Curvature-Density Compiler Gains an Explicit Error Bound, Finite Legal Dictionary Theorem Established: The Constant-Width Target Space Is Compressed, for the First Time, into an Enumerable Finite Dictionary

Round 01 (AMRAL-LUC-FC-R01) had already compressed the original target family down to centered unit constant-width bodies, producing an infinite-dimensional but legal curvature-density domain R (0≤r≤1, r(θ+π)=1−r(θ), no first harmonic), and left behind one exact gate: can this still-infinite-dimensional legal domain be built into a finite, verifiable dictionary in which every element keeps convexity and constant width, together with an explicit Hausdorff error? Round 02 (AMRAL-LUC-FC-R02, 2026-09-18) answers YES. Theorem 4.1 proves that circular convolution against any non-negative, normalized, even 2π-periodic kernel preserves convexity, constant width 1, the Steiner gauge, and the curvature-density box constraint exactly; this round instantiates that kernel as the squared-Fejér/Jackson-type kernel J_N (degree 2N), and derives the round's first constructive quantitative theorem — an explicit error bound E_N=π²(4N+1)/[2(2N²+4N+3)(N+1)]=O(N⁻²) (about 0.1110078205 at N=8, falling to about 0.0001489913 at N=256). Because plain Fourier-coefficient rounding can still break convexity, this round adds a safety interiorization h_{N,δ}=(1−δ)J_Nh+δ/2 (leaving a curvature safety margin of δ/2), then quantizes the degree≤2N odd Fourier coefficients onto a grid of spacing q, proving that whenever qA_N≤δ/2 (with A_N=4(N−1)N(N+1)/3) the quantized shape is still a legal unit constant-width body, with total error ε_{N,δ,q}=E_N+δ/2+(N−1)q. This yields Theorem 20.1 (the Constructive Finite Legal Dictionary Theorem): for any N≥2, 0<δ<1, and suitably small q, there exists a genuinely finite legal dictionary D_{N,δ,q} such that every centered unit constant-width body K has some K̂∈D_{N,δ,q} with d_H(K,K̂)≤ε_{N,δ,q}, and for any ε>0 the parameters can be chosen explicitly to drive the error below ε — upgrading Round 01's ABSTRACT FINITE NET to a CONSTRUCTIVE LEGAL FINITE NET. This round's sanity check (test family r=1/2+0.25cos3θ+0.15sin5θ, with N=8, δ=0.1) shows an actual smoothing error of only about 0.006394669, well below the theoretical bound E_8≈0.111007820, with total support error about 0.00946175 — confirming the derivation while showing the constants still have plenty of room to be tightened. The round also introduces a signed placement margin M_U(K) to replace Round 01's non-negative Ψ_U, proves its shape-Lipschitz property |M_U(K)−M_U(L)|≤d_H(K,L), and gives Rule A (a universal-cover certificate) and Rule B (a non-universality witness) as the two halves of a finite dictionary certification sandwich. None of this round's core theorems depend on large-scale computation, however; three follow-on tasks — C02-1 (a dictionary-growth census), C02-2 (a Jackson-compiler adversarial benchmark), and C02-3 (coefficient-quantization branch compression) — are all explicitly marked COMPUTE-DEFERRED. The document is also explicit that this round does not claim to solve the Lebesgue universal covering problem, does not produce any new numerical bound on a_Leb, and does not replace the 2026 finite-certificate routes of Mishra or Zeng: the round's own verdict is CONSTRUCTIVE FINITE SHAPE COMPILER: CLOSED, while GLOBAL FINITE CLOSURE remains STILL OPEN, because the continuous degrees of freedom in orientation, support direction, active-branch saturation, and the candidate-cover boundary have not yet been touched. Research direction and methodology are due to Neo.K; this round's AI collaborating researcher and primary executor was Aletheia / ChatGPT, GPT-5.6 Sol.