Æthra ∼ Veylith: A Sharp Alphabet Law for Sparse Affine-Hyperplane Erasure Recovery, with Exact Moment Certificates, Explicit Inverses, and Noncommuting Packet Codes

This self-contained research release determines the exact minimum symbol alphabet needed to recover every affine-hyperplane erasure in a broad, explicitly defined class of spatial convolution codes. It combines a necessary-and-sufficient recovery criterion, a matching impossibility theorem and construction, sparse encoding, explicit decoding, and companion results showing when noncommuting packet maps succeed where commuting maps cannot. The principal contribution is a complete solution to the alphabet-size optimization problem within the stated convolution-and-linear-relabeling architecture, in every dimension. The storage model places \(2^d\) source symbols and \(2^d\) parity symbols on the two layers of the binary affine space \( F_2^d× F_2\). Symbols belong to \(K=GF(2^m)\). For an arbitrary kernel \(f\) and an invertible binary linear transformation \(T\), encoding is defined by \[ y(v)=∑_u f(v+Tu)x(u). \] A single fixed encoder must recover every possible source assignment after any affine hyperplane is erased. Each such loss removes exactly half of all physical sites, and the guarantee also covers every subset of a protected hyperplane. The central theorem establishes the sharp threshold \[ mₘᵢₙ(d)= d2 (d≥2), mₘᵢₙ(1)=2. \] Below this threshold, no kernel and no invertible linear relabeling in the defined scalar architecture can provide universal hyperplane recovery, including when dense kernels are allowed. At and above the threshold, explicit constructions achieve the guarantee. This supplies both sides of the optimization problem: an impossibility bound for the entire stated class and a constructive family attaining it. The main structural discovery is that universal recoverability depends only on the kernel’s zeroth, first, and second moments. Writing \[ σ=∑_w f(w), μ=∑_w f(w)w, M=∑_w f(w)wwT, \] recovery holds if and only if \(σ≠0\) and \[ aT(σ M+μμT)T- Ta≠0 \] for every nonzero binary normal \(a\). The potentially large physical recovery systems are therefore classified by a field-valued quadratic certificate. The necessity direction is constructive: when the certificate fails, the theory identifies an unrecoverable hyperplane and a nonzero source difference that produces identical surviving observations. A degree-counting argument supplies the alphabet converse, while explicit even- and odd-dimensional constructions establish attainability. The attaining family also has sparse updates and an exact, factored inverse. A changed source symbol affects \(3d/2\) parity symbols, and complete encoding and recovery require \(O(d2^d)\) field operations and \(O(2^d)\) working storage. After contributions from surviving source symbols are subtracted, the inverse uses at most three dependency layers of sparse convolutions. A separate geometric argument proves that at least \(d+1\) parity updates are necessary for any systematic encoder in the fixed physical layout. Explicit examples simultaneously attain the minimum alphabet and this absolute update bound in dimensions 2 through 6. The general minimum update support at the sharp alphabet remains open beyond those established cases. A companion packet theory replaces scalar ray coefficients with arbitrary rectangular linear maps, including noncommuting maps. It gives a universal full-column-rank criterion for the associated matrix pencil and a finite recovery procedure. When the geometric field \(Q\) is contained in the coding field \(K\), the exact minimum parity-packet length is \[ Lₘᵢₙ = max( ,+1/[K:Q] ). \] The release also establishes a complete existence law for normalized commuting square-packet maps and exhibits a concrete separation: on GF(4) geometry with three standard rays, explicit binary four-bit maps recover every prescribed hyperplane, although every normalized commuting four-bit construction fails. In this setting, the complete binary packet-length law is \(Lₘᵢₙ()=max(,3)\). The theoretical results are accompanied by exact computational evidence. A separately implemented moment audit checks 2,304 arbitrary encoders against 47,872 directly constructed physical recovery matrices, with zero reported mismatches. Additional records include 2,665 packet-operator coordinate-basis checks, 359 packet-decoder hyperplane roundtrips, all 63 nonzero GF(4) combinations of the four-bit matrix construction, and direct rank checks for its 84 physical hyperplanes. The package distinguishes mathematical proofs, exhaustive finite certificates, and implementation experiments, and provides the programs and result records needed to reproduce the reported checks. A byte-level construction is executed at \(d=16\) over GF(256), with 65,536 data sites, 65,536 parity sites, and 24 parity updates per changed data byte. A deterministic 65,021-byte payload is restored byte for byte after two separate losses of 65,536 physical sites: erasure of the entire data layer and an oblique cut erasing 32,768 source bytes and 32,768 parity bytes. All 65,535 nonzero moment normals are checked for this instance. The theorem covers all 262,142 affine hyperplanes; the large-payload experiment executes the two specified cuts. The scientific significance lies in turning a geometric recovery requirement into an exact algebraic design rule. The work identifies the minimum alphabet, certifies arbitrary encoders within the stated class, constructs sparse encoders meeting the alphabet bound, and supplies explicit inverse operators and failure witnesses. The packet results additionally isolate a concrete benefit of noncommutativity under fixed geometry, packet width, and update support. These results provide a rigorous framework for studying storage layouts with structured spatial erasures and the relationship between symbol size, update locality, and recovery complexity. The guarantee is specific to affine-hyperplane losses and their subsets. The byte construction has an explicit uncorrectable 25-site pattern consisting of one source site and its 24 affected parity neighbors. The alphabet optimum applies to the stated convolution-and-relabeling class, and the packet theorems retain their specified field and characteristic assumptions. Classical group-algebra, dyadic inversion, Chevalley-Warning, rank-metric, and binary matrix-space ingredients are credited in the manuscript; the claims concern the precise recovery equivalences, optima, constructions, and separations developed in this release. The deposit includes a 39-page standalone manuscript with complete proofs, editable LaTeX sources, Python codec implementations, independently written audits, raw JSON results and execution logs, figures, reproduction instructions, and integrity manifests. The codecs and verification programs require Python 3.10 or newer and use only the standard library. No earlier document or external legacy research package is required to understand or reproduce the stated results.