Foundations of Warp-Drive Physics (I): Nonnegative-Energy Initial Data for Warp Bubbles Explicit Constructions of Ordinary-Matter Initial Data and Sharp Shaping Windows

Abstract Static warp bubbles canberealizedwithout negative energy. Weexhibit a wholeclass of warp bubbles built solely from ordinary matter satisfying the weak energy condition. Their full-spacetime realizations are stratified by symmetry and charge: spherically symmetric neutral, continuously charged, and arbitrarily shaped charged configurations yield exact static solutions, while a non-CF Weyl class yields zero-charge textured weak-field bubbles (numerically scanned). The shaping capability is sharply locked, its cost is computable, and the configuration can be reconstructed from the target by inversion. The stratification is sharp: to have texture one must leave the conformally flat class; to have arbitrary shape one must pay charge (e/m = 1); to be neutral one must accept either spherical symmetry (exact) or the weak field (textured). The conclusions follow from one unified structure: the Xinxiang Boundary Encoding Algebra (AECM, §2–§7). Its claim is that, in the constraint sector, the nonlinearity of the Einstein equations is re-encoded as a linear elliptic encoding of an effective source σ together with the convex constraint σ ≥ 0. In the time-symmetric conformally flat sector of general relativity (Kij = 0, i.e. an initial slice with neither expansion nor shear in the cabin), the constraint equations reduce to a single Lichnerowicz equation; the exterior geometry is uniquely determined by the boundary data on the diagnostic surface through elliptic development, and every global quantity is read off mode by mode alongtheDirichlet–Neumannspectrum. Themapfromsourcetoboundarydatahasonlyfouressential questions (image, kernel, singular values, adjoint), and all four have explicit answers; their physical projections are the four verdicts Q1–Q4 below. The same encoding principle holds at the level of exact static spacetimes: the exterior spacetime of an equipotential-shell bubble (metric, all multipole moments, mass, redshift) is fully encoded by a single scalar boundary datum (Theorem 11.10). Three verdicts follow. Feasible: sources satisfying the slice prescription ρ ≥ 0 (a necessary condition for the full-spacetime WEC) can actively shape the exterior vacuum geometry through multipole distributions; the attainable range of the quadrupole shaping ratio is [−1 2,1] at the universal level and tightens to the sharp window [−1 5, 2 5] at the spectrally pure level, with both endpoints saturated pointwise by explicit thin-shell configurations in the weak-∗ closure sense. Obstructed: within the conformally flat class, quadrupole-textured initial data admit no smooth small-amplitude timelike static extension (Theorem 10.6): the obstruction tensor is explicitly nonzero on every massive quadrupole exterior (Byxx = q(q − 12m)/3888 for all q= 0). Realized: the spherically symmetric neutral bubble (Theorem 11.1) is an exact static solution satisfying the full-spacetime WEC; the charged Majumdar–Papapetrou bubble holds for arbitrary smooth shapes, with a strictly Minkowski, zero-tidal interior, at the cost of the extremal charge-to-mass ratio e/m = 1. General static flat-cavity bubbles are unified by the capacity law MADM = f(N)C(Σ) (neutral branch f = ln(1/N), charged branch f = 1/N −1),andtheisoperimetric lower bound M ≥ f(N)(3V /4π)1/3 is saturated on the sphere. The core delivered here is a theory of physical realizability: given a target (mass, texture, redshift), we provide the unique structure, the construction algorithm, and the cost Allconclusionsonmomentwindows,texturepricing,andshapingverdictsrestonthetimesymmetricslice: thisisboththephysical requirementofatime-orderedcabininawarpship(the conditionKij=0removesexpansionandshear)andthemathematicalentrypoint thatmakesthe problemanalyticallysolvable. Anamedexample: thehabitableflatcavity.Theorem5.30givesanexplicitconstruction(aflat, habitableinterior, anexteriorcarryingthefullmultipoletexture, andashell layerwithρshell ≥0 pointwise). Positionintheseries. This isPaper Iofaseries: themomentwindows, texturepricing, and shapingverdictsprovedherearethedesignconstraintsfortheengineering-gradecompletiondevelopedinthesubsequentpapers—dynamicalevolution(openproblems3,3bis),full-spacetimeenergyconditionverificationandstability(openproblems14, 17, 27), thevectorboundary-datachannel (openproblem6),andtheremainingescaperoutes(openproblems20,21). Keywords: time-symmetricinitialdata; constraintequations; conformalflatness;Dirichlet-toNeumannoperator;multipolemoments;Hawkingmass; thinshell;Majumdar–Papapetroububble; conformastaticrigidity;realizabilitytheory;Weylclass;conductorpotential;radialdegeneracy;Newtoniancapacit