A Low-Coupling Circle of Harmonic Tensors Selected by the Contact Direction and Long-Tail Return Memory: An Interim Report on the Hayashi Harmonic-Sphere Model (v0.6)

Interim report (preprint), v0.6. This is an interim report on a research model that the author calls the Hayashi Harmonic-Sphere Model. Naming the model does not mean that it is complete or verified. What is constructed here is a contact model on the sphere of internal shapes S⁴ ⊂ Sym₀(3) with the positive-definite Euclidean metric (the Qn long-tail model). Every derivative, elapsed time, and decay variable is the observed time.

Pillar 1 gives conditions under which slow internal return memory may be omitted and the prior harmonic-sphere description recovered: exact return by an isometric intertwining map (Proposition 1), finite-time error bounds of order ε or ε² under weak coupling with a positive decay floor (Proposition 2), and an error bound for replacing a short memory kernel by its direct-current (DC) coefficient, that is, the area of the kernel (Proposition 3).

Pillar 2 shows how the low-coupling circle selected by the contact direction makes the memory structurally different from local or fixed finite-exponential approximations: the rotation-covariant contact map G_n(Q) = g Qn and its zero circle with normal Gram eigenvalues (1/2, 1/2, 2/3) (Proposition 4); the t^−3/2 long tail of the uniform-readout return memory kernel with an exact incomplete-Gamma bound on the residual (Theorem 5); the same tail in the exact solution of a passive parent coupled system (Theorem 6); a finite DC integral with a divergent first memory moment and a non-analytic real-Laplace square-root correction (Proposition 7, Theorem 8, Proposition 9); the fixed ratios 825/64 and 11/3 between the uniform and the degree-4 harmonic readouts (Theorem 10); the exponential cutoff of the tail by an angular spread of the contact direction through the exact identity K_ε(t) = e^−δt K₀((1−3ε)t) (Theorem 11); and the sign inversion between a friction memory tail and the velocity impulse response tail (Proposition 12).

This is an interim report on a mathematical model built under explicitly stated conditions. It does not claim experimental proof of the null time or of a null spacetime metric, nor universal superiority over the full Boltzmann equation or over generalized Langevin equations with general continuous spectra. The new model has not yet been compared independently with measurements.

Files. Japanese and English main-text PDFs (manuscript_ja.pdf, manuscript_en.pdf; each figure is placed on its own landscape page, with its caption at the top of the next page), a guide (README.md, in Japanese), one ZIP archive (contact_tail_v06_public_files.zip) with the 87 public files (Markdown manuscripts, 16 figures in SVG and PNG in Japanese and English, figure sources, the scripts that compute the numbers in the text and figures, and the self-contained verification code) together with their MD5/SHA-256 lists, and MD5/SHA-256 manifests of the uploaded files (MANIFEST_md5.txt, MANIFEST_sha256.txt).

日本語要旨(第0.6版):粒子を一つの向きから押すと、内部の変形模様のうち、その向きとほとんど結び付かない模様が円周状に並び、押した影響が長く残る。残り方はべき乗で、指数関数のように同じ時間の幅ごとに同じ割合で減るのではない。本報告は、この長い響きを無視してよい条件(第1の柱:命題1〜3)と、無視できないときの予言(第2の柱:命題4・7・9・12と定理5・6・8・10・11)を、一つの幾何学模型から導く中間報告である。著者はこの研究模型を林調和球模型と呼ぶが、名前は研究目標を指し、模型の完成や実証を意味しない。本稿で構成したのは、正定値の S⁴ 上の Qn 長尾模型である。第三者の人間による査読は経ていない。