"The universe is three-dimensional. Now we know why."

Author: Hugo Hertault — Tahiti, French Polynesia Series: Dark Geometry — Book III of V Companion to: Informational Relativity (I), Informational Geometry (II), The Holographic Fibration (IV)

Books I and II derived ~170 quantitative predictions from a single integer: d = 3. But they never explained why d = 3.

Book III answers that question. The dimension is not an axiom — it is derived as a theorem. Spacetime is not fundamental — it emerges as the continuum limit of a quantum-information network. The Hertault Axiom is not postulated — it is established (up to a stated functional-analytic limit) from that network.

This repository contains the verification code and the scorecard — not the book text (376 pages, archived on Zenodo).

34 predictions specific to Book III · mean error 0.42% · 40 decades in energy · zero free parameters beyond d = 3 and M_Pl.

Spacetime is the low-energy limit of a MERA tensor network of qutrits (χ = d = 3) with beam-splitter isometries at the Hertault angle θ_H = 35.26°. The dimension d = 3 is not an input but an output: the unique dimension in which the network is simultaneously anomaly-free, topologically knot-stable, algebraically unique, cosmologically viable, spinor-consistent, and RT–Fisher consistent. The holographic fibration ℋ = M⁴ ×_σ ℱ emerges as the continuum limit, and the Hertault Axiom e^{4σ} = ℐ is established as that limit up to the C*-limit (Corollary 7.24; the remaining step is stated as open).

𝒩_H ──► d = 3 ──► θ_H ──► h₃ ──► SM ──► masses ──► cosmology

The ratio β = cos²θ_H = 2/3 is the Hertault beam splitter — the same number giving Ω_Λ/Ω_m = 2, Q_Koide = 2/3, the holographic exponent, and the area-law entropy exponent. One number, everything.

d = 3 is the unique dimension satisfying all six simultaneously. Number-theoretic unifier: 4/d! = (d−1)/d has d = 3 as its only positive-integer solution. Condition (vi) is independent of the others: the algebraic beam-splitter variance 2/d² equals the RT quantum Fisher information (d−1)/d² only at d = 3 (both equal 2/9, the Koide phase). The theorem is graded a strong conjecture (verified d = 1–5); a rigorous proof for all d remains open.

Fermions = prime knots; generation k = c(K) mod 3; gauge bosons = braids; confinement = topological closure; CPT = mirror-knot symmetry. ER = EPR is a structural identity in the tensor network (two regions are entangled iff geometrically connected through the network), with exact partition function Z_ER(T) = 2 sinh(βT/2)/(β/2). The Page curve follows in three lines from knot unknotting, the total knot invariant conserved. (The knot–particle map beyond crossing number is an open problem.)

e^{4σ} = ℐ

Established as the continuum limit of 𝒩_H, up to the C-limit*, via five steps:

1. GH convergence: d_GH(𝒩_H, ℋ) ≤ 3.30 ℓ_P
2. Holonomy convergence (operator norm): ‖H_N − H_∞‖ ≤ 2.68/N
3. Double-limit SOT: ‖ω_{N,L}(O) − ω_∞(O)‖ ≤ C·N^{−0.057}, L = ⌊log₃ N⌋
4. Axiom verification: ω_∞ satisfies A1–A4
5. Identification: Book II Uniqueness Theorem ⟹ e^{4σ} = ℐ

Spectral gap Δ ≈ 0.061, consistent with the SOT exponent (Δ/ln 3 = 0.0555 ≈ 0.057). The step beyond the C*-limit is stated explicitly as open — the corollary is titled "partial resolution of the MERA-to-Axiom convergence."

Algebraic: Var(L_BS) = 2/χ² = 2/9 = β(1−β), unique for d = 3. Geometric (RT): ε = ∮ A_{u(1)} = F_Q/4 = Var(N̂) = β(1−β) = 2/9, on the u(1) sector of h₃ over S². The two expressions agree only at d = 3. Full Hertault phase: arg(b) = 2π/3 + 2/9 = 2.317 rad.

The fibre = the Dark Boson = the ER bridge. The Axiom at the IR boundary:

Λ/M_Pl⁴ = ℐ₀ = e^{−T} = β^L = 1/S_cosmo ≈ 10^{−122}

verified as an exact rational identity (Λ · S_cosmo = M_Pl⁴, i.e. β·(3/2) = 1). The angle θ_H ensures V₀ = β²/4 = 1/9 > 0: the wormhole is traversable at any depth without exotic matter, which is what lets the cascade operate for any inherited Λ. Distribution P(Λ) ∝ Λ^{−2/3}.

Total informational time τ ≈ 70 Planck units, corresponding to N_steps ≈ 10^{122} network updates — exactly the cosmological horizon entropy. Primordial entropy, derived exactly: S₀ = 16π² ≈ 158 nats = 228 bits. Inflation = MERA unfolding (ΔL ≈ 12 layers); scale-invariance follows as a theorem from MERA self-similarity.

Gravitons emerge as entanglement ripples in a three-level hierarchy (inter-site / inter-layer / inter-mode). The conformal ghost is absent before quantisation as a structural consequence of the axiom (Tier A): two second-class Dirac constraints remove σ from the physical phase space (n_phys^conf = 0). One-loop corrections are suppressed by e^{−π/α_*} ≈ e^{−41.9}. Newton's constant (from the beam splitter, Book IV): G₄ = sin²θ_H/(8π M₅³), with sin²θ_H = 1/d holding only at d = 3.

In topological Kondo insulators YbB₁₂ and SmB₆: Fibonacci frequency ratios (3:2, 5:3, 8:5, … → φ), surface-to-bulk conductivity ratio 2:1, transport scaling exponent β = 2/3.

Four persistent anomalies (low quadrupole, quadrupole–octupole alignment, hemispherical power asymmetry of order 6–7%, no correlations above ~60°) are presented as consistent topological imprints of the MERA bounce — these are post-dictions. The one genuine falsifiable prediction: all four anomaly axes lie within ~20° of a single cascade direction n̂_cascade, probability ≲ 1% under ΛCDM. Testable by CMB-S4 / LiteBIRD (~2030).

Selected predictions (full list in docs/scorecard.csv):

The five major quantum-gravity programmes are presented not as rivals but as projections of the single Hertault angle — each computes one trigonometric shadow of θ_H; their convergence is argued as a structural theorem (Chapters 20–25).

Every canonical number is reproduced by a script before being quoted. Both scripts end in ALL PASS.

pip install -r requirements.txt
cd code
for f in *.py; do python "$f"; done   # each script must end in ALL PASS

Dependencies: numpy, mpmath.

Part I    — The Constraints
  1. What the Model Demands
  2. The Conformal Mode Is Not a Degree of Freedom
Part II   — The Dimensional Selection Theorem
  3. The Problem of Dimension
  4. The Entry Corridor
Part III  — The Information Network
  5. Tensor Networks and Holography
  6. The Hertault Network 𝒩_H
  7. Emergence of the Hertault Axiom        (Corollary 7.24)
Part IV   — Topology of the Fibre
  8. Knots, Particles, and the Origin of Matter
Part V    — The Cascade
  9. The Cascade of Nested Horizons
  10. Time as Information Flow
Part VI   — Collapse, Singularity, and Decollapse
  11. The Fate of the Singularity
  12. Geometric Singularity Avoidance
  13. Decollapse: The Geometry Runs Backward
Part VII  — The Multimode Boson
  14. The Spectral Structure of the Hertault Field
  15. Resonances in Condensed Matter
  16. Black Holes as Informational Engines
  17. The Network View of the Early Universe
Part VIII — Entanglement and Gravity
  18. Entanglement as the Foundation
  19. Comparison with Other Approaches
Part IX   — The Convergence of Quantum Gravity
  20. The Five Projections of the Hertault Angle
  21. Asymptotic Safety: The Ultraviolet Fixed Point
  22. Loop Quantum Gravity: The Discrete Sections
  23. String Theory and the Spectral Dimension
  24. The Holographic Principle: UV–IR Mixing
  25. Why the Convergence Is a Theorem
Part X    — Predictions and Tests
  26. Predictions Unique to the Quantum Geometry
Part XI   — Open Problems
  27. The Frontier
Part XII  — Compilation
  28. The Complete Derivation Chain
  29. Conclusion
  30. Computational Methods and Verification
  31. Summary of Book I Results Used
  32. Summary of Book II Results Used
  33. Notation and Conventions

Established: the MERA→Axiom convergence up to the C-limit (Corollary 7.24); the Koide phase ε = 2/9 via RT on the u(1) sector.*

Every quantitative claim is classified by evidential tier — A algebraic/exact, B derived with stated approximations, C conjecture/order of magnitude — and every canonical number is reproduced by a script before it is quoted. Observational comparisons are made like for like (S₈ against the observable the surveys report).

See CITATION.cff.

@book{hertault2026quantum,
  author    = {Hertault, Hugo},
  title     = {Quantum Geometry: Topological Informational Gravity},
  series    = {Dark Geometry},
  volume    = {III},
  year      = {2026},
  publisher = {Self-published (KDP)},
  address   = {Tahiti, French Polynesia},
  doi       = {10.5281/zenodo.18929646}
}

Code: MIT (see LICENSE). The books are separate works, licensed CC BY 4.0.

e^{4σ} = ℐ

One equation. Zero free parameters. Everything. The universe is three-dimensional — now we know why.