with RT as two-sided emission uncertainty:

| Source | EHT \(n_\sigma\) (RT typ) | ngEHT \(n_\sigma\) (RT typ) | Auto \(>3\sigma\) ngEHT? |
|--------|-------------------------:|---------------------------:|:------------------------:|
| Sgr A* | **~0.78** | **~1.12** | **No** |
| M87* | **~0.54** | **~1.10** | **No** |

| Instrument | Verdict |
|------------|---------|
| **Current EHT** | Still **does not select** NZ (RT floors dominate) |
| **ngEHT ~0.5 µas** | Still a **TARGET**; with RT ~4% floors, \(n_\sigma\sim 1\) — **not** automatic discovery |
| Geometric-only multi-\(\sigma\) (T-SH old) | **Optimistic upper bound** only |

---

## 7. NZ GRMHD + spinning NZ — OPEN checklist

| Prerequisite | Status |
|--------------|--------|
| Freeze spinning NZ metric (slow-rotation / KS-like / other) | **DESIGN FROZEN** · Class A dual-book · SNZ-a `PASS_SNZ_A` · `a_geom_spin2_spinning_nz_metric_design.md` |
| Photon critical curve \(b_{\mathrm{eff}}(a)\) in that metric | **DONE** · SNZ-c |
| Emission model + radiative transfer | **DONE as GRRT** · T-RT named \(n(r)\) · **not** BHAC |
| Synthetic imaging / ring fit | **PARTIAL** · 1-D \(I(b)\) + static curves · no EHT pipeline |
| In-repo GRMHD fluid evolution | **Not present** |

**Forbidden now:** invent NZ Kerr coefficients; claim “we ray-traced NZ GRMHD.”

---

## 8. Gates

| Gate | Status |
|------|--------|
| PASS_KERR_A0 | **PASS** |
| PASS_KERR_ENVELOPE | **PASS** |
| PASS_RT_ENVELOPE | **PASS** |
| PASS_NZ_SPIN_OPEN | **PASS** |
| PASS_NO_FALSE_GRMHD | **PASS** |
| PASS_NO_DISCOVERY | **PASS** |
| PASS_INTEGRITY | **PASS** |

**Status:** `PASS_T_SH_KERR_RT_ENVELOPE`

---

## 9. Relation to T-SH pack

| Layer | Role |
|-------|------|
| T-SH geometric \(b_c\) | Static NZ vs Schw TARGET |
| T-SH plasma honesty (§5) | Flat 1.5 µas floor |
| **T-SH-K (this)** | GR Kerr band + literature RT + NZ GRMHD OPEN |
| **T-RT GRRT** | In-repo geodesic + named emission · `PASS_T_RT_GRRT` · not fluid GRMHD |
| True NZ GRMHD | Still **OPEN** · external cube / public pipeline only |

---

## 10. One-line

> **§T-SH-K: Kerr \(b_{\mathrm{eff}}\in[4.48,5.20]\); NZ static above Kerr band; RT 4–8% floor; NZ GRMHD OPEN; no false GRMHD; no EHT discovery; no promote.**