# H-D15 — Sub-luminal Worldline + Regularised Field Self-Force

### True β control · dual cancel kinematics-dependent · LW quiets · freeze packaging honesty · hold OPEN

**CPT Framework — Dynamical hold deep probe**  
**Date:** 2026-08-04  
**Status:** **H-D15 BANKED** · sub-luminal β control **PASS** · dual cancel **fails** under β_max<1 · freeze **unchanged as packaging** · continuum hold **OPEN**  
**Script:** `hold_hd15_subluminal_reg_field_sf.py` · data `hold_hd15_subluminal_reg_field_sf.json`  
**Priors:** H-D14 · H-D14b · H-D13 · core v2.10.3 freeze  

Tags: **[E]** · **[M]** · **[O]** · **[F]** · **[E honesty]** · **[E process]**

---

## 0. One-line

> **With honest sub-luminal control (`mode=max`, max|β|=β_dial<1), full Liénard–Wiechert nonlocal residual quiets to \(R_F\sim 10^{-3}\), but **no** book — Weber, retarded Coulomb nloc, or Detweiler–Whiting proxies — recovers a dual-band \(F_N^{\mathrm{nloc}}\) zero: Weber \(\mu^\star\) stays \(\approx -0.37\!\to\!-0.46\) for β_max∈[0.1,0.9]. Dual cancel at \(\mu^\star\!\approx\!-0.66\) (H-D14b) is a **near-null renorm packaging effect**, not a sub-luminal continuum property. Freeze remains Weber@ILC-1/3 packaging; continuum hold OPEN.**

---

## 1. Bug fixed · setup

### 1.1 Renorm wipe (integrity)

Upstream `build_tube` always re-renorms tube velocities to **mean |v|=1**, wiping any sub-luminal dial on the centreline. First H-D15 run showed `maxβ=2.304` independent of dial.

**Fix:** local `build_tube_beta` + `renorm_to_speed(..., mode="max")` so **max|v|=β_dial** on centreline and tube.

| β_dial (max) | max|β| tube | mean|β| tube |
|-------------:|------------:|-------------:|
| 0.1 | 0.100 | 0.043 |
| 0.5 | 0.500 | 0.217 |
| 0.9 | 0.900 | 0.391 |

**Finding HD15-0 [E honesty]:** Prior null renorm (|v| mean→1) leaves **max|β|~2.3** on Hopf — superluminal kinematics (H-D14 note confirmed). True sub-luminal requires **max** renorm or mean dial ≲0.43.

### 1.2 Books

| Book | Rule |
|------|------|
| **WEBER** | Freeze Weber fixed-arc nloc @ ILC-1 |
| **COUL_NLOC** | Retarded Coulomb fixed-arc nloc (REG4) |
| **LW_VEL / LW_FULL** | Liénard–Wiechert \(E_{\mathrm{vel}}\) / full, fixed-arc nloc |
| **DW_COUL / DW_VEL** | Proxy \(F_{\mathrm{ret}}-F_{\mathrm{sing}}\) (inst. Coulomb local subtract) |

Seat: Hopf dual, \(\varepsilon=0.07\), \(\mu=-0.80\), \(n_t=28\), \(n_s=4\), subsample 2, \(f_L=0.25\).

**Not claimed:** continuum Detweiler–Whiting / MiSaTaQuWa theorem.

---

## 2. Seat residuals vs β_max (μ=−0.80)

| β_max | WEBER \(R_F\) | COUL \(R_F\) | LW_VEL | DW_COUL | LW_FULL |
|------:|-------------:|------------:|-------:|--------:|--------:|
| 0.1 | \(1.23\times 10^{-3}\) | \(1.23\times 10^{-3}\) | \(1.33\times 10^{-3}\) | \(9.2\times 10^{-4}\) | quiet |
| 0.5 | \(1.22\times 10^{-3}\) | \(1.23\times 10^{-3}\) | \(1.75\times 10^{-3}\) | \(9.2\times 10^{-4}\) | \(1.74\times 10^{-3}\) |
| 0.9 | \(1.20\times 10^{-3}\) | \(1.23\times 10^{-3}\) | \(2.23\times 10^{-3}\) | \(9.2\times 10^{-4}\) | quiet |

**Finding HD15-1 [E]:** Under true β<1, **full LW nloc quiets** (\(R_F\sim 10^{-3}\)) — H-D13/14 bare blow-up was largely a superluminal-kinematics artefact.

**Finding HD15-2 [E]:** Coulomb / DW_COUL residuals are **β-independent** (force is position-only; lag/positions barely reweight on this seat). Weber residual softens slightly with β.

---

## 3. Dual-band μ zeros @ β_max=0.5

| Book | \(\mu^\star\) (fit) | Zero in dual band \([-1.1,-0.6]\)? |
|------|-------------------:|:----------------------------------:|
| **WEBER** | **−0.403** | **NO** |
| COUL_NLOC | −0.371 | NO |
| DW_COUL | −0.479 | NO |
| LW_VEL | −0.264 | NO |
| DW_VEL | −0.125 | NO |

### AA/AB anatomy (β_max=0.5, μ=−0.90)

| Book | \(F_N^{AA}\) | \(F_N^{AB}\) | AB/AA | \(\langle W\rangle_{AA}\) | \(\langle W\rangle_{AB}\) |
|------|-------------:|-------------:|------:|--------------------------:|--------------------------:|
| Weber | \(-1.24\times 10^{-3}\) | \(+2.77\times 10^{-3}\) | −2.23 | **1.17** | **1.11** |
| Coulomb | \(-1.06\times 10^{-3}\) | \(+2.57\times 10^{-3}\) | −2.43 | ≡1 | ≡1 |
| DW_Coul | \(-1.37\times 10^{-3}\) | \(+2.57\times 10^{-3}\) | −1.88 | — | — |

Compare H-D14b under **null renorm** (mean|β|→1, max~2.3):

| Book | \(\langle W\rangle_{AA}\) | \(\langle W\rangle_{AB}\) | \(\mu^\star\) | dual? |
|------|--------------------------:|--------------------------:|-------------:|:-----:|
| Weber (null) | **4.70** | **3.35** | **−0.66** | **YES** |
| Coulomb | 1 | 1 | −0.37 | NO |
| Weber (β_max=0.5) | 1.17 | 1.11 | −0.40 | NO |

**Finding HD15-3 [E honesty]:** The \(W_{AA}>W_{AB}\) differential that moves \(\mu^\star\) into the dual band is a **high-β / near-null renorm effect**. At sub-luminal β_max=0.5, \(W\approx 1.1\) and Weber collapses toward Coulomb.

---

## 4. μ* vs β_max

| β_max | \(\mu^\star_W\) | \(\mu^\star_C\) | mean|β| |
|------:|---------------:|---------------:|--------:|
| 0.1 | −0.372 | −0.371 | 0.043 |
| 0.3 | −0.383 | −0.371 | 0.130 |
| 0.5 | −0.403 | −0.371 | 0.217 |
| 0.7 | −0.430 | −0.371 | 0.304 |
| 0.9 | −0.461 | −0.371 | 0.391 |

**Finding HD15-4 [E]:** Weber \(\mu^\star\) **drifts deeper** with β_max but **never enters** dual packaging band for β_max≤0.9. Coulomb \(\mu^\star\) is **frozen** at −0.371.

**Finding HD15-5 [M]:** Reaching \(\mu^\star\!\approx\!-0.66\) requires the packaging null-renorm kinematics of H-D7d–H-D14b (mean|β|→1, max superluminal). Dual cancel is therefore **packaging-seat dependent**, not a sub-luminal continuum identity.

---

## 5. Plateau · mesh · scorecard (β_max=0.5)

| Book | CV_ε | Plateau | Dual0 | mesh-hi | soft | strong |
|------|-----:|:-------:|:-----:|:-------:|:----:|:------:|
| WEBER | 0.14 | YES | NO | YES | NO | NO |
| COUL_NLOC | 0.14 | YES | NO | YES | NO | NO |
| DW_COUL | 0.39 | NO | NO | NO | NO | NO |
| LW_VEL | 0.09 | YES | NO | — | NO | NO |
| LW_FULL | — | — | NO | — | NO | NO |

No soft or strong winner under honest sub-luminal β.

---

## 6. Integrity ledger

| ID | Claim | Verdict |
|----|-------|---------|
| **HD15-a** | max|β|<1 on all seats (mode=max) | **PASS** |
| **HD15-b** | Weber dual-band zero under β_max∈(0,1) | **FAIL** |
| **HD15-c** | Coulomb nloc gains dual-band zero under β<1 | **FAIL** |
| **HD15-d** | DW_COUL recovers dual cancel @β_max=0.5 | **FAIL** |
| **HD15-e** | Full LW nloc quiet under β_max=0.5 | **PASS** |
| **HD15-f** | Field book replaces Weber@ILC freeze | **FAIL** |
| **HD15-g** | Continuum DW theorem | **NOT_CLAIMED** |

---

## 7. Freeze status

| Item | Status |
|------|--------|
| **Packaging freeze v2.10.3** | **Unchanged** — pack_no_log + Weber fixed-arc nloc@ILC-1/3 on **null renorm** packaging seat |
| **Honesty addendum** | Dual-band cancel \(\mu^\star\!\approx\!-0.66\) **requires** that packaging's high-β kinematics; under sub-luminal β_max≤0.9, \(\mu^\star_W\!\gtrsim\!-0.46\) |
| **Field replacement** | **No** — DW / LW / Coulomb do not recover dual cancel |
| **Continuum hold** | **OPEN** |

---

## 8. What this means scientifically

1. **H-D14b mechanism stands** as an explanation of the *packaging* dual cancel: \(W\) differential under ILC + null renorm.
2. **H-D15 shows the cost of honesty on kinematics:** if the worldline is forced sub-luminal (max|β|<1), that \(W\) differential collapses and dual cancel leaves the packaging dual band.
3. **LW is not intrinsically huge** once max|β|<1 — H-D13/14 blow-up was partly kinematics.
4. **No free lunch:** field regularisation under honest β quiets residual but does **not** hand dual cancel; Weber under honest β also loses dual cancel.

---

## 9. Non-claims

- Not continuum Detweiler–Whiting / MiSaTaQuWa.
- β is a **packaging dial**, not a free SM parameter.
- ILC-1 remains discrete packaging, not a retarded Green function.
- Null renorm of the freeze seat is **not** replaced; H-D15 diagnoses its kinematics cost.

---

## 10. Next doors

| ID | Work |
|----|------|
| **H-D16?** | Darwin–Weber coefficient / \(W\) expansion match at controlled β |
| Optional | Soften dual-band definition vs packaging \(\mu\); joint energy–force under max renorm |
| Continuum | True DW singular field on a rigorously sub-luminal closed worldline |

---

## 11. Agent one-liner

> **H-D15: max|β| control PASS; LW quiets under β<1; Weber μ*≈−0.40 @β_max=0.5 (dual cancel FAIL); dual cancel of H-D14b is near-null packaging; freeze stays Weber@ILC-1/3 with kinematics honesty note; hold OPEN.**