# universal_fixed_point_framework **Repository Path**: dpsnet/universal_fixed_point_framework ## Basic Information - **Project Name**: universal_fixed_point_framework - **Description**: 本项目实现元通用不动点函子范畴框架理论，实现谱去递归函子与原创谱静默机制。集成分形 RKHS、Clifford 谱、Kerr 黑洞、全息 CFT、BSM 新粒子、EFT 转化等模块，配套完整理论论文，全部数值实验可复现。 - **Primary Language**: Python - **License**: CC-BY-4.0 - **Default Branch**: master - **Homepage**: None - **GVP Project**: No ## Statistics - **Stars**: 0 - **Forks**: 0 - **Created**: 2026-07-14 - **Last Updated**: 2026-08-31 ## Categories & Tags **Categories**: Uncategorized **Tags**: None ## README # Fractal Spectral De-recursion Theory · Universal Fixed-Point Categorical Framework (MUFPF) > **Research Goal**: To build a sufficiently abstract mathematical language that allows recursive systems from different domains—fractals, neural networks, renormalization groups, quantum gravity, the Standard Model, etc.—to be described, compared, and transformed within a unified spectral framework. --- **Latest (2026-07-31)**: **RAP-Errata v0.6 released** — all 37 papers status complete (31 stable, 6 new), **zero warnings, zero pending**. Parameter count reduced to **0 free parameters + 1 external scale $M_{\text{Pl}}$**. B2 continuum limit (fractal attractor → smooth $\mathbb{R}^4$ quasi-symmetric embedding) theoretically closed. New papers: Paper XXXV (category-theoretic origin of gravity) and Paper XXXVII (open problems survey). **CoherenceToBranching.lean §11 outward proof formalized** (dimension gap ln 15 < 3 + layer orthogonality S₄/c₁ = e³). **Path B complete**: 8 core modules re-formalized in Agda 2.8.0, all type-checked (proof-assistant cross-validation). See `universal_fixed_point_framework/paper/RAP_勘误与立场声明.md`. --- ## 1. Project Overview This project consists of two interrelated research layers: | Layer | Location | Role | |-------|----------|------| | **Original numerical implementation layer** | Root directory `.` | Early-stage concrete numerical fits and experimental validations for the Standard Model mass spectrum, NTK spectral optimization, etc. | | **Universal fixed-point categorical framework** | `universal_fixed_point_framework/` | Category-theoretic interdisciplinary unification framework via fixed-point axioms. **Latest: RAP-Errata v0.49 — 48 papers, 0 free parameters + 1 external scale $M_{\text{Pl}}$**. | Core idea: treat "recursive iteration" as an **object-level evolution rule**, and its corresponding "operator semigroup spectrum" as a **spectral-level static structure**. The two are related systematically through a spectral de-recursion functor. --- ## 2. Core Theoretical Skeleton ### 2.1 Three-layer Axiomatic System | Layer | Content | Modifiability | |-------|---------|---------------| | **Meta-axioms** | Existence and naturality of the recursive-system category, the spectral category, and the spectral de-recursion functor | Not modifiable by instances | | **Structural theorems** | Contraction mappings, fixed-point equations, spectral correspondence natural equivalence, RKHS convergence rates | Derived from meta-axioms | | **Instance hypotheses** | Standard Model = Cl(1,7), string theory = Cl(9,1), NTK = lazy training limit, etc. | Replaceable; do not feed back to upper layers | Core rule: **a poor fit at the instance layer does not refute the upper layers.** ### 2.2 Key Mathematical Structures - **Recursive-system category** $\mathbf{Rec}$: objects are self-similar evolution systems; morphisms are structure-preserving maps that commute with the evolution rule. - **Spectral category** $\mathbf{Spec}$: objects are positive spectral operators on Hilbert spaces; morphisms satisfy the spectral intertwining condition. - **Spectral de-recursion functor** $D: \mathbf{Rec} \to \mathbf{Spec}$: maps recursive evolution to the exponential evolution of an operator semigroup. - **Spectral correspondence natural equivalence** $\eta_R: \mu \mapsto e^{-\mu}$: a natural bijection between compression spectra and operator spectra. - **Orbit functor** $O$: encodes symmetry weights under gauge-group actions. - **Universal fixed-point equation** $\mathcal{F}[\mathcal{V}] = \mathcal{V}$: a unified form of subsystem fixed-point equations. - **Dual-track Koopman operator**: unconditional definition on $\ell^\infty(X)$ + spectral correspondence on $L^2$/$C(X)$ (`DynSys.lean`) - **Lean 4 formalization**: 74 core modules, `lake build` clean (0 errors, 8 warnings). **Only 3 active `sorry`s**: 1 conceptual feature in `HigherSpCategory.lean:103` (exchange law deviation = gravity), 2 pending Mathlib `Matrix.Spectrum` in `DeviationBound.lean:386/412`. 10 core theorem modules fully machine-proven (zero `sorry`). **New in CoherenceToBranching.lean §11**: `dimension_gap` + `outward_proof_maps_to_orthogonal_layer` (outward proof formalization). - **Category verification (Phase 60 🆕)**: `python -m verify.run_all` — 8/8 PASS. Path C complete (Python executable categorical semantics). **Path B complete (2026-07-31)**: 8 core modules re-formalized in Agda 2.8.0 (`agda_formalization/`), `Everything.agda` type-checks, theorem signatures match Lean one-to-one (proof-assistant cross-validation). See `roadmap/phase60_category_verification.md`. ### 2.3 Key Physical Correspondences - Standard Model mass spectrum ← fractal compression spectrum - Gravitational spectrum ← spectrum of spacetime curvature operators - String scattering spectrum ← genus spectrum of topological recursion - Holographic entropy ← area law in the spectral-measure framework --- ## 3. Directory Structure ``` . ├── README.md # Chinese overview ├── README_EN.md # English overview ├── Clifford值分形RKHS构造.md # Core mathematical construction (1600+ lines, Chinese) ├── docs/ │ ├── 研究目标整理.md # Checklist of work needed for top-tier journals │ └── 分形谱去递归理论研究路线图.md # Full research roadmap (v2.1) ├── universal_fixed_point_framework/ # Universal fixed-point categorical framework │ ├── README.md # Framework roadmap and progress overview │ ├── paper/ # 37 papers │ │ ├── paper1_*.md — paper29/ # Papers I–XXIX: foundation, physics, formalization │ │ ├── paper30_dH_structural_analysis.md # d_H structural analysis │ │ ├── paper31_mass_delta_directionality.md # 🆕 Mass-Δ directionality │ │ ├── paper32_silence_spacetime.md # 🆕 Cl(1,7) spectral silence & spacetime │ │ ├── paper33_origin_of_3.md # 🆕 Origin of "3" │ │ ├── paper34_continuum_limit.md # 🆕 Continuum limit (B2 closure) │ │ ├── paper35_gravity_origin.md # 🆕 Category-theoretic origin of gravity │ │ ├── paper37_open_problems.md # 🆕 Open problems survey │ │ ├── RAP_勘误与立场声明.md # RAP-Errata v0.49 │ │ └── RAP_盲登记协议.md # RAP-Registry v0.49（与勘误 1:1 同步） │ ├── src/ # Core code implementations │ │ ├── rec_category.py # Rec category │ │ ├── spec_category.py # Spec category │ │ ├── decursion_functor.py # Spectral de-recursion functor D │ │ ├── spectral_correspondence.py # Spectral correspondence natural equivalence │ │ ├── orbit_functor.py # Orbit functor O │ │ ├── fixed_point_solver.py # Universal fixed-point equation solver │ │ ├── spectral_silence.py # Spectral silence: an alternative to compactification │ │ ├── theory_transformation.py # Theory-transformation framework │ │ ├── eft_equivalence_framework.py # EFT equivalence framework │ │ ├── rkhs_*.py # RKHS convergence-rate theory │ │ ├── bsm_*.py # BSM new-physics predictions and experiment interfaces │ │ ├── kerr_*.py # Kerr black holes and gravitational waves │ │ ├── holographic_entropy.py # Holographic entanglement entropy │ │ ├── complex_cft_phase_transition.py # Complex CFTs and holographic phase transitions │ │ ├── ntk_fractal_bidirectional.py # NTK-fractal bidirectional transformation │ │ └── ... # 40+ additional modules │ ├── notes/08_first_principles/ # Research notes (v1.48) │ ├── formal_proof/MUFPFormalization/ # Lean 4 formalization: 9 core modules │ ├── roadmap/ # Phase roadmap documents │ ├── paperX_*.py # Numerical verification scripts │ └── run_all_tests.py # Full regression test suite ├── complete_chain_derivation.py # Forward chain from Clifford algebra to SM masses ├── sm_mass_complete_v5.py # v5.0 Standard Model mass-spectrum prediction ├── final_sm_prediction.py # Final SM mass-prediction pipeline ├── v5_final.py / v52_*.py # v5.x analysis tools └── final_sm_prediction_results.txt # Final prediction results ``` --- ## 4. Current Research Status ### 4.1 Completed (development stage) **Mathematical theory** - [x] Python prototype of Rec/Spec categories and $D$ functor + adjoint $D \dashv R$ triangle identities - [x] Spectral correspondence $\lambda = e^{-\mu}$ as categorical natural equivalence (braided extension) - [x] Orbit functor in 12+ instances - [x] RKHS convergence rates: strong/weak/non-separated, measure-theoretic proof, high-dimensional IFS - [x] Singular continuous spectrum, spectral silence, theory transformation, EFT equivalence - [x] Dual-track Koopman existence ($\ell^\infty(X)$ unconditional definition + spectral correspondence) **Physical applications** - [x] GR+SM unified spectral correspondence (partial), $G_N$ from spectral intertwining - [x] BSM new physics prediction ($L_4 \approx 1470$ GeV) with HL-LHC/FCC-hh interface - [x] Kerr non-equatorial chaos and NR ringdown comparison - [x] Holographic entanglement entropy, complex CFT phase transitions, N=4 SYM TBA **48 papers** - [x] Paper I v2.35: Fractal spectral de-recursion theory (categories/IFS/spectral measures/Clifford/RKHS) - [x] Paper II v2.22: Physics applications (SM/BSM/Kerr/holographic entropy/dark matter) - [x] Paper III v1.1: Spectral classification completeness (three-layer + BPS numerical verification + Lean) - [x] Paper IV v1.1: Stretched Horizon → D-brane BH entropy unification (with duality extensions) - [x] Paper V–XVI: Spectral dynamics, fluid, thermodynamics, black holes, QFT, quantum gravity, cross-domain - [x] Paper XVII v1.8: Zero-parameter predictions (RAP errata-compliant: 15 strict + 14 partial + 7 frozen) - [x] Paper XVIII: Spectral Newtonian mechanics - [x] Paper XIX–XXIX: Formalization extensions (spectral gap, Grothendieck fibration, quantum chemistry, BCS μ*, spectral bond rigidity) - [x] Paper XXX: $d_H$ structural analysis & machine verification - [x] **Paper XXXI 🆕**: Mass-Δ directionality (J1-J3 formal propositions + Lean proof) - [x] **Paper XXXII 🆕**: Cl(1,7) spectral silence & 4D spacetime emergence (8 theorems, machine-proven) - [x] **Paper XXXIII 🆕**: Origin of "3" — unified 3 theorem, inequality chain, Bott-Moran bridge - [x] **Paper XXXIV 🆕**: Continuum limit — fractal attractor → smooth spacetime (B2 theoretical closure) - [x] **Paper XXXV 🆕**: Category-theoretic origin of gravity — exchange law deviation = gravity - [x] **Paper XXXVII 🆕**: Open problems survey — A/B/C classification, hierarchy distance, Bott-Moran bridge **Lean 4 formalization** - [x] 74 core modules, `lake build` clean (0 errors, 8 warnings) - [x] 10 core theorem modules fully machine-proven (zero `sorry`) - [x] Active `sorry`: only `spExchangeLaw` (conceptual feature) + 2 pending Mathlib `Matrix.Spectrum` **Category-theoretic verification (Phase 60 🆕)** - [x] **Path C complete**: `python -m verify.run_all` — 8/8 PASS - [x] V1 Sp strict 4-category | V2 D functor faithful | V3 D ⊣ R triangles - [x] V4 Spectral correspondence natural | V5 Unified 3 theorem - [x] V6 Inequality chain | V7 c₁