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 capa
bility 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 ini
tial 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 ex
act 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 con
dition for the full-spacetime WEC) can actively shape the exterior vacuum geometry through mul
tipole 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 satu
rated 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 time
like 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 symmet
ric 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,andshapingverdictsrestonthetime
symmetricslice: 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-gradecompletiondevel
opedinthesubsequentpapers—dynamicalevolution(openproblems3,3bis),full-spacetimeenergy
conditionverificationandstability(openproblems14, 17, 27), thevectorboundary-datachannel
 (openproblem6),andtheremainingescaperoutes(openproblems20,21).
 Keywords: time-symmetricinitialdata; constraintequations; conformalflatness;Dirichlet-to
Neumannoperator;multipolemoments;Hawkingmass; thinshell;Majumdar–Papapetroububble;
 conformastaticrigidity;realizabilitytheory;Weylclass;conductorpotential;radialdegeneracy;New
toniancapacit