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.
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.**