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Pull request overview
This PR synchronizes the English documentation with the Japanese version by refining the cosmological-constant discussion, adding a translated Appendix E, and introducing new translated sections that formalize assumptions, audit coincidence rarity, and clarify identification postulates around the spinor correspondence.
Changes:
- Updates §7.3 in the cosmological constant chapter to reframe the coefficient discussion as a non-unique “algebraic representation” informed by the coincidence audit.
- Extends the English appendices from A–D to A–E by adding a translated numerical confirmation appendix about the flag stabiliser (U(2)).
- Adds three new translated English documents: formal system inventory (§12), coincidence rarity audit (§13), and spinor/internalisation & identification postulates (§14).
Reviewed changes
Copilot reviewed 5 out of 5 changed files in this pull request and generated 6 comments.
Show a summary per file
| File | Description |
|---|---|
| docs/en/04_cosmological_constant.md | Renames/reframes §7.3 to reflect the coincidence audit’s limitations on the coefficient narrative. |
| docs/en/appendix.md | Adds Appendix E with numerical confirmation results and updates appendices header to A–E. |
| docs/en/09_formal_system.md | New English translation formalizing Axiom Ω and enumerating independent axioms/assumptions. |
| docs/en/10_coincidence_audit.md | New English translation quantifying coincidence rarity via exhaustive expression-space scans. |
| docs/en/11_spinor_internalization.md | New English translation clarifying what is proved vs postulated for the (\mathbf{16}) correspondence and identification postulates. |
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| ← [§12: Formalisation of Axiom Ω](09_formal_system.md) | Next → [§14: Internalisation of the $\mathbf{16}$ Representation Correspondence](11_spinor_internalization.md) | ||
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| This document counts whether the principal numerical coincidences are reproducible by post-hoc selection, by exhaustive scan of the expression space. The method is as follows: enumerate all natural expressions mechanically generatable from the same materials as those used by the theory, and count how many fall near the observed values. The rarer a coincidence is within that space, the less it can be explained by selection freedom. |
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| The two identities for $\ell_{1}$ and $\Delta\ell$ are the unique solutions within the scan space at observational precision and are statistically non-trivial. The $\Lambda$ coefficient lacks discriminating power. The accidentality hypothesis is quantitatively disfavoured for the former two, and cannot be excluded for the coefficient story. | ||
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| The verification code is included as [coincidence_audit.py](coincidence_audit.py). All procedures — expression-space generation, counting, and tolerance scanning — are reproducible from the code. |
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| ← [§13: Coincidence Rarity Audit](10_coincidence_audit.md) | [README (top)](README.md) | ||
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| This document separates the spinor-side argument that was delegated externally in §11.5 into the part that can be written using established mathematics and the part that must be stated explicitly as identification postulates, assigning numbers to the latter. |
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| Branching rules: under $\mathfrak{so}(10) \supset \mathfrak{su}(5) \supset \mathfrak{su}(3) \oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1)$, one has $\mathbf{16} = \mathbf{1} \oplus \bar{\mathbf{5}} \oplus \mathbf{10}$, and the hypercharge is uniquely fixed by the branching rules up to normalisation of the abelian direction. | ||
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| Uniqueness of the abelian direction: by Theorem 6.10.3, the spectrum $\{0, \pm 1, \pm 2\}$ of the centre of the flag stabiliser $U(2)$ on $\mathrm{Im}\,\mathbb{O}$ agrees exactly with the lepton pattern, and the normalisation is determined by integer normalisation. The lepton components of $\bar{\mathbf{5}} \oplus \mathbf{1}$ — namely $L: Y = -1$ and $\nu^{c}: Y = 0$ — and $e^{c}: Y = +2$ in $\mathbf{10}$ are consistent including the conjugates of this assignment. |
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| Postulate R3 (abelian-direction identification): the hypercharge direction is the centre of the flag stabiliser. | ||
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| Status: elevated to theorem (Theorem 6.10.3; uniqueness follows from the lepton-pattern agreement). |
| | $\mathrm{Stab}(\mathbb{H})$ (preserves $\mathrm{span}\{e_{1}, e_{2}, e_{3}\}$) | 6 | | ||
| | Intersection $\mathrm{Stab}(i) \cap \mathrm{Stab}(\mathbb{H})$ | 4 | | ||
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| The intersection is closed under the bracket; the derived subalgebra is three-dimensional and the centre is one-dimensional (both confirmed with threshold $10^{-9}$; residuals are at machine precision). The eigenvalues of the central generator on $\mathrm{Im}\,\mathbb{O}$ are $0$ (the $e_{1}$ direction), $\pm i/\sqrt{3}$ ($\mathrm{span}\{e_{2}, e_{3}\}$), and $\pm i/(2\sqrt{3})$ (each doubly degenerate, $\mathrm{span}\{e_{4}, \ldots, e_{7}\}$); the norm of the off-diagonal blocks is below $10^{-15}$. The charge ratio $2:1$ is confirmed (see [§6.10](03_fundamental_equation.md#610-derivation-of-the-electroweak-group-and-normalisation-of-hypercharge)). |
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04_cosmological_constant.mdの§7.3のタイトルを「Algebraic Derivation」から「Algebraic Representation」に変更し、日本語版にある稀少性監査への参照・注釈・識別力が弱いという断り書き・同値な読みの一つという留保を英語版に反映しました。
appendix.mdでは付録A〜Dに加え、日本語版にある付録Eを英語訳して追加しました。
09_formal_system.md、10_coincidence_audit.md、11_spinor_internalization.mdは日本語版から英語に翻訳した新規ファイルです。