https://scalarresonance.org/studies/ Mathematical and theoretical studies from the Scalar Resonance Research Program. en Wed, 29 Apr 2026 00:00:00 GMT https://scalarresonance.org/papers/galois-solids.pdf https://scalarresonance.org/papers/galois-solids.pdf Wed, 29 Apr 2026 00:00:00 GMT Spherical polyhedra constructed from canonical embeddings of algebraic number fields. The icosahedron is the Galois solid of ℚ(φ) under A₅; the Alfredohedron is the Galois solid of ℚ(α) under A₄. https://scalarresonance.org/papers/morphic-sumset-growth.pdf https://scalarresonance.org/papers/morphic-sumset-growth.pdf Tue, 28 Apr 2026 00:00:00 GMT The sumset G + G of a geometric progression {1, r, r², …, r^(N−1)} has collisions if and only if r is a morphic number, identifying these ratios as the additive-minimal positive reals. https://scalarresonance.org/papers/metallic-mean-gap-ratio.pdf https://scalarresonance.org/papers/metallic-mean-gap-ratio.pdf Tue, 28 Apr 2026 00:00:00 GMT For irrationals α with constant continued fraction [0; a, a, a, …], the gap ratio of the Kronecker sequence {nα mod 1} takes exactly (a+1) distinct values forming an arithmetic sequence — a complete characterization of the noble numbers via the Three-Distance Theorem. https://scalarresonance.org/papers/hash-optimality.pdf https://scalarresonance.org/papers/hash-optimality.pdf Sun, 01 Mar 2026 00:00:00 GMT Knuth's recommendation to use 1/φ as a hash multiplier proven optimal via the Three-Distance Theorem. https://scalarresonance.org/papers/sturmian-compression.pdf https://scalarresonance.org/papers/sturmian-compression.pdf Sun, 01 Mar 2026 00:00:00 GMT The Fibonacci word eventually maximizes the LZ78 phrase count among all Sturmian sequences. The class of asymptotically co-maximal rotations is exactly the noble numbers. https://scalarresonance.org/papers/markov-dissonance.pdf https://scalarresonance.org/papers/markov-dissonance.pdf Sun, 01 Mar 2026 00:00:00 GMT The classical Lagrange spectrum read as a complete musical hierarchy of anti-consonance. φ sits dramatically isolated at the apex — 20.9% more anti-consonant than √2 (the tritone, rank 2).