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2026
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| Online Access: | https://doi.org/10.5281/zenodo.20388451 |
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| author | Somma, Massimo Michele Edoardo |
| author_facet | Somma, Massimo Michele Edoardo |
| contents | <p>This paper records a canonical decomposition of the golden–plastic compositum induced by the two normalized relative trace projections onto the golden and plastic subfields. Because the golden and plastic fields are linearly disjoint and the compositum has degree six, these two projections commute, are idempotent, and have product equal to the rational trace projection.</p> <p>The compositum splits accordingly into four canonical sectors: the rational sector, the golden trace-zero sector, the plastic trace-zero sector, and the mixed sector given by their tensor product. For a SATOR mass monomial in the golden and plastic units this yields an exact four-sector decomposition into rational, golden-only, plastic-only, and mixed components. The mixed component factorizes exactly as the product of the golden monomial minus half the corresponding Lucas number and the plastic monomial minus a third of the corresponding Perrin number, so it vanishes if and only if one of the two exponents vanishes. In the canonical SATOR thirteen-row internal set the mixed-null rows are therefore exactly the three normal rows with vanishing golden exponent.</p> <p>The closing sections make the cyclic class-field action explicit. After passage to the splitting field, the plastic trace-zero plane carries the two-dimensional irreducible rational representation of the cyclic group of order three, and a Fourier/Gauss basis diagonalizes the cyclic action after adjoining a primitive cube root of unity. The Fourier-conjugate product of the two plastic modes is exactly three, and its mixed lift has product fifteen-fourths. Two further exact identities in the plastic field are recorded — the sum of the cubes of the two Fourier plastic modes equals twenty-seven, and a specific quadratic polynomial expression in the plastic generator equals sixty-nine — exposing precise arithmetic fingerprints of the cyclic trace-zero sector.</p> <p>The contribution is mathematical and structural: it isolates the canonical four-sector splitting of the compositum and the exact factorization of the mixed sector on golden–plastic monomials.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20388451 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | The SATOR Matrix - Bi-Trace Decomposition of the Golden-Plastic Compositum. Canonical projections, norm/trace separation, cyclic plastic modes, and the mixed sector. Somma, Massimo Michele Edoardo <p>This paper records a canonical decomposition of the golden–plastic compositum induced by the two normalized relative trace projections onto the golden and plastic subfields. Because the golden and plastic fields are linearly disjoint and the compositum has degree six, these two projections commute, are idempotent, and have product equal to the rational trace projection.</p> <p>The compositum splits accordingly into four canonical sectors: the rational sector, the golden trace-zero sector, the plastic trace-zero sector, and the mixed sector given by their tensor product. For a SATOR mass monomial in the golden and plastic units this yields an exact four-sector decomposition into rational, golden-only, plastic-only, and mixed components. The mixed component factorizes exactly as the product of the golden monomial minus half the corresponding Lucas number and the plastic monomial minus a third of the corresponding Perrin number, so it vanishes if and only if one of the two exponents vanishes. In the canonical SATOR thirteen-row internal set the mixed-null rows are therefore exactly the three normal rows with vanishing golden exponent.</p> <p>The closing sections make the cyclic class-field action explicit. After passage to the splitting field, the plastic trace-zero plane carries the two-dimensional irreducible rational representation of the cyclic group of order three, and a Fourier/Gauss basis diagonalizes the cyclic action after adjoining a primitive cube root of unity. The Fourier-conjugate product of the two plastic modes is exactly three, and its mixed lift has product fifteen-fourths. Two further exact identities in the plastic field are recorded — the sum of the cubes of the two Fourier plastic modes equals twenty-seven, and a specific quadratic polynomial expression in the plastic generator equals sixty-nine — exposing precise arithmetic fingerprints of the cyclic trace-zero sector.</p> <p>The contribution is mathematical and structural: it isolates the canonical four-sector splitting of the compositum and the exact factorization of the mixed sector on golden–plastic monomials.</p> |
| title | The SATOR Matrix - Bi-Trace Decomposition of the Golden-Plastic Compositum. Canonical projections, norm/trace separation, cyclic plastic modes, and the mixed sector. |
| url | https://doi.org/10.5281/zenodo.20388451 |