H12_Analytic_Fourier_README.md
Hilbert 12: analytic Fourier branch
H12_Analytic_Fourier_README.md
1 # Hilbert 12 Ambient Cubic Fourier Attempt
2
3 Author: Komorebi P.
4
5 This package contains a new exact algebraic continuation of the pure cubic Hilbert 12 program together with an explicit meromorphic gamma candidate and numerical diagnostics.
6
7 ## Included files
8
9 - `H12_Analytic_Fourier_Resolvent_Research_Note.md` — mathematical setup, proofs, conditional analytic targets, and scope.
10 - `H12_Analytic_Resolvent_Verifier.py` — exact SymPy polynomial audits and independent mpmath fifth-smoothed gamma series.
11 - `H12_Analytic_Resolvent_Verification_Log.txt` — accepted exact checks, gamma data, and negative control.
12
13 ## Reproduce
14
15 Requires Python 3 and `sympy` and `mpmath`.
16
17 ```bash
18 python H12_Analytic_Resolvent_Verifier.py --exact-only
19 python H12_Analytic_Resolvent_Verifier.py --dps 64 --nbase 4000 --n10 19000 --n23 34000 --negative-control
20 ```
21
22 All work is offline.
23
24 ## Established
25
26 - Exact order three Möbius action extending the Galois action on the cubic roots.
27 - An invariant quotient coordinate and a one parameter cyclic cubic pencil.
28 - Exact perfect square discriminant and the three deformed Fourier identities.
29 - A canonical rational analytic gamma candidate and an explicit scalar obstruction.
30 - An algebraic reduction to gamma chart orbit identities plus trace normalization.
31 - Numerical agreement at conductor three, with a separate negative control at one quarter.
32
33 ## Not established
34
35 - The analytic gamma chart orbit relations.
36 - The exact trace normalization of those analytic values.
37 - The vanishing of the gamma scalar obstruction at one third.
38 - The elliptic gamma/class field equality or Hilbert's twelfth problem.
39
40 The numerical log does not constitute an interval proof. The analytic series tail has a mathematical bound, but the floating point operations themselves are not directed interval computations. The first attempted numerical run had prematurely rounded `1/3`. The accepted log was generated after correcting this precision issue.