Discrete 3D+3D Temporal Geometry: A Single-Axiom Unified Framework for Galactic Dynamics, Cosmology, Particle Physics, and Quantum Coherence

Authors/Creators: Calzighetti, Simone (Project leader)

How I Built a Single-Axiom Geometric Theory of Spacetime — and Why I'm Asking the Scientific Community to Falsify It

My name is Simone Calzighetti. I am not a professor, nor a university researcher. I don't have a PhD. I am a theoretical physics enthusiast — and on September 14, 2025, I had an intuition about discrete mathematics and three-dimensional space that has evolved into a complete theoretical framework.

I have developed a geometric theory of spacetime with six dimensions and metric signature (−,+,+,+,−,−): three spatial and three temporal (3D+3D). Two of the temporal dimensions are compactified on a torus T² with modular parameter τ = i/φ (where φ is the golden ratio), at scales L₂ ≈ 9.5 light-years and L₃ ≈ 6.0 light-years. The ordinary temporal dimension remains non-compactified. This geometry generates emergent gravitational effects that explain phenomena attributed to dark matter and dark energy — without exotic particles, through pure geometry.

With the help of artificial intelligence — Lucy (Claude, Anthropic) as primary co-author and derivation engine, Vega (GPT, OpenAI) as mandatory adversarial Red Team reviewer, Gemini (Google) for observational validation, and Copilot (Microsoft) for implementation support — I have constructed what I believe to be a complete, falsifiable, zero-free-parameter theory of nature.

The Single Axiom and Its Theorem Chain

The entire framework descends from one geometric postulate:

[POST 1 — Determinacy Postulate] The modular parameter of the compact temporal torus T² is uniquely fixed by the SO(3,3) symmetry of the 6D Einstein–Hilbert action. The unique solution is:

τ = i/φ, φ = (1 + √5) / 2

This is not an assumption chosen to fit data. It is the unique solution to the self-consistency condition P(θ*) = 1/D = 1/6, where D = 6 is the total number of spacetime dimensions. The proof reduces to a single quadratic equation x² − x − 1 = 0, whose unique positive root is the golden ratio.

From τ = i/φ alone — with zero free parameters — the following theorem chain is derived (all SymPy residuals = 0):

τ = i/φ
  ↓
A = [[1,1],[1,0]]          Fibonacci matrix (companion of x²−x−1=0)
  ↓
K = I + A² = [[3,1],[1,2]] Q-sector kinetic matrix:  tr(K) = det(K) = 5
  ↓
W = uᵀKu = 7 = 2 + 5      Coherent-mode rigidity  (u = (1,1)ᵀ)
  ↓
(I_E, det M) = (19, 73)   Einstein invariant + Bridge matrix determinant
  ↓
Ω_geom = 19/73 ≈ 0.2603   Geometric dark matter/energy density
  ↓
A = 133/2628               Kernel amplitude  (exact rational, gcd = 1)
  ↓
μ(k,a) = (133/2628)·S(a)/[1+(k/k_μ)²]   Modified gravity kernel
  ↓
CLASS verified:  R = μ_phys/μ_th = 1.000 ± 0.003   (71 independent points)

Every arrow is a proven theorem. No number is inserted by hand.

Dimensional Unification: D = 6 from First Principles (April 2026)

The most recent advance closes the last foundational gap: the integer D = 6 itself is now derived — not postulated — from four independent physical principles, each selecting D = 6 uniquely.

Cosmological-Topological Duality (CTD). The dark energy fraction Ω_geom has two independent representations: an IR one from the Friedmann sector (denominator 2D² + 1) and a UV one from the K-matrix intersection form (denominator D² + 6D + 1). Demanding that both give the same unique value forces 2D² + 1 = D² + 6D + 1, i.e. D² = 6D, whose only positive solution is D = 6.

Lovelock–Gauss-Bonnet bound. D = 6 is the minimum even dimension where the quadratic Lovelock invariant L₂ is dynamical and the cubic L₃ is a topological Euler density — the structural requirement for the topological protection g²Λ = 2π/W proved in Paper C.

E₂ modular anomaly. The Eisenstein series E₂ is the unique quasi-modular form whose anomaly E₂(−1/τ) = τ²E₂(τ) + 12τ/(2πi) contains the factor 12 = 2D. This anomaly breaks SL(2,ℤ) invariance and enables the Coleman-Weinberg mechanism to fix τ. For D ≥ 8, the corresponding E_w is fully modular — no vacuum selection is possible.

Spectral identity. ζ(2) = π²/6 = π²/D. The Basel sum equals the inverse of the total dimension only for D = 6.

Algebraic Quartic Closure. The Universal Identity tr(K) = det(K) = 4 + n² (proved via Cayley-Hamilton for any 2×2 matrix A with det(A) = −1, where K = I + A²) combined with det(K) = D − 1 gives D = 5 + n². Substituting into D² = 6D yields the quartic n⁴ + 4n² − 5 = 0, which factors as (n² − 1)(n² + 5) = 0, forcing n = 1 and hence tr(A) = 1. The Primitive Generator Principle — previously an axiom — becomes a theorem.

The complete chain is now:

Physical Principles (CTD / Lovelock / E₂ / ζ(2))
  ↓
D² = 6D  →  D = 6  →  W = D+1 = 7
  ↓
Algebraic Quartic: n⁴+4n²−5 = 0  →  n = 1  →  tr(A) = 1
  ↓
A = [[1,1],[1,0]]  →  K = [[3,1],[1,2]]  →  eigenvalue = φ
  ↓
τ = i/φ  →  all predictions

ZERO parameters. ZERO assumptions. Pure algebra from D² = 6D.

The Dynamical System (DynSys — March 2026)

The cosmological evolution of the Q-field is governed by the autonomous dynamical system:

u' = ξ(ξ − 3u) / [2(1 − u)]

where u = Ω_Q/Ω_m and ξ = ln(1+z). This ODE has a unique attractor at u* = ξ/3, yielding the cosmological transition redshift:

z_tr = e^(36/53) − 1 ≈ 0.972

derived entirely from the algebraic chain τ = i/φ, with no free parameters. The DynSys formulation replaces the earlier phenomenological activation function S(a) with a first-principles evolution equation, completing the theoretical closure of the cosmological sector.

The Work Done

With the AI collaboration described above, I have developed:

100+ academic papers (~2200+ pages) covering: mathematical foundations, galactic dynamics, gravitational lensing, cosmic web structure, 6D thermodynamics, quantum decoherence, black holes, chronology protection, baryogenesis, gravitational waves, complete fermion spectrum, gauge couplings, neutrino masses, UV completion, warp engine geometry, SMBH formation, N-body cosmological simulations, nuclear physics, and atomic physics.

All 42 Standard Model and cosmological parameters derived from geometry with zero free parameters:

• Fine structure constant: α⁻¹ = 137.036 (< 0.01% precision)
• Weinberg angle: sin²θ_W = (3−φ)/6 = 0.2295 (0.7% precision)
• Strong coupling: α_s = 1/(2φ³) = 0.1180 (0.1% precision)
• Electron mass: m_e = 0.510 MeV (0.18% precision)
• W boson mass: m_W = 80.4 GeV (< 0.1% precision)
• Higgs mass: m_H = 126.8 GeV (1.3% precision)
• CKM phase: δ_CKM = π/φ² = 68.75° (0.7% precision)
• Neutrino mass ratio: Δm²₃₂/Δm²₂₁ = 32.66 (obs: 32.58, 0.27% precision)
• Dark matter density: Ω_geom = 19/73 ≈ 0.260 (1.8% from Planck)
• Dark energy equation of state: w₀ = −0.80 (geometric origin, no Λ)
• Average precision across all 42 parameters: ~1.2%

Empirical validation on real observational data:

• SPARC galaxy rotation curves: 175 galaxies, median RMS = 15 km/s, zero free parameters per galaxy
• WALLABY PDR2 HI survey: independent confirmation on 187 galaxies, 15.0 km/s RMS
• HALOGAS deep HI survey: extended rotation curve validation to outer galactic radii
• SLACS gravitational lensing: 4.0σ detection consistent with Q-field dark matter
• NANOGrav 15-yr pulsar timing: detected periods T₂ = 30 yr and T₃ = 19 yr consistent with compactification scales derived from τ = i/φ
• DESI DR1 BAO: χ² = 14.00, ΔBIC = +25.59 relative to ΛCDM
• Solar System constraints: full compatibility with Cassini, Lunar Laser Ranging, Mercury perihelion
• Cosmic web structure: characteristic scale λ₁₃ = 0.856 Mpc following the φ-ladder hierarchy

N-body cosmological simulations (GADGET-4):

• First full-scale 384³ simulation with the 3D+3D gravity kernel μ(k,a)
• Power spectrum enhancement: P₃D₊₃D(k)/P_ΛCDM(k) = +8.0% to +10.0% at z = 0
• Enhancement grows from zero at z ~ 3 (Q-field activation), consistent with theoretical prediction δP/P ~ 2A·S(a) ~ 10%
• σ₈ ratio: σ₈^(3D+3D)/σ₈^(ΛCDM) = 1.045 — partially alleviates S8 tension by factor ~2 (from 0.065 to 0.034) with zero free parameters
• 480³ run completed: ⟨R⟩ = 1.093 (+9.3% power spectrum enhancement)
• 512³ convergence run completed: confirms numerical convergence of all results
• Cosmic web density maps and evolution visualizations included in repository

Rigorous mathematical foundations:

• UV completion via asymptotic safety with Gaussian-like fixed point (only 2 relevant operators)
• Parameter Regress Theorem (Theorem 2.5): formal proof using the Implicit Function Theorem that 4D theories have an irreducible floor of d_free ≥ 19 free parameters
• Complete 6D Lagrangian with Kaluza–Klein reduction to the Standard Model
• Boltzmann-complete kernel theorem: CLASS v3.3.4 verification R = 1.000 ± 0.003 on 71 independent (k,a) points
• Uniqueness theorem: τ = i/φ is the unique modular parameter satisfying all physical constraints simultaneously (four independent no-go theorems)
• IBM Quantum hardware verification: structural constant k = 1.003 ± 0.097 (4 backends)
• Spectral action via Kreĭn-space heat-kernel expansion
• Variational V-tree formulation for systematic perturbative expansion
• Dimensional Unification Theorem (Paper CI v3.2): D² = 6D derived from 4 independent physical principles; algebraic quartic eliminates all axioms
• Topological Protection Theorem (Paper C): ζ(3) cancels exactly on T²; g²Λ = 2π/W is Gauss-Bonnet type invariant
• Non-Renormalization Theorem (Paper XCIX): 7 of 9 geometric numbers exactly protected to all loop orders; remaining 2 shift by rational multiples of 1/W
• NLO Hierarchy Correction (Paper B3): ΔΛ = 1/28 = 1/[4W] reduces hierarchy error from 3.66% to 0.019%; φ cancels exactly at NLO

Geometric warp engine (speculative sector):

• The 6D metric admits a modified Alcubierre-type solution where the Q-field provides the exotic energy density
• Energy requirement reduced by geometric factor relative to standard Alcubierre drive
• Nonlinear stability analysis completed (v3.0), energy budget analysis (v4.0)
• Classified as speculative — included for completeness, not claimed as engineering blueprint

Pre-Registered Predictions for Upcoming Experiments

Fifteen specific predictions were deposited on Zenodo before the corresponding observational data releases, to prevent post-hoc fitting:

If any one kill switch is triggered at > 5σ, the corresponding sector of the theory is falsified.

Why I Cannot Validate It Alone

I lack academic credentials to guarantee absolute mathematical correctness of every step. I have no access to experimental instruments for independent tests. I have worked with AI as my primary collaborator — which introduces risks of systematic biases that might escape my notice.

What I Am Asking

I explicitly request the scientific community to:

1. Verify the mathematical derivations
2. Criticize the physical assumptions
3. Refute the predictions with data
4. Improve the framework where necessary
5. Falsify the theory using the kill switches provided in this repository

I have uploaded all documents, codes, simulations, and predictions. Every claim is accompanied by explicit formulas and verifiable data. Key predictions were deposited on Zenodo before the release of observational data.

I am not seeking confirmation: I am seeking truth.

If this theory is wrong, I want to know why — and learn from the mistake. If it is correct, I want it to be rigorously tested. If it is partially valid, I want it to be refined by minds more expert than mine.

Repository Contents

This release contains 25 thematic archive folders plus root documents:

Root Documents

• README (this file), LICENSE (MIT), canonical parameter dictionary (SSoT), falsification criteria
• Reading Guide v4.0 — complete navigation guide with 5 reading paths, paper dependency map, updated with D² = 6D
• Symbol Book v5.0 — comprehensive dictionary of all symbols, parameters, and conventions (17 sections)
• Logical Chain Paper — complete causal flow from Determinacy Principle to observables

01 — Foundations Core

Foundational papers: logical inevitability of 6D, Parameter Regress Theorem, uniqueness theorems, axiom-to-theorem chain, complete derivation of τ = i/φ

02 — Galactic Dynamics

SPARC validation (175 galaxies), WALLABY confirmation (187 galaxies), NGC 3198 case study, robustness analysis, HALOGAS extended validation, three-scale verification

03 — Cosmology & Large Scale

Cosmic web structure, dark energy dynamical system (DynSys), cosmological constant solution, DESI BAO fit, high-redshift validations, correlated observables

04 — Particle Physics & Standard Model

Fermion mass hierarchy, neutrino masses, CKM/PMNS derivation, gauge coupling unification, baryon radii predictions, golden ratio nuclear/leptonic relations

05 — Mathematical Theorems

Fibonacci decomposition lemma, α⁻¹ = 137 closure proof, spectral theory, Kreĭn-space formulation, variational V-tree, Ω-bridge theorems

06 — Advanced Topics

6D thermodynamics, chronology protection, quantum decoherence, dimensional reduction, electromagnetic vertex derivation

07 — UV Completion & QFT

Asymptotic safety analysis, Gaussian fixed point proof, renormalization group flow, 2 relevant operators

08 — Gamma-Ray Astrophysics

AGN outflow quantization, discrete velocity structure in AGN, breathing mode predictions

09 — Supplementary Materials

Mathematical appendices, screening derivation, addenda, correction documents

10 — High-Redshift Validations

DynSys onset verification, observational paper on high-z behaviour, transition redshift confirmation

11 — Last Gap Solved

Papers completing the final theoretical gaps in the framework

12 — Dark Energy

Complete dark energy sector: DynSys formulation, w(z) evolution, baryogenesis connection

13 — Gravitational Effects & Relativity (GER)

Gravitational wave signatures, SLACS lensing analysis, J1453g IMF prediction, cosmic birefringence

14 — Atomic Physics

Golden ratio relations in atomic structure, nuclear/leptonic coupling patterns

15 — Foundations & Axioms

Extended axiomatic foundation, uniqueness of (3,1,1) signature, torus uniqueness theorem

16 — Ω_geom Bridge

Bridge matrix derivation, 19/73 density parameter proof, connection to cosmological observables

17 — Fibonacci & Golden Ratio

Deep structure of φ in the framework, Fibonacci matrix properties, golden ratio across physical sectors

18 — Uniqueness Theorems

Four independent no-go theorems, uniqueness of τ = i/φ, impossibility of alternative solutions

19 — Dynamical Systems

Complete DynSys sector: autonomous ODE, fixed-point analysis, z_tr = 0.972 derivation, phase-space portraits

20 — Spectral & Kreĭn Theory

Heat-kernel expansion, spectral action, Kreĭn-space formulation, functional analysis foundations

21 — V-tree & Variational

Variational V-tree formulation, systematic perturbative expansion, loop corrections

22 — Recent Papers (XCIII–XCV+)

Latest papers: advanced topics, completions, new derivations

23 — Simulations & Images

GADGET-4 cosmic web visualizations, power spectrum plots, density maps at z = 0, 1, 2, 3, cosmic evolution images (384³, 480³, 512³)

24 — Reference Tools

Python codes for SPARC fits, CLASS integration, GADGET-4 kernel, Euclid mock testing, IBM Quantum circuits, SALO optimization algorithms

25 — Dimensional Unification & QFT Protection (NEW — April 2026)

Four papers closing the foundational chain:

• Paper CI v3.2 — Dimensional Unification Theorem: all geometric constants from D = 6 (zero parameters, zero assumptions, algebraic quartic closure, Dimensional Selection from 4 physical principles)
• Paper C — Topological Protection of the CW Hierarchy: ζ(3) cancellation on T², g²Λ = 2π/W as Gauss-Bonnet invariant
• Paper XCIX — Non-Renormalization Theorem: 7/9 geometric numbers exactly protected to all orders
• Paper B3 NLO — Hierarchy Correction: ΔΛ = 1/28 reduces error from 3.66% to 0.019%

Reproducibility

All papers in PDF and Markdown format. All codes executable with standard Python (numpy, scipy, matplotlib, astropy, sympy). GADGET-4 modifications fully documented with build instructions. CLASS v3.3.4 integration scripts included.

A Note on AI Collaboration

This work was developed through intensive human-AI collaboration. Every derivation was independently verified by at least two AI systems, with mandatory adversarial review (Red Team) by a separate AI. All numerical results are reproducible with the provided Python codes. The use of AI as a derivation engine is disclosed transparently — it is a tool, not a substitute for scientific judgment. The final responsibility for every claim rests with the human author.

Science is not certainty. It is openness, dialogue, falsifiability. This is my contribution — offered in complete transparency, with humility, and with the hope that someone with the appropriate expertise will take a look.

The truth belongs to science. We are asking the community to test it.

Simone Calzighetti 3D+3D Laboratory, Abbiategrasso, Italy simone.calzighetti@3dplus3d.it www.3dplus3d.it

April 2026 — Theory Origin: September 14, 2025

README v9.0 — Updated April 21, 2026 with Dimensional Unification (D² = 6D) and Folder 25