# Results — P-HI: maximum tangential drag on neutral HI from the paired disk-loop

### Axisymmetric loop \(a_\varphi=0\) · leftover \(J_z\times B_r\) diskwide \(\sim 0.4\%\) of \(v^2/R\) · stolen \(|J\times B|\) into \(\varphi\) with \(B_{\mathrm{tot}}\) is the scare, not the circuit · ion-neutral is a **rate**, not a halo · no SPARC \(I\) · no promote

**CPT Framework — BCJ / Book C / GAL-B-PERP / G2**  
**Date:** 2026-09-20  
**Status:** **BANKED** · `PASS_P_HI` · ceiling named · **cannot** flatten \(v(r)\)  
**Script:** `scripts/investigations/s3_geometry/p_hi_tangential_drag.py`  
**Data:** `scripts/investigations/s3_geometry/data/p_hi_tangential_drag.json`  
**Parents:** BCJ · P-CR · P-WAVE E4 · GAL-B-PERP · GAL-ROT-ELEC

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

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## 0. One-line

> **The paired disk-loop does not tangentially drag HI at halo order.** \(J_\varphi\times B_z\) is **radial**. \(a_\varphi=0\) for an axisymmetric loop. A sustained \(\varphi\) force needs leftover \(J_z\) or \(J_r\) (unshielded remainder) against the in-plane field: disk-wide that is \(\sim 0.16\)–\(0.40\%\) of \(v^2/R\) at \(8.5\,\mathrm{kpc}\), \(220\,\mathrm{km\,s^{-1}}\). Stealing the whole \(|J\times B|=B^2/(\mu_0 h)\) into \(\varphi\) and using **total** \(5\,\mu\mathrm{G}\) reaches \(\sim 35\%\) — wrong direction and the wrong \(B\) for this circuit. Concentrating leftover \(I_{\mathrm{net}}\) in a \(100\,\mathrm{pc}\) channel can hit \(\sim 14\)–\(34\%\) **locally**; filling \(f\sim 0.2\) brings the disk mean back to a few per cent. Ion-neutral \(\nu_{ni}\Delta v\) looks huge because it is a **coupling time** (\(\sim 500\,\mathrm{yr}\)), not a sustained acceleration. Once \(v_i\sim v_n\), the force is \(J\times B/\rho\). Not SPARC. Not a halo. No promote.

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## 1. What “maximum” is allowed to mean

Orbit: \(R=8.5\,\mathrm{kpc}\), \(v=220\,\mathrm{km\,s^{-1}}\), \(a_{\mathrm{cent}}=v^2/R=1.85\times 10^{-10}\,\mathrm{m\,s^{-2}}\). Named \(n_{\mathrm{HI}}=0.2\,\mathrm{cm^{-3}}\), \(h=0.3\,\mathrm{kpc}\), pitch \(11^\circ\). Ampère leftover \(I\sim BR/\mu_0\) (\(5\,\mu\mathrm{G}\Rightarrow 1.0\times 10^{17}\,\mathrm{A}\), the BCJ class).

HI is neutral. The only licensed transfer is ion–neutral coupling (P-WAVE E4). **Sustained** \(a_\varphi\) cannot exceed \(|J\times B|_\varphi/\rho_{\mathrm{HI}}\). \(\nu_{ni}\Delta v\) is how fast the slip is erased, not the orbital force.

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## 2. Geometry first

Disk-loop (GAL-B-PERP): \(B_\perp=B_z\), current azimuthal.

| \(J\times B\) | Direction |
|---------------|-----------|
| \(J_\varphi\times B_z\) | **radial** |
| \(J_\varphi\times B_r\) (pitch) | **vertical** |
| \(J_\varphi\times B_\varphi\) | \(0\) |
| \(J_z\times B_r\) or \(J_r\times B_z\) | **tangential** — leftover pair, not the loop |

**Finding H1 [E algebra].** Axisymmetric disk-loop: \(a_\varphi=0\). Tangential drag is not the leading Book C force on this circuit.

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## 3. Ceilings (named, not a fit)

| Ceiling | What it assumes | \(a_\varphi/a_{\mathrm{cent}}\) |
|---------|-----------------|--------------------------------|
| Loop, axisymmetric | geometry | **\(0\)** |
| Leftover \(J_z\times B_r\), disk-wide, \(B_z=0.3\,\mu\mathrm{G}\), \(B_{\mathrm{reg}}\) pitch | unshielded remainder | \(\mathbf{0.16\%}\) |
| Same, \(B_{\mathrm{tot}}\) pitch | remainder × total in-plane | \(0.40\%\) |
| Leftover concentrated in \(100\,\mathrm{pc}\) channel, \(B_{\mathrm{reg}}\) | local MAX, still \(I_{\mathrm{net}}\) | \(14\%\) |
| Same, \(B_{\mathrm{tot}}\) | local MAX | \(34\%\) |
| Those locals × filling \(f=0.2\) | disk mean | \(\sim 7\%\) |
| **Stolen** \(|J\times B|\) into \(\varphi\), \(B_z=0.3\,\mu\mathrm{G}\) | wrong direction | \(0.13\%\) |
| Stolen, \(B_{\mathrm{reg}}=2\,\mu\mathrm{G}\) | wrong direction | \(5.6\%\) |
| Stolen, \(B_{\mathrm{tot}}=5\,\mu\mathrm{G}\) | wrong direction **and** turbulent \(B\) | \(\mathbf{35\%}\) |
| Stolen \(B_{\mathrm{tot}}\) × \(f=0.2\) | scare × filling | \(7\%\) |

**Finding H2 [E].** The licensed leftover, disk-wide, is **\(<1\%\)** of centripetal. It does not replace the halo.

**Finding H3 [E honesty].** The \(35\%\) scare is three violations at once: (i) dump radial \(J\times B\) into \(\varphi\), (ii) use total synchrotron \(B\) as the loop field, (iii) put all HI in the current layer. Drop any one and the ratio collapses. Concentrating **\(I_{\mathrm{leg}}\)** (undoing the pair) is the July wire. Forbidden.

**Finding H4 [E].** A \(100\,\mathrm{pc}\) channel can feel \(O(10\%)\) locally. That is a filament, not a flat \(v(r)\). Volume filling \(f\sim 0.2\) (P-WAVE E2) puts the disk mean back to a few per cent.

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## 4. Ion-neutral is not a second halo

\(\nu_{\mathrm{in}}=n_{\mathrm{HI}}\langle\sigma v\rangle\sim 4\times 10^{-10}\,\mathrm{s^{-1}}\), \(\nu_{ni}=(\rho_i/\rho_n)\nu_{\mathrm{in}}\sim 6\times 10^{-11}\,\mathrm{s^{-1}}\). Coupling time \(\sim 500\,\mathrm{yr}\). \(\nu_{ni}v_A/a_{\mathrm{cent}}\sim 10^4\) looks like a miracle force. It is the rate at which slip **vanishes**. After that, HI and ions share \(J\times B/\rho\), which is the table above.

**Finding H5 [E honesty].** Fully coupled (P-WAVE E4 at 300 Myr) means the neutrals **follow the ions**, not that \(\nu\Delta v\) stays on as a centripetal budget.

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## 5. Integrity

| Gate | Verdict |
|------|---------|
| \(a_\varphi\) of the axisymmetric loop | **\(0\)** |
| Licensed leftover disk-wide | \(\boldsymbol{\sim 10^{-3}\,a_{\mathrm{cent}}\)** |
| Stolen \(B_{\mathrm{tot}}\) scare | **not the circuit** |
| Flatten HI \(v(r)\) / retire halo | **NO** |
| SPARC \(I\) / undo pairing | **FORBIDDEN** |
| \(\nu\Delta v\) as orbital \(a\) | **FAIL_IDENTIFY** |
| G1 | **untouched** |
| Chip / promote | **NO** |

**Exit:** `PASS_P_HI`

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## 6. One-line for agents

> **§P-HI: disk-loop \(a_\varphi=0\); leftover \(J_z\times B_r\) disk-wide \(\sim 0.4\%\) of \(v^2/R\); stolen \(B_{\mathrm{tot}}\) \(35\%\) is the wrong direction and the wrong \(B\); \(\nu\Delta v\) is a coupling rate; cannot flatten HI; chips OPEN; no promote.**