Discrete 3D+3D Temporal Geometry: A Single-Axiom Unified Framework for Galactic Dynamics, Cosmology, 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.

The Work Done

With the AI collaboration described above, I have developed:

90+ academic papers (~1500 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, and UV completion.

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
• 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 at 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 in progress

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)

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:

• Verify the mathematical derivations
• Criticize the physical assumptions
• Refute the predictions with data
• Improve the framework where necessary
• 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:

• Root documents: README, LICENSE (MIT), canonical parameter dictionary (SSoT), falsification criteria
• Foundational papers: logical inevitability of 6D, Parameter Regress Theorem, uniqueness theorem
• Mathematical core: axiom-to-theorem chain, Fibonacci decomposition lemma, α⁻¹ = 137 closure, complete 6D Lagrangian, spectral theory, cosmological kernel theorem, Boltzmann-complete kernel theorem
• Galactic dynamics: SPARC validation (175 galaxies), WALLABY confirmation (187 galaxies), NGC 3198 case study, robustness analysis, HALOGAS validation
• Cosmology: cosmic web, cosmological constant solution, dark energy dynamical system, DynSys fixed points, 15 pre-registered predictions, DESI BAO fit
• N-body simulations: GADGET-4 modified gravity implementation, power spectrum analysis, σ₈ tension results
• Particle physics: fermion mass hierarchy, neutrino masses, CKM/PMNS derivation, baryogenesis, proton decay
• Reproducibility: Python codes for SPARC fits, CLASS integration, GADGET-4 kernel, Euclid mock testing, IBM Quantum circuits

All papers in PDF and Markdown format. All codes executable with standard Python (numpy, scipy, matplotlib, astropy, sympy).

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