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Warp Factory cross-check (Task 2A.9b / TRUST_AUDIT #3)

Scope

Independent numerical reproduction of Fuchs et al. 2024 (arXiv:2405.02709) Fig. 10 via Warp Factory (Helmerich et al. 2024, arXiv:2404.03095) on MATLAB R2023a Update 8 (Parallel Computing Toolbox + default toolboxes). Closes:

  • TRUST_AUDIT.md row #3 (Fuchs existence anchor) — B → A.
  • ROADMAP.md Task 2A.9b (mass-to-velocity scaling, Warp Factory cross-check).

This was the last deferred TRUST_AUDIT item.

Slice scope

Same as the rest of MATTER_SHELL_PATH.md:

  • 4D General Relativity, signature $(-,+,+,+)$.
  • Static spherical anisotropic-fluid shell on $R_1 \le r \le R_2$ with TOV-iterated $T^{\hat\mu\hat\nu} = \mathrm{diag}(\rho, P_1, P_2, P_3)$.
  • Minkowski interior ($r < R_1$), Schwarzschild exterior ($r > R_2$), comoving frame ($v_{\rm shell} = 0$).
  • Alcubierre-class shift $\beta^x(r),\hat x$ with Fuchs bump function $f(r)$ supported on the shell domain; $\beta^x_{\rm interior} = -v_s$, $\beta^x_{\rm exterior} = 0$.
  • Fixed canonical params: $R_1 = 10$ m, $R_2 = 20$ m, $M = 4.49 \times 10^{27}$ kg, $\beta_{\rm warp} = 0.02 c$, $\sigma = 0$, smoothing $= 4000$.

What was done

  1. Cloned Warp Factory v1.0 into F:\science-projects\WarpFactory\ (out-of-tree; do not commit the dependency itself).
  2. Reproduced Fig. 10 at canonical params via fuchs_fig10_repro.m — a headless wrapper around metricGet_WarpShellComoving + evalMetric that exports the energy tensor and the four pointwise energy-condition arrays (NEC/WEC/DEC/SEC) on a 300 × 300 × 5 grid (in-plane spacing 0.2 m).
  3. κ-bracket sweep via kappa_sweep.m: held $(M, R_2, \beta)$ fixed and varied $\Delta = R_2 - R_1 \in {1, 1.5, 2, 3, 5, 7, 10}$ m to bracket the numerical $\Delta_{\min}$ where DEC first achieves in-shell pass-fraction = 1.

Artifacts in warp_factory_repro/:

  • fuchs_fig10_repro.m, kappa_sweep.m — the MATLAB scripts.
  • fuchs_repro.mat — energy tensor + four EC arrays at canonical params.
  • fuchs_repro_{rho,nec,wec,dec,sec}.png — slice plots through the equator.
  • kappa_sweep.mat — pass-fractions and min(DEC|shell) as a function of $\Delta$.

Result 1 — Fuchs Fig. 10 reproduced

At canonical params:

Energy condition In-shell pass fraction
NEC 1.0000
WEC 1.0000
DEC 1.0000
SEC 1.0000

Visual signature of the DEC slice (fuchs_repro_dec.png): a uniform DEC-positive annulus from $R_1 = 10$ m to $R_2 = 20$ m, white interior (Minkowski) and white exterior (Schwarzschild vacuum), DEC value $\sim 9 \times 10^{39}$ at the shell mid-radius. Matches Fuchs et al. 2024 Fig. 10 panel structure on visual inspection. The Fuchs existence claim is independently confirmed.

Result 2 — κ-bracket cross-check (2A.9b proper)

Setup: hold $M = 4.49 \times 10^{27}$ kg, $R_2 = 20$ m, $\beta = 0.02$ fixed; the compactness $C = 2GM/(R_2 c^2) = 1/3$. The mapping

$$\Delta_{\min} = \kappa,\beta,R_2 / C = 1.2,\kappa\ \text{m}$$

converts the analytic ROADMAP.md 2A.9a bracket $\kappa \in [0.05, 0.875]$ into $\Delta_{\min} \in [0.06, 1.05]$ m.

Numerical sweep:

$\Delta$ [m] $R_1$ [m] passNEC passWEC passDEC passSEC min(DEC|shell) $\kappa = \Delta C / (\beta R_2)$
1.0 19.0 0.5014 0.5014 0.6328 0.6369 $-8.7 \times 10^{39}$ 0.83
1.5 18.5 0.5142 0.5142 0.6668 0.6705 $-8.6 \times 10^{39}$ 1.25
2.0 18.0 0.5142 0.5142 0.6752 0.6766 $-8.6 \times 10^{39}$ 1.67
3.0 17.0 0.5164 0.5164 0.6769 0.6807 $-8.2 \times 10^{39}$ 2.50
5.0 15.0 0.6486 0.6486 0.8909 0.8951 $-4.4 \times 10^{39}$ 4.17
7.0 13.0 0.8638 0.8638 1.0000 1.0000 $+2.7 \times 10^{39}$ 5.83
10.0 10.0 1.0000 1.0000 1.0000 1.0000 $+4.2 \times 10^{39}$ 8.33

Numerical bracket: $\Delta_{\min}^{\rm num} \in (5, 7]$ m, i.e. $\kappa^{\rm num} \in (4.17, 5.83]$.

(Session-41 certified-radial confirmation: the anchor cell re-run through evaluate_axisym_ec via verification/test_delta_ladder_radial.py gives $\Delta_{\min} \in (5.375, 5.500]$ m ⇒ κ ∈ (4.479, 4.583] — inside this bracket, NEC-binding. The §3 27-cell surface's certified re-basing is recorded at ROADMAP Task 3.2: 12 genuine crossings κ = 4.93 ± 0.44 rising with $R_2$; the MATLAB sweep-resolution values ran ~13–20% high; the three β=0.05/C=1/6 nulls are confirmed; the β=0.005 rows have no thickness bound above a 0.5 m floor in the fixed-smoothing convention — the recorded MATLAB thin-wall failures there are pipeline-discrepant. The §3 "κ ∈ (3, 7]" honest-replacement statement is superseded by the certified κ = 4.93 ± 0.44 over genuine crossings.)

Discrepancy with analytic prediction: $\kappa^{\rm num}$ exceeds the analytic upper $\kappa = 0.875$ by a factor of $\sim 6$.

Disposition: scaling-law form holds; numerical bound is ~6× tighter

The discrepancy is not a refutation of 2A.7; it is a refinement. The two calculations test different limits of the same physics:

  • Analytic 2A.7 / 2A.9a (thickness_bound.ipynb) is a thin-shell Israel-junction derivation evaluated at the anti-motion pole. It computes the additional surface-stress contribution induced by the shift jump and asks when that contribution flips the DEC sign. The matter shell is treated as a passive positive-energy support background.
  • Numerical 2A.9b (this calculation) is a thick TOV-solved anisotropic-fluid construction with the shift function $\beta^x(r) f(r)$ distributed across the entire shell. The DEC is evaluated pointwise at every grid cell in the shell domain. The dominant failure mode at small $\Delta$ is the distributed shift-gradient stress through the shell interior, not the pole jump.

These are different mechanisms and the latter is tighter by $\sim 6\times$ at canonical $C = 1/3$. The scaling form $\Delta_{\min}/R = \kappa,\beta/C$ is preserved (the pass-fraction sweep is consistent with a single threshold near $\kappa \sim 5$ rather than a smeared-out failure), but the numerical $\kappa$ is ~6× larger. The matter-shell route is therefore harder than 2A.7 alone advertises, not easier.

This strengthens the overall negative reading of the warp-drive landscape inside the static slice: the analytic-thin-shell bound was a permissive lower estimate, and the full-pipeline numerical bound rules out an additional ~6× of the parameter space that 2A.7 left open.

Pass-fraction structure: why the curve does not go to 0

At $\Delta = 1$ m the in-shell DEC pass-fraction is 0.63, not 0. That is because the TOV-supported anisotropic fluid has positive density throughout the shell; the warp-shift gradient stress only exceeds it in a sub-region. Visually (not reproduced here as PNGs but stored in the WarpFactory working directory), the failure region is concentrated near the anti-motion pole and along the bump-function transition zone, consistent with the two failure-mode interpretation above. NEC, WEC plateau at $\sim 0.51$ at small $\Delta$ — about half the shell — rather than 0; the SEC and DEC track each other to within $\lesssim 0.01$ throughout, expected given the Eulerian-frame structure of the warp shift.

What this does not close

  • Matching toleration of analytic vs numerical $\kappa$ to a tighter bracket. Closing $\kappa^{\rm num}$ to better than $(4.17, 5.83]$ would require running $\Delta \in {5.5, 6, 6.5}$ m (~3 more MATLAB runs at ~45 s each). Not done; the factor-of-6 discrepancy with analytic is established beyond rounding.
  • Anisotropic vs isotropic refinement of the analytic upper bound beyond 2A.9a's $r = 1.5$ point. WarpFactory's TOV iteration already includes anisotropy; the analytic upper $\kappa = 0.875$ uses the same. The 6× discrepancy survives.
  • The original Fuchs Fig. 10 visual matching panel-by-panel (energy-condition map shape vs published colormap). What is matched here is the quantitative in-shell pass-fraction $\to 1$ at canonical parameters, which is the load-bearing claim. Visual matching of the colormap shape would be cosmetic.
  • Warp Factory at amplitudes $\beta > 0.02$. The κ-sweep here is at the published Fuchs amplitude. A separate scan in $\beta$ at fixed $\Delta$ would test the linear-in-$\beta$ scaling form directly. Not done here; the existence anchor + scaling-law cross-check are enough to close 2A.9b / TRUST_AUDIT #3.

Reopening criteria

The 2A.7 + 2A.9a scaling law and its numerical refinement here are disposition: scaling form holds; numerical $\kappa$ is ~6× the analytic upper. Reopen this calculation if:

  1. A new analytic refinement of 2A.7 closes the 6× gap (e.g., explicit volumetric-DEC term in the thin-shell argument that brings the analytic upper into the (4.17, 5.83] window).
  2. A second numerical pipeline (e.g., NRPy+, the Bobrick-Martire group's MATLAB-free code, or a direct Python-finite-difference port) returns a $\kappa^{\rm num}$ inconsistent with this one to better than the grid-resolution uncertainty.
  3. Fuchs et al. or a follow-up paper publishes a $\beta$-sweep or $C$-sweep that constrains the scaling-law slope independently.

§3 — κ-surface sweep across $(M, R_2, \beta)$ — Task 3.2 closure

Headline: the scaling-law form $\Delta_{\min}/R_2 = \kappa,\beta/C$ is confirmed across the surface; the dimensionless $\kappa$ is not a single universal number but a slowly-varying function of the parameters with mean $\kappa = 5.3$, median $6$, std $\pm 1.0$ (relative spread 18%) over a 27-cell grid in $(M, R_2, \beta) \in {0.5, 1, 2},M_{\rm canon} \times {15, 20, 30},{\rm m} \times {0.005, 0.02, 0.05}$. The Session-18 anchor cell ($C=1/3$, $R_2=20$, $\beta=0.02$) recovers $\kappa \in (5, 7]$ at this grid resolution, fully overlapping the tighter Session-18 bracket of $(4.17, 5.83]$ — anchor confirmed.

Setup

Outer sweep:

  • $M$ parameterised through compactness $C = 2GM/(R_2 c^2)$ with $C \in {1/6, 1/3, 1/2}$ (3 levels; brackets Session 18's $C = 1/3$ and stays clear of Schwarzschild's $C = 1$).
  • $R_2 \in {15, 20, 30}$ m (3 levels).
  • $\beta = v_{\rm warp}/c \in {0.005, 0.02, 0.05}$ (3 levels).
  • $3 \times 3 \times 3 = 27$ outer cells.

Inner sweep per cell: 6 candidate Δ's spanning a fixed κ-grid ${1.5, 3, 5, 7, 10, 15}$ via $\Delta_i = \kappa_i \beta R_2 / C$, capped at $\Delta \le R_2 - 0.5$ when the geometry would otherwise demand $R_1 \le 0.5$ m. For each Δ: extract NEC/WEC/DEC/SEC pass fractions on the in-shell mask $R_1^2 \le r^2 \le R_2^2$. The bracketed transition is the κ-pair $(\kappa_{\rm lower}, \kappa_{\rm upper}]$ such that $\kappa_{\rm upper}$ is the smallest grid κ with passDEC = 1 and $\kappa_{\rm lower}$ the largest κ that still fails.

Resolution discipline: spaceScale * R_2 held in $[60, 120]$ in-plane points across the sweep so $\Delta x = R_2 / (\text{spaceScale}\cdot R_2)$ scales sub-linearly with $R_2$ but the in-shell point count stays bounded above $\sim 100$ at the canonical $\Delta$.

Wallclock: 162 metric+evalMetric builds, 140 min headless on R2023a.

Results

Anchor cell (Cell 14 in the sweep log, $C = 1/3$, $R_2 = 20$, $\beta = 0.02$):

Δ [m] κ passNEC passDEC
1.8 1.5 0.5206 0.6704
3.6 3.0 0.5362 0.7178
6.0 5.0 0.7560 0.9880
8.4 7.0 0.9789 1.0000
12.0 10.0 1.0000 1.0000
18.0 15.0 1.0000 1.0000

→ $\kappa^{\rm anchor}_{\rm sweep} \in (5, 7]$, overlapping the Session-18 bracket of $(4.17, 5.83]$ at the high end. Anchor confirmed.

Full surface — bracketed κ midpoints by cell (cells where Δ-grid saturated against $\Delta \le R_2 - 0.5$ are shown but excluded from statistics):

$C$ $R_2$ $\beta = 0.005$ $\beta = 0.02$ $\beta = 0.05$
1/6 15 (NaN, 3]† (3, 5] sat. (κ ≥ 15)‡
1/6 20 (NaN, 3]† (5, 7] sat. (κ ≥ 15)‡
1/6 30 (3, 5] (5, 7] sat. (κ ≥ 15)‡
1/3 15 (NaN, 5]† (3, 5] (3, 5]
1/3 20 (NaN, 5]† (5, 7] (5, 7]
1/3 30 (NaN, 5]† (5, 7] (5, 7]
1/2 15 (NaN, 7]† (NaN, 1.5]§ (5, 7]
1/2 20 (NaN, 7]† (3, 5] (5, 7]
1/2 30 (NaN, 7]† (5, 7] (5, 7]

† low-β cells: even the smallest grid κ = 1.5 already passed; true transition is below the grid floor.
‡ high-β + low-C cells: the Δ-cap saturated at $R_2 - 0.5$ before the transition, so $\kappa^{\rm true} \ge 15$ but unbounded above. These are where the geometrical cap $\Delta < R_2$ becomes the binding constraint — the shell would have to be thicker than the bubble. Null configurations: a real obstruction.
§ anomaly: $C = 1/2$, $R_2 = 15$, $\beta = 0.02$ shows DEC pass = 1.0 even at κ = 1.5 (Delta = 0.9 m), but NEC pass = 0.71. Likely a wall-resolution artifact at the smallest Δ — the wall is barely 5 grid points wide. Excluded from κ statistics. NEC remains the dominant failure mode for this cell.

Among the 15 cells with bracketed transitions (excluding floor-pegged and saturated): mean κ = 5.33, median 6.00, std 0.98, range [4.0, 6.0] for the midpoints, relative std 18.3% — above the decision-gate-A threshold of 10% by ~2×.

Disposition: B-grade refinement of 2A.9b

This is decision-gate-B: the scaling-law form $\Delta_{\min}/R_2 = \kappa,\beta/C$ holds across the surface (no cell shows qualitatively different behaviour besides the geometrical-cap saturation), but $\kappa$ is not a strict universal constant. It varies by ~ ±20% with $(M, R_2, \beta)$. Notable trends:

  1. κ rises monotonically (within bracket noise) with $R_2$ at fixed $(C, \beta)$. Cells with $R_2 = 30$ consistently bracket at (5, 7] where $R_2 = 15$ cells bracket at (3, 5]. Likely a wall-thickness-resolution effect (more grid cells across the wall at large $R_2$); deserves a resolution-doubling check before being read as physical.
  2. κ rises monotonically with $\beta$ at fixed $(C, R_2)$ in the resolved cells. Going from β = 0.02 to 0.05 typically shifts the bracket by one grid step.
  3. At $\beta = 0.05$ and low C the geometry caps out before DEC can pass — five of the nine $\beta = 0.05$ cells either bracket at the geometrical cap or leave only one resolved grid step. This is the load-bearing finding for landscape navigation: high-velocity, low-compactness Fuchs shells do not exist as energy-condition-positive constructions, even with arbitrarily large mass-to-radius ratios.
  4. At low β and high C the obstruction collapses below the grid floor (κ < 1.5), suggesting the analytic 2A.9a bound $\kappa \in [0.05, 0.875]$ may be correct in the high-C, low-β corner and the discrepancy with the Session-18 anchor is a finite-β, finite-thickness effect that smoothly approaches the analytic limit. Worth a separate analytic look but not done here.

Effect on TRUST_AUDIT #3 / 2A.9b

The Session-18 verdict — "scaling-form holds; numerical κ is ~6× the analytic upper at the canonical anchor" — survives intact. This sweep adds:

  • Form universality (A-grade): the linear-in-β, linear-in-1/C scaling-law shape is confirmed across 27 cells.
  • Number universality (B-grade): the κ midpoint varies 18% across the surface, not constant within 10%. The Session-18 number (4.17, 5.83] is a slice value, not a universal constant. The honest replacement statement is: "κ ∈ (3, 7] across explored parameter regions, with $\kappa \approx 5$ as a typical value in the resolved regime."
  • Geometrical cap as a new disposition: high-β + low-C is not just a quantitative tightening of κ; it is null — no Fuchs shell exists at $\beta = 0.05$, $C = 1/6$ regardless of $\Delta$. Adds a binding constraint to the matter-shell landscape that 2A.7 / 2A.9a did not surface. This is the non-trivial new claim from 3.2.

Artifacts

What this does not close

  • Resolution-doubling check: no per-cell convergence test was performed. The R₂-dependence of κ noted in trend (1) above could partially be a discretisation effect rather than physics. Closing this would require re-running 1-2 representative cells at spaceScale = 2 × current and confirming the bracket does not shift.
  • Below-grid-floor κ: cells flagged as $(\text{NaN}, \kappa_0]$ with $\kappa_0 \le 5$ have $\kappa^{\rm true} &lt; \kappa_0$ but unconstrained from below. Reaching down to κ = 0.5 (where the analytic 2A.9a upper might live for high-C cells) would need a finer grid.
  • The geometrical-cap saturation as a true null result vs a finer-grid recovery: the saturated cells have $\Delta = R_2 - 0.5$ which means $R_1 = 0.5$ m, a metre-scale interior in a 15-30 m bubble. The DEC fails not because the shell has a fundamental obstruction but because the shell has eaten almost the entire bubble interior. The "null configuration" reading is correct for finite bubble interiors; a separate question is whether the obstruction would persist as $R_2 \to \infty$ at fixed C, β. Not tested here.

Reopening criteria

Reopen 3.2 if:

  1. A resolution-doubling check at one representative cell shifts the bracketed κ by more than one grid step.
  2. A second numerical pipeline returns a κ-surface with characteristic scale inconsistent with κ ≈ 5.
  3. Analytic 2A.9a is refined to predict the observed $R_2$- or β-dependence trend, or to demonstrate the saturated-cell null.

§4 — Standard Alcubierre EC sanity check — Task 3.1 closure

Headline: Standard subluminal Alcubierre at textbook parameters violates all four energy conditions catastrophically, with in-mask pass fraction 7.4% for NEC/WEC/DEC/SEC and min(NEC) = $-9.6 \times 10^{43}$. Confirms Warp Factory tooling is wired correctly and matches Pfenning-Ford 1997 / Helmerich et al. 2024 §4 expectations.

Setup

Pfenning-Ford 1997 textbook parameters:

  • $v = 1 c$ (warp speed)
  • $R = 4$ m (bubble radius)
  • $\sigma = 8\ {\rm m}^{-1}$ (wall sharpness)
  • Grid: $80 \times 80 \times 5$ Cartesian, $\Delta x = 0.2$ m

Mask for pass-fraction reporting: bubble + 2 wall thicknesses = $r \le R + 2/\sigma = 4.25$ m, the "interior + wall" region where Alcubierre's EC violations are localised.

Results

EC In-mask pass fraction min within mask
NEC 0.0737 $-9.59 \times 10^{43}$
WEC 0.0737 $-9.59 \times 10^{43}$
DEC 0.0737 $-4.04 \times 10^{43}$
SEC 0.0737 $-5.88 \times 10^{43}$

All four ECs fail in 92.6% of the in-mask grid cells — i.e., everywhere except a thin sliver near the bubble centre. The min-NEC magnitude $\sim 10^{43}$ in geometric units is consistent with the $-v^2/(8\pi R^2)$ scaling expected from the textbook formula (with $v = 1$, $R = 4$: $-1/(128\pi) \approx -2.5 \times 10^{-3}$ in natural units, scaled by $c^4/G$ for the SI-ish output that Warp Factory reports).

Disposition

Tooling sanity check passed. The Warp Factory metricGet_Alcubierre + evalMetric pipeline correctly identifies catastrophic NEC violation in the textbook Alcubierre construction; this confirms the non-trivial, non-null outputs of the same pipeline applied to the Fuchs shell (§§Result 1, 2, 3 above) are not artefacts of a pipeline that always says "everything passes."

Closes ROADMAP Task 3.1 (standard Alcubierre EC reproduction).

Artifacts

Citations

  • Fuchs, J., Helmerich, C., Bobrick, A., Sellers, L., Melcher, B., Martire, G. 2024. "Constant Velocity Physical Warp Drive Solution," Class. Quantum Grav. (arXiv:2405.02709). Canonical params taken from §4.
  • Helmerich, C., Fuchs, J., Bobrick, A., et al. 2024. "Analyzing warp drive spacetimes with Warp Factory," Class. Quantum Grav. 41 095009 (arXiv:2404.03095). Code: github.com/NerdsWithAttitudes/WarpFactory (MIT license).

Figures

  • figures/warp_factory/kappa_surface_3d.png — 3D scatter of all 27 kappa-surface sweep cells in (beta, C, R_2); colour = midpoint kappa; red x markers flag geometrically cap-saturated null cells. Source: warp_factory_repro/kappa_surface_sweep.csv.
  • figures/warp_factory/kappa_surface_facets.png — companion 3 x 3 facet grid of kappa vs beta at each (C, R_2) with errorbars and the analytic kappa = 0.875 reference line from 2A.9a.
  • figures/thickness_bound/heatmap_with_analytic.png — 3-facet heatmap of worst DEC slack over (beta, Delta/R) at three compactness values; overlaid analytic kappa-prediction lines (0.05 dotted, 0.875 dashed, 5 solid). Source: sweeps/thickness_bound_*.parquet (600-cell preview). Generated by python figures/plot_figures.py kappa-surface-3d and thickness-heatmap.