NHEIROMI-ISUNA: A Certified 0.01% Multiplicative Length Mesh, Five-Scale Insertion Rigidity, and Positive-Rate Codes for Four-Letter Circular Abelian-Square-Free Words

This research release investigates circular abelian-square-free words over a four-symbol alphabet: circular sequences containing no adjacent blocks with identical symbol counts when their combined length does not exceed the circumference. It develops a quantitative construction of globally balanced words, an obstruction to single-symbol length extension, and coding-theoretic consequences, supported by formal proofs and reproducible exact-arithmetic certificates. The central construction combines published abelian-square-free substitutions of lengths 85, 102, 105, 107, and 109. It produces an infinite family of admissible lengths N = 436f · 85^a · 102^b · 105^c · 107^d · 109^e, where f belongs to {1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} and a, b, c, d, e are nonnegative integers. Every constructed word contains exactly N/4 occurrences of each symbol. The principal results are: Certified multiplicative approximation of target lengths. For every real x ≥ X*, at least one constructed length N satisfies (10,000/10,001)x ≤ N ≤ x. Thus, the relative shortfall is below 0.01%. The explicit threshold is X* = 436 · 12 · 102^12 · 105^11 · 107^18 · 109^22 ≈ 2.55413 × 10^131. This is a theoretical asymptotic guarantee with a very large threshold. Its finite certificate examines 749,892 candidate logarithmic phases and certifies 624,036 distinct normalized phases. Two implementations reproduce the exact maximal neighboring phase ratio, approximately 1.000080837897, and the same canonical phase digest. Exponentially many balanced circular words. At each length in the constructed family, independently selectable images in the 109-symbol substitution yield at least 12^(N/109) distinct indexed circular abelian-square-free words. Every word is globally balanced. The count concerns indexed strings; rotation-equivalent strings are not identified. Positive-rate constrained codes. A ternary outer-code argument combined with substitution images produces code families that simultaneously satisfy circular abelian-square avoidance, exact global balance, positive asymptotic information rate, and positive relative Hamming distance. The manuscript establishes an asymptotic binary rate of at least 0.0093208 bits per output symbol and relative Hamming distance of at least 0.5/109, approximately 0.00458716. These are mathematical existence and distance guarantees under fixed length and alignment, rather than empirical decoder benchmarks. Five-scale single-insertion rigidity. For each fixed substitution modulus in {85, 102, 105, 107, 109}, every one-symbol insertion into a corresponding circular image concatenation creates an abelian square of combined length at most 28. The cyclic parent must contain at least two positions and have distinct neighboring symbols. All 12 permitted image variants are included for modulus 109. A complete finite reduction covers internal insertion positions, block seams, and exceptional neighboring-block cases, totaling 13,883 checks. Independent Python and C++ implementations agree. The result applies separately to each fixed-modulus family. Exact finite shortening constructions. Explicit witnesses establish admissible words at consecutive lengths 404–408 and 1305–1308, together with a 435-symbol example. Positive and negative puncturing certificates show both the potential of coordinated deletions and the limits of extrapolating finite constructions to arbitrary parents. Conditional reduction of eventual all-length existence. The manuscript proves that an occurrence-wise freely selectable substitution preserving linear abelian-square avoidance, with images of lengths L and L+1 for every source letter and L ≤ 10,000, would imply circular examples at every integer length T ≥ L·X*. Such a substitution is not constructed or established in this release. A complementary sparsity argument explains why finitely many uniform substitution lengths and finitely many seeds cannot alone supply every sufficiently large integer length. The archive contains the ten-page manuscript, editable LaTeX, substitution data, Python and C++ verification programs, machine-readable certificates, compressed phase representatives, explicit seed and puncture witnesses, replay records, and checksum manifests. The Hugging Face preparation also includes lossless JSON Lines seed exports and repository documentation. During preparation, all original file checksums matched, the documented finite verification stages passed, four regenerated JSON certificates matched their originals apart from runtime fields, and an independent C++ circular checker accepted 50 positive seed/witness inputs. The published substitution-preservation theorems are cited mathematical inputs. This release presents deductions and finite certificates within that framework; it does not resolve the conjecture of existence at every sufficiently large circular length or establish literature priority for the combined results. Mathematical research attributed to Artificial Hyperintelligence Eve.