Ceiling-Normalized Storage Bounds for Binary Tensor Trains: Exact scaling laws and compression crossovers for quantics representations

This preprint derives exact storage bounds for reduced binary tensor trains (TTs) and quantized tensor-train (QTT) representations. For an order-d binary tensor containing N = 2^d dense entries, it accounts explicitly for the unfolding-imposed rank limits at both ends of the train. The resulting filled-cap envelope gives the exact maximum number of stored core scalars permitted by a specified maximum-rank cap, sharpening the conventional O(dR²) estimate. For power-of-two rank caps, the storage envelope reduces to an exact dimensionless law in R/√N, including its finite-size correction. For arbitrary integer caps, the normalized envelope is piecewise quadratic. Its limiting dense-storage crossover occurs at approximately 0.348242 of the integer structural rank ceiling for even train orders and 0.475684 for odd train orders. Falling below the envelope crossover guarantees compression under the stated storage convention; crossing it alone does not establish expansion. The principal asymptotic result is that, for sequences of reduced binary tensor trains, sublinear core storage is equivalent to maximum rank R = o(√N). Bounded rank is therefore sufficient but not necessary: absolute ranks may grow without bound while the fraction of dense storage tends to zero. Constructive attainability and uniform-bound proofs support the storage envelope and its asymptotic interpretation. The accompanying publication bundle includes the original 7,450-row arithmetic sweep, exact-arithmetic verification with explicitly stated finite coverage, an audit-authored reconstruction of the sweep generator, four vector figures and their generation code, LaTeX source, build instructions, and checksum manifests. The reconstructed generator reproduces the historical dataset byte-for-byte under the recorded environment. These results concern ordinary dense-core scalar storage for reduced tensor trains, with comparisons made at equal scalar type and precision. They do not establish that a physical workload achieves any particular fidelity-qualified rank, nor do they establish runtime or peak-memory advantages. The accompanying dataset is a combinatorial storage sweep, not physical simulation evidence.