Edition README
Seeded Two-Axis Calculus
README.md
1 # Komorebi Seeded Two-Axis Calculus — v0.11.0
2
3 Author: Komorebi P.
4 Working formalization, 27 September 2026.
5
6 ## Current mathematical result
7
8 This release extends the cumulative manuscript to continuous representations of an actual profinite source. Its principal result is a complete finite-matrix criterion for unconstrained continuous representations
9
10 rho_n : Z_p^r -> GL_2(Z/p^n).
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12 Let a be the minimum scalar depth of the generator matrices. All-scalar inputs lift. Otherwise subtract their scalar parts, divide by p^a into matrices modulo p^(n-a), and select a normalized generator that is nonscalar modulo p. The representation lifts over Z_p exactly when every normalized generator commutes with that selected one. A cyclic vector gives explicit linear polynomials producing exactly commuting integral lifts.
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14 A uniform extension horizon is 2n-1, sharp for n >= 2 and r >= 2. In arbitrary matrix rank, the same argument applies when a normalized generator is cyclic modulo p. Separate results cover continuous H^2 obstructions, source inflation, reconstruction from towers, and free pro-p sources. Coefficient reduction and source inflation have opposite variance.
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16 These are written proofs and authoring-session self-review. They are not Lean-certified or independently reviewed. No claim of historical novelty, general Galois lifting, geometric local conditions, or automorphic existence is made.
17
18 ## Files
19
20 - `Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.pdf`: cumulative manuscript, 106 pages.
21 - `Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex`: standalone LaTeX source.
22 - `proofs/profinite_lifting.tex`: new K72–K82 proof block.
23 - `tools/lift_abelian_rank2.py`: exact positive/negative certificate command.
24 - `examples/`: positive and negative JSON inputs and their certificates.
25 - `checks/`: all six exact regression suites, runner, and actual reports.
26 - `audit/`: source diff, integration script, proof/self-review records, final compile log, source and provenance checks.
27 - `research/records/`: durable research records, indexed by `RESEARCH_LOG.md`.
28 - `previous/`: exact supplied v0.10.0 source, PDF, and full package. Earlier history is retained inside that package.
29 - `SHA256SUMS.json`: checksums of every other file in this release.
30
31 ## Use the rank-two certificate tool
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33 Python 3.10 or newer is sufficient for the new standard-library tool and its dedicated suite. The full inherited suite was run with Python 3.13.5 and SymPy 1.14.0.
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35 From the extracted release directory:
36
37 ```bash
38 python tools/lift_abelian_rank2.py examples/positive_rank2.json
39 python tools/lift_abelian_rank2.py examples/negative_rank2.json
40 ```
41
42 A custom input has this form:
43
44 ```json
45 {
46 "p": 3,
47 "n": 3,
48 "generators": [
49 [[1, 3], [9, 1]],
50 [[7, 12], [9, 7]]
51 ]
52 }
53 ```
54
55 The command checks that the prime and precision are valid, that the matrices commute at that precision, and that their residual orders are p-powers. `LIFTABLE` returns exact integral matrix entries (units in Z_p, not necessarily in GL_2(Z)). `NOT_LIFTABLE` returns a nonzero normalized commutator. Malformed input exits with code 2. Both positive and negative mathematical certificates are successful command outcomes with exit code 0.
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57 Only the stated unconstrained GL_2 problem is certified. A supplied determinant, Galois condition, Hodge condition, ramification restriction, or finite-image requirement is not checked or preserved automatically.
58
59 ## Reproduce
60
61 Install the dependencies listed in `requirements.txt` in a suitable Python environment. With a standard LaTeX installation providing the packages in the source:
62
63 ```bash
64 python checks/run_all.py
65 python audit/integrate_v0110.py
66 latexmk -pdf -outdir=build -interaction=nonstopmode -halt-on-error Komorebi_Seeded_Two_Axis_Calculus_v0.11.0.tex
67 ```
68
69 Copy the last TeX engine log to `audit/latex_final.log` before running `python audit/check_source.py` to audit a fresh build. The distributed source check refers to the supplied final log and run reports. It verifies preservation, references, hashes, and reproducible source generation, not mathematical truth.
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71 The full runner gives each script a 40-second timeout, sets single-thread numerical-library environment variables, removes a previous report before running its script, and records before/after script hashes. It imposes no memory cap. The standalone new suite is:
72
73 ```bash
74 python checks/check_profinite_lifting.py --output checks/profinite_results.json
75 ```
76
77 Re-running a test updates evidence timestamps and therefore changes checksums. The release checksum inventory describes the delivered files, not later regenerated outputs.
78
79 ## Boundaries and next target
80
81 A continuous obstruction can be nonzero at one finite quotient and vanish after inflation to a finer quotient. A genuine relation of the fixed source still has to hold. Free pro-p and abelian pro-p sources therefore have different lifting behavior. The main abelian theorem retains exact commutativity without erasing coefficient information.
82
83 Remaining targets are specified noncommutative profinite presentations, noncyclic higher-rank centralizers, and additional arithmetic deformation conditions. The manuscript does not turn a finite-precision witness into an automorphic representation.