17 sections found. Current publication documents are listed before historical material.
E8 and the Gaussian code
theoremTheorem lemma[theorem]Lemma corollary[theorem]Corollary remark remark[theorem]Remark
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documents/note_e8_gaussian_code/preamble.tex · lines 1 to 46
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E8 and the Gaussian code
Build the [formula in source] lattice by Construction A over the extended Hamming code [formula in source] , placed equivariantly with respect to four fixed coordinate pairs, and let [formula in source] be the complex structure that rotates each pair ( [formula in source] , [formula in source] ). Then [formula in source] becomes a unimodular Hermitian [formula in source] -lattice of rank [formula in source] , and red
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documents/note_e8_gaussian_code/sections/001-section-the-three-objects-label-sec-setup.tex · lines 1 to 46
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E8 and the Gaussian code
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documents/note_e8_gaussian_code/sections/001-section-the-three-objects-label-sec-setup.tex · lines 47 to 48
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E8 and the Gaussian code
Let [formula in source] be the extended Hamming code [formula in source] — the unique doubly-even self-dual binary code of length [formula in source] , permutation equivalent to the first-order Reed–Muller code [formula in source] MS77. Its automorphism group has order [formula in source] , so [formula in source] admits exactly [formula in source] distinct placements (coordinate images) inside [formula in source] . F
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documents/note_e8_gaussian_code/sections/001-section-the-three-objects-label-sec-setup.tex · lines 49 to 73
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E8 and the Gaussian code
Let [formula in source] be the Construction-A lattice of [formula in source] CS99, with the standard inner product [formula in source] of [formula in source] . Then [formula in source] , and since [formula in source] is doubly even, writing [formula in source] with [formula in source] a codeword lift gives so every nonzero [formula in source] has [formula in source] : the minimum of [formula in source] is [formula in
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documents/note_e8_gaussian_code/sections/001-section-the-three-objects-label-sec-setup.tex · lines 74 to 89
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E8 and the Gaussian code
Let [formula in source] rotate each [formula in source] pair, [formula in source] , and let [formula in source] be the pair [formula in source] -cycle [formula in source] , [formula in source] , fixing [formula in source] . Then (machine checks I0.2–I0.3; Lean theorems J_sq, J_skew, sigma_cubed_ambient, J_sigma_commute, lattice_J, lattice_sigma) [formula in source] where the last two use precisely the [formula in sou
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documents/note_e8_gaussian_code/sections/001-section-the-three-objects-label-sec-setup.tex · lines 90 to 135
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E8 and the Gaussian code
All checks of this section are from gaussian_code_bridge_probe.py ( [formula in source] checks, [formula in source] failures, verdict gaussian-code-bridge-exact; exact integer and Fraction arithmetic throughout, sympy only for the Smith normal form), with the finite certificates re-proved by the Lean kernel in GaussianCodeBridge.lean.
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documents/note_e8_gaussian_code/sections/001-section-the-three-objects-label-sec-setup.tex · lines 136 to 226
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E8 and the Gaussian code
Since [formula in source] is [formula in source] -linear and preserves [formula in source] , it preserves [formula in source] and descends to an [formula in source] -linear map [formula in source] on [formula in source] . (Well-definedness is itself a kernel certificate: the transport matrix of [formula in source] into Smith coordinates has even top-right block, Lean theorem sigma_label_block.)
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documents/note_e8_gaussian_code/sections/001-section-the-three-objects-label-sec-setup.tex · lines 227 to 326
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E8 and the Gaussian code
The exponents of [formula in source] are the totatives [formula in source] of the Coxeter number [formula in source] . As a multiplicative group this set has element orders [formula in source] in that listing — explicitly the pairs (unit, order) [formula in source] , [formula in source] , [formula in source] , [formula in source] , [formula in source] , [formula in source] , [formula in source] , [formula in source]
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documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 1 to 93
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E8 and the Gaussian code
All checks of this section are from quartic_half_probe.py ( [formula in source] checks, [formula in source] failures, verdict quartic-half-alive; exact Gaussian-integer and Gaussian-rational arithmetic, no floats anywhere), run in the standard doubled-coordinate model of [formula in source] (roots [formula in source] and [formula in source] with an even number of minus signs; norm [formula in source] ), which by Rema
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documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 94 to 106
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E8 and the Gaussian code
The exponents of [formula in source] are [formula in source] , the totatives of the Coxeter number [formula in source] Bou68. The character [formula in source] (residue mod [formula in source] ) splits them: the exponents [formula in source] are [formula in source] , and shifted by [formula in source] , [formula in source] the degree quadruple of the rank- [formula in source] complex reflection group [formula in sour
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documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 107 to 133
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E8 and the Gaussian code
For [formula in source] let [formula in source] , summed over the [formula in source] roots, and let [formula in source] denote the restriction to [formula in source] , computed in the holomorphic weights [formula in source] , [formula in source] . Write [formula in source] .
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documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 134 to 210
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- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
- cross reference join OrbitPacket.lean
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- explicit script identifier v634_st31_structure.py
literal short identifier uniquely resolves in local script registry
- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
- cross reference join SquareParity.lean
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
- cross reference join G31WordOrders.lean
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- explicit lean symbol QuarticHalf.lean
exact texttt symbol under explicit Lean/theorem context
E8 and the Gaussian code
A structure claim needs controls that destroy it. Four are run (probe II4, all firing as demanded): Honest negative first. The natural candidate control — flipping the orientation of one pair, [formula in source] — does not fire: [formula in source] with the even sign flip [formula in source] , so [formula in source] preserves all [formula in source] roots. No signed-permutation complex structure can break root compa
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documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 211 to 252
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E8 and the Gaussian code
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documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 253 to 254
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E8 and the Gaussian code
@lccl@ probe & checks & verdict & arithmetic\\ gaussian_code_bridge_probe.py & [formula in source] & gaussian-code-bridge-exact & exact [formula in source] / Fraction; SNF via sympy\\ quartic_half_probe.py & [formula in source] & quartic-half-alive & exact [formula in source] / [formula in source] ; no floats\\ Both probes cross-validate their own machinery: the class map is computed twice (Hermite-normal-form labels
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documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 255 to 294
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E8 and the Gaussian code
The algebraic cores are formalized in two Lean 4 modules over Mathlib Moura21,mathlib, committed to the repository's Lean project ( lake build green, no sorry, no native_decide): GaussianCodeBridge.lean ( [formula in source] theorems) and QuarticHalf.lean ( [formula in source] theorems), [formula in source] kernel-checked theorems in total. The finite certificates are proved by the kernel-checked decide tactic over e
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documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 295 to 328
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E8 and the Gaussian code
What is established is finite algebra, fully certified: the pair [formula in source] carries a canonical four-bit quotient on which the [formula in source] roots equidistribute as [formula in source] messages [formula in source] [formula in source] roots ( [formula in source] Gaussian lines) each, with a distinguished coordinate block; the clock [formula in source] equips the quotient with the family/anchor semantics
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Source location and explicit evidence links
documents/note_e8_gaussian_code/sections/002-section-explicit-character-basis-of-the-eight-coxeter-ch.tex · lines 329 to 396
These are recorded citation relations. A source citation or topic match does not establish a mathematical or physical result.
- cross reference join OrbitPacket.lean
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- explicit file citation GaussianCodeBridge.lean
literal source filename/path in TeX
- explicit script identifier v634_st31_structure.py
literal short identifier uniquely resolves in local script registry
- explicit script identifier v626_e8_code.py
literal short identifier uniquely resolves in local script registry
- explicit script identifier v638_code_semantics.py
literal short identifier uniquely resolves in local script registry
- cross reference join HammingCode.lean
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- cross reference join SquareParity.lean
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- explicit file citation gaussian_code_bridge_probe.py
literal source filename/path in TeX
- explicit file citation QuarticHalf.lean
literal source filename/path in TeX
- cross reference join GaussianCodeBridge.lean
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- cross reference join G31WordOrders.lean
TeX and evidence source explicitly name the same Python identifier; statement equivalence not inferred
- explicit file citation quartic_half_probe.py
literal source filename/path in TeX