65 sections found. Current publication documents are listed before historical material.
Standard Model and flavor structure
Mehta–Seshadri supplies the equivalence-class framework — stable parabolic bundles [formula in source] irreducible unitary representations — but it does not by itself give ``parabolic-stable [formula in source] transport-minimal'' or the hexagonal word-lengths; that is the TFPT-specific step. The family connection [formula in source] provides the representation, its weights are the cusp exponents, the stable degree i
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documents/tfpt_2_standard_model/sections/006-section-neutrinos.tex · lines 348 to 377
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Standard Model and flavor structure
The metric-sector field equation itself carries no free dimensionless Newton coupling under those premises: fixed- volume entanglement equilibrium (Jacobson 2015) forces the full covariant Einstein equation [formula in source] with both coefficients fixed ( [formula in source] , [formula in source] from [formula in source] ; v358/ v359). What this subsection details is the complementary spectral-action / inflation la
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documents/tfpt_2_standard_model/sections/007-subsection-2-gravity-the-classical-covariant-field-equat.tex · lines 1 to 50
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Standard Model and flavor structure
The key reduction: [formula in source] needs neither the full OS package nor the mass gap. It follows from three purely structural facts plus reflection positivity on the admissible branch:
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documents/tfpt_2_standard_model/sections/007-subsection-2-gravity-the-classical-covariant-field-equat.tex · lines 51 to 91
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Standard Model and flavor structure
This is the constructive QFT closure of the admissible physical sector — not the unprojected ambient theory: the unprojected ambient metric measure is not constructed here and remains the [formula in source] frontier item ( ambient QG). On the admissible sector, the input previously listed as ``open'' — the parent-domination / mass-gap lemma from (3) — is not analytically open: it is computed by the explicit discrete
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documents/tfpt_2_standard_model/sections/007-subsection-2-gravity-the-classical-covariant-field-equat.tex · lines 92 to 179
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Standard Model and flavor structure
Here the hardening is empirical and concrete. The Lean 4 proof ( lean4-carrier-rigidity, toolchain v4.29.1 [formula in source] Mathlib) builds cleanly: @lY@ Verified (Lean, 0 sorry) & Content \\ Polarization.sixY_carrier_polynomial & [formula in source] as a corollary of orthogonal idempotents \\ Rigidity.unique_carrier_pair & [formula in source] unique (integer rigidity) \\ Hypercharge.trace_Y, _carrier_polynomial &
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documents/tfpt_2_standard_model/sections/007-subsection-2-gravity-the-classical-covariant-field-equat.tex · lines 180 to 243
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Standard Model and flavor structure
The same numbers, now as closed expressions instead of distributed theorems:
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documents/tfpt_2_standard_model/sections/007-subsection-2-gravity-the-classical-covariant-field-equat.tex · lines 244 to 278
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Standard Model and flavor structure
tocsectionConclusion The simplification is substantial: the discrete Standard-Model structure (charges, families, flavor operators) is derived under the axioms; [formula in source] D existence, chiral measure, mirror gap, couplings and neutrino dynamics are open. The [formula in source] -ladder formula [formula in source] with one seed [formula in source] and one fixed residue matrix whose spectral invariants ( [form
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documents/tfpt_2_standard_model/sections/007-subsection-2-gravity-the-classical-covariant-field-equat.tex · lines 279 to 304
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Standard Model and flavor structure
This note closes the last structural gap of the [formula in source] compiler by deriving the charged-fermion transport word-lengths [formula in source] , [formula in source] , [formula in source] — equivalently the residue triple [formula in source] and its [formula in source] orbit — from first principles, rather than from the legacy resolvent construction. The correct framework is parabolic Higgs/bundle theory on t
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documents/tfpt_2_standard_model/sections/007-subsection-2-gravity-the-classical-covariant-field-equat.tex · lines 305 to 349
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Standard Model and flavor structure
The family curve is the rigid four-punctured sphere [formula in source] , [formula in source] (the flavor construction above). The flavor bundle is a rank- [formula in source] parabolic bundle [formula in source] on [formula in source] . At each puncture the local monodromy is the order- [formula in source] cusp class with eigenvalues [formula in source] , [formula in source] ; by the Mehta–Seshadri dictionary the co
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documents/tfpt_2_standard_model/sections/008-section-the-setup-parabolic-data-on-the-family-curve.tex · lines 1 to 49
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Standard Model and flavor structure
For a unitary (flat) family local system, [formula in source] is parabolic-polystable of parabolic degree zero (Mehta–Seshadri). With full flags and weights [formula in source] at each of the four punctures, The balanced (stable) underlying splitting is therefore [formula in source] .
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documents/tfpt_2_standard_model/sections/008-section-the-setup-parabolic-data-on-the-family-curve.tex · lines 50 to 86
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Standard Model and flavor structure
Constructing the rigid [formula in source] -symmetric local system explicitly (four monodromies [formula in source] , [formula in source] , [formula in source] ), the closure [formula in source] is solvable to machine precision, and the unique rotation [formula in source] has eigenvalues [formula in source] : the [formula in source] Coxeter element. \ This independently reproduces the corner-rotation [formula in sour
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documents/tfpt_2_standard_model/sections/008-section-the-setup-parabolic-data-on-the-family-curve.tex · lines 87 to 114
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Standard Model and flavor structure
Status (no drift). The splitting type is now closed: the unique partition [formula in source] fixes the anchor and [formula in source] fixes the residue branch (verified, below and v4_flavor_matrix.py), and the solar texture is built ( v9_neutrino_texture.py). The roadmap below is kept as the derivation record of how [formula in source] is obtained; it is no longer an open project. The only residual is the parabolic
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documents/tfpt_2_standard_model/sections/008-section-the-setup-parabolic-data-on-the-family-curve.tex · lines 115 to 172
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Standard Model and flavor structure
The global (Fuchsian/Hecke) computation does resolve one integer datum exactly. Work in the rotation eigenbasis [formula in source] and take the [formula in source] -symmetric Fuchsian residues [formula in source] with [formula in source] (the cusp exponents). Then the symmetrisation collapses exactly, because [formula in source] for [formula in source] (off-diagonal terms are summed fourth roots of unity). The expon
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documents/tfpt_2_standard_model/sections/008-section-the-setup-parabolic-data-on-the-family-curve.tex · lines 173 to 272
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Standard Model and flavor structure
Every residue decomposes uniquely into a cusp index [formula in source] (the [formula in source] family phase, [formula in source] ) and a sheet parity [formula in source] (the [formula in source] deck involution), via the bijection [formula in source] : [formula in source] The full transport table in these parabolic coordinates is l|ccc|ccc & 3c|cusp index [formula in source] (g1,g2,g3) & 3csheet [formula in source]
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documents/tfpt_2_standard_model/sections/008-section-the-setup-parabolic-data-on-the-family-curve.tex · lines 273 to 316
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Standard Model and flavor structure
We now state the result as a theorem: under three precisely-stated hypotheses — two of which are already established in this note — the entire transport matrix (anchors and the six non-anchor residues) is forced. The combinatorial core is proved by complete enumeration.
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documents/tfpt_2_standard_model/sections/008-section-the-setup-parabolic-data-on-the-family-curve.tex · lines 317 to 378
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Standard Model and flavor structure
The anchor vector [formula in source] — the absolute value of the splitting type [formula in source] — is the smallest parabolic object that encodes the three operative structures at once: The [formula in source] norm is the glue/anti-degree, the [formula in source] norm [formula in source] is the [formula in source] positive-root hexagon, and the product is the sheet parity.
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documents/tfpt_2_standard_model/sections/009-section-the-parabolic-anchor-compiler.tex · lines 1 to 13
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Standard Model and flavor structure
With [formula in source] fixed by parabolic degree zero, the entire integer landscape collapses: So [formula in source] and the spinor block [formula in source] are all functions of [formula in source] with the single new ingredient [formula in source] derived from parabolic degree zero (not posited).
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documents/tfpt_2_standard_model/sections/009-section-the-parabolic-anchor-compiler.tex · lines 14 to 27
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Standard Model and flavor structure
The flavor budget splits into anchor [formula in source] two equal Coxeter copies: The residue row sums and full row sums read named objects, and the generation (column) sums are [formula in source] . \ Finally the scalaron exponent is the parabolic anti-degree plus the family rank, linking the stable family bundle [formula in source] directly to the inflation scale. The [formula in source] residue sum is the project
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documents/tfpt_2_standard_model/sections/009-section-the-parabolic-anchor-compiler.tex · lines 28 to 50
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Standard Model and flavor structure
/ / in 0/-2/blue!60!black, 1/-1/green!50!black, 2/-1/green!45!black (0,- 0.62) rectangle (2.6,- 0.62+0.5); at (1.3,- 0.62+0.25) [formula in source] ; at (1.3,0.95) splitting type\\[-1pt] [formula in source] ; at (1.3,-2.05) degrees [formula in source] anchor [formula in source] \\[-1pt] [formula in source] ;
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documents/tfpt_2_standard_model/sections/009-section-the-parabolic-anchor-compiler.tex · lines 51 to 99
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Standard Model and flavor structure
Hypotheses (H1),(H3) plus the unique sum- [formula in source] set fix the up and lepton sectors and reduce (H2) to a single binary choice in the down sector: [formula in source] versus its distinct-distance sibling [formula in source] (both contain the anchor [formula in source] , both have row sum [formula in source] , so sum rules do not separate them). The residue matrix [formula in source] (rows [formula in sourc
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documents/tfpt_2_standard_model/sections/009-section-the-parabolic-anchor-compiler.tex · lines 100 to 197
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