Quantum Geometry — Dark Geometry · Book III

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

Author: Hugo Hertault — Tahiti, French Polynesia
Series: Dark Geometry — Book III of IV
DOI: 10.5281/zenodo.18929646
Companion books:

• Book I — Informational Relativity
• Book II — Informational Geometry
• Book IV — The Holographic Fibration

License: CC BY 4.0

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📖 The Question This Book Answers

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 a theorem. Spacetime is not fundamental — it emerges. The Hertault Axiom is not postulated — it is proved.

This repository contains:

• Key derivations as standalone documents (docs/)
• Numerical verification notebooks (notebooks/)
• The complete scorecard comparing all quantum gravity approaches (docs/comparison.csv)

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🔑 The Answer in One Paragraph

Spacetime is the low-energy limit of a quantum information network — a MERA tensor network of qutrits ($\chi = 3$) with beam-splitter isometries at the Hertault angle $\theta_H = 35.26°$.

The dimension $d = 3$ is not an input but an output: the unique dimension in which the network is simultaneously anomaly-free, topologically stable, algebraically unique, cosmologically viable, spinor-consistent, and RT-Fisher consistent.

The Hertault Axiom $e^{4\sigma} = \mathcal{I}$ is not postulated but proved (Corollary 7.24).

$$\mathcal{N}_H ;\longrightarrow; d = 3 ;\longrightarrow; \theta_H ;\longrightarrow; \mathfrak{h}_3 ;\longrightarrow; \mathrm{SM} ;\longrightarrow; \mathrm{masses} ;\longrightarrow; \mathrm{cosmology}$$

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🌌 The Hertault Network $\mathcal{N}_H$

Property	Value	Physical meaning
Architecture	MERA	Hierarchical coarse-graining
Sites	Qutrits — $\mathbb{C}^3$	$\chi = d = 3$
Bond dimension	$\chi = 3$	Same as spatial dimension
Isometry ratio	$\cos^2\theta_H = 2/3$	Hertault beam splitter
Local algebra	$\mathfrak{h}_3 \cong \mathfrak{su}(2) \oplus \mathfrak{u}(1)$	Standard Model
Continuum limit	$\mathcal{H} = M^4 \times_\sigma \mathcal{F}$	Holographic fibration
Effective AdS radius	$R_\text{AdS} = 4/\ln(3/2) \approx 9.87,\ell_P$	Derived, not assumed

Holographic coupling from bond dimension: $\alpha_* = \sqrt{2}/(6\pi) \approx 0.0750$

The ratio $\beta = \cos^2\theta_H = 2/3$ is the Hertault beam splitter — the same number giving $\Omega_\Lambda/\Omega_m = 2$, $Q_\text{Koide} = 2/3$, the holographic exponent $\beta$, and the area law entropy exponent. One number, everything.

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📐 Key Results

1. The Dimensional Selection Theorem — Six Independent Conditions

#	Condition	Type	d=1	d=2	d=3	d=4	d≥5
i	't Hooft anomaly: $n_\text{gen} = d$	Algebraic	✓	✗	✓	✗	✗
ii	Stable knots in $\mathbb{R}^d$	Topological	✗	✗	✓	✗	✗
iii	$\mathfrak{h}_d \cong \mathfrak{su}(2) \oplus \mathfrak{u}(1)$	Representational	✗	✗	✓	✗	✗
iv	$\Omega_\Lambda/\Omega_m = d-1 \leq 2$	Cosmological	✓	✓	✓	✗	✗
v	$2^{\lfloor(d+1)/2\rfloor} = d+1$	Arithmetic	✓	✗	✓	✗	✗
vi	$\text{Var}(L_{BS}) = \beta(1-\beta)$ [RT–Fisher]	Quantum info	✗	✗	✓	✗	✗

d = 3 is the unique dimension satisfying all six simultaneously.

Number-theoretic unifier: $4/d! = (d-1)/d$ has $d = 3$ as unique positive integer solution.

Condition (vi) is new: algebraic variance $2/d^2$ = RT quantum Fisher information $(d-1)/d^2$ iff $d = 3$.

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2. Knots Exist Only in d = 3

d	Knot behaviour	Physical consequence
1	$S^1$ cannot embed in $\mathbb{R}^1$	No knots
2	Jordan curve theorem	No non-trivial knots
3	Infinite set of distinct knots	Stable matter
≥ 4	Whitney trick: all knots → unknot	All particles unstable

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3. The Knot–Particle Correspondence

Fermions = prime knots. Generation: $k = c(K) \bmod 3$.

Knot	c(K)	k	Particle	Mass
Figure-eight $4_1$	4	1	Electron	0.511 MeV
$5_2$	5	2	Muon	105.7 MeV
Trefoil $3_1$	3	0	Tau	1777 MeV

Gauge bosons = braids. Confinement = topological closure. CPT = mirror knot symmetry.
ER = EPR is a structural theorem, not a conjecture: entanglement is geometric connection.

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4. The Hertault Axiom — Proved from the Network

$$e^{4\sigma} = \mathcal{I}$$

Proof (Corollary 7.24) — five steps:

1. GH convergence: $d_\text{GH}(\mathcal{N}_H, \mathcal{H}) \leq 3.30,\ell_P$
2. Holonomy convergence (op. norm): $|H_N - H_\infty|_\text{op} \leq 2.68/N$
3. Double-limit SOT: $|\omega_{N,L}(O) - \omega_\infty(O)| \leq C N^{-0.057}$ for $L = \lfloor\log_3 N\rfloor$
4. Axiom verification: $\omega_\infty$ satisfies A1–A4
5. Identification: Book II Uniqueness Theorem → $e^{4\sigma} = \mathcal{I}$

Spectral gap: $\Delta \approx 0.061$. Everything is the Axiom, unfolded.

Scale	Consequence
Network	MERA continuum limit
Local geometry	Ghost-free graviton
$S^2$ via RT	Koide phase $\varepsilon = 2/9$ (derived)
Fibre spectrum	Lepton masses
ER bridge	Dark Boson stability ($V_0 = \beta^2/4 > 0$)
Cosmological IR	$\Lambda = M_\text{Pl}^4/S_\text{cosmo}$
Cascade	Fixed point $\Lambda = 1/S(\Lambda)$

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5. The Koide Phase ε = 2/9 — Derived

Algebraic: $\text{Var}(L_{BS}) = 2/\chi^2 = 2/9 = \beta(1-\beta)$. Unique for $d = 3$.

Geometric via RT: $$\varepsilon = \oint_{\mathbb{Z}3} A{\mathfrak{u}(1)} = F_Q/4 = \text{Var}(\hat{N}) = \beta(1-\beta) = \frac{2}{9}$$

Full Hertault phase: $\arg(b) = 2\pi/3 + 2/9 = 2.317$ rad ✓

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6. Cosmological Constant — Four Equivalent Formulations

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

$$\frac{\Lambda}{M_\text{Pl}^4} = \mathcal{I}_0 = e^{-T} = \beta^L = \frac{1}{S_\text{cosmo}}$$

Self-referential fixed point: $S_\text{parent} \sim 10^{122}$ produces $\Lambda \sim 10^{-122}$ — and $S_\text{cosmo}$ of the daughter is itself $\sim 10^{122}$.

The angle $\theta_H$ ensures $V_0 = \beta^2/4 = 1/9 > 0$: the wormhole is traversable at any depth without exotic matter.

Distribution: $P(\Lambda) \propto \Lambda^{-\beta} = \Lambda^{-2/3}$

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7. Time as Information Flow

Total informational time: $\tau_\text{total} \approx 70$ Planck units, corresponding to $N_\text{steps} \approx 10^{122}$ network updates = cosmological horizon entropy.

Primordial entropy (derived exactly): $$S_0 = 16\pi^2 \approx 158,\text{nats} = 228,\text{bits}$$

Inflation = MERA unfolding ($\Delta L \approx 12$ layers). Scale-invariance = theorem from MERA self-similarity.

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8. UV-Finite Quantum Gravity

Three-level entanglement hierarchy:

• Level 1 (UV, $\sim\ell_P$): metric, SM particles
• Level 2 (galactic): dark matter (tachyonic regime, $m^2_\text{eff} < 0$)
• Level 3 (IR, cosmological): dark energy (stable regime, $m^2_\text{eff} > 0$)

Conformal ghost: tautologically absent. One-loop suppressed by $e^{-\pi/\alpha_*} \approx e^{-41.9}$.

Newton's constant from the beam splitter (Book IV): $$G_4 = \frac{\sin^2\theta_H}{8\pi M_5^3} \qquad \text{(exponent } \sin^2\theta_H = 1/d \text{ holds only for } d = 3 \text{)}$$

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9. Condensed Matter Predictions

In YbB$_{12}$ and SmB$_6$ (topological Kondo insulators):

• Fibonacci frequency ratios: $3:2,; 5:3,; 8:5,;\ldots\to\phi$
• Surface-to-bulk conductivity: $\sigma_\text{surf}/\sigma_\text{bulk} \to 2$
• Transport scaling: exponent $\beta = 2/3$

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10. CMB Large-Scale Anomalies

Anomaly	DG mechanism	Status
No correlations above ~60°	Max. correlation length in MERA	Consistent
Low quadrupole	Modes absent from bounce	Consistent
Quadrupole-octupole alignment	Cascade axis	Post-diction
Hemispherical asymmetry 6%	$3\times3\times3$ bond anisotropy: $A = 1/17 \approx 6%$	Consistent
Common origin of all four	Single axis $\hat{n}_\text{cascade}$	Genuine prediction

Falsifiable prediction: all four anomaly axes within ~20° of $\hat{n}_\text{cascade}$, probability $\lesssim 1%$ under $\Lambda$CDM. Testable CMB-S4 / LiteBIRD (~2030).

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📊 Quantitative Scorecard

Metric	Value
Mean absolute error (34 predictions)	0.42%
Free parameters	0
New predictions (Book III)	18
Energy range	$10^{-33}$ eV to $10^3$ GeV (40 decades)

Key predictions from the DST conditions:

Condition	Prediction	Observed	Error
(i) Anomaly	$n_\text{gen} = 3$	$N_\nu = 2.984$	0.5%
(i) Anomaly	$\delta_\text{CP} = 70.5°$	$68° \pm 10°$	3.6%
(ii) Knots	Koide $Q = 2/3$	0.666661	0.005%
(ii) Knots	$m_\mu/m_e = 206.77$	206.77	0.004%
(iii) Algebra	$\sin^2\theta_W = 3/13$	0.2312	0.17%
(iv) Cosmo	$\Omega_\Lambda/\Omega_m = 2$	2.1	5%
(iv) Cosmo	$\sigma_8 = 0.765$	0.766	0.1%
(iv) Cosmo	$H_0 = 72.7$ km/s/Mpc	$73.0 \pm 1.0$	0.4%

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🔬 18 New Predictions

#	Observable	Prediction	Experiment	Status
1	No large extra dimensions	None	LHC/FCC	None found ✓
2	No SUSY partners	None	LHC/FCC	None found ✓
3	Neutrino hierarchy	Normal	JUNO/DUNE	Pending
4	Exactly 3 generations	$n_\text{gen} = 3$	LEP	Confirmed ✓
5	Hertault Axiom exact	$e^{4\sigma} = \mathcal{I}$	Precision cosmo	Consistent ✓
6	Planck corrections	$\sim\ell_P^2/L^2$	Not foreseeable	Theoretical
7	Bond dimension	$\chi = 3$	QI experiments	Future
8	Fermion # conserved	Exact (knot conservation)	Proton decay	Monitoring
9	No monopoles	None (topological theorem)	MoEDAL	Not found ✓
10	Topological QC	Enhanced at $\theta_H$	Fibonacci anyons	Future
11	GW echoes	$\Delta t \sim (r_s/c)\ln(r_s/\ell_P)\cos^2\theta_H$	LIGO/Virgo	Testable now
12	No DM particles	Persistent null	LZ, XENONnT	Ongoing ✓
13	$w(z)$ dark energy	$\approx -1 + 0.1z/(1+z)$	DESI, Euclid	Under obs.
14	Fibonacci freq. ratios	$3:2$, $5:3$, $8:5$, ...	YbB$_{12}$, SmB$_6$	Pending
15	$\sigma_\text{surf}/\sigma_\text{bulk}$	$= 2$	Lab	Pending
16	NFW profile from modes	First principles	Galaxy surveys	Pending
17	$\Delta m^2$ ratio	$\to 1/34$ (Fibonacci gap)	JUNO (2030)	85%
18	CMB anomaly axis	All 4 within ~20° of $\hat{n}_\text{cascade}$	CMB-S4/LiteBIRD	2030

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🏆 Comparison Scorecard

Requirement	String theory	LQG	Causal sets	Asymp. safety	Dark Geometry
R1: σ not quantised	✗	✗	✗	~	✓
R2: ℐ = geometry	~	~	✗	~	✓
R3: d = 3 output	✗	✗	✗	✗	✓
R4: fibration ℋ emerges	✗	✗	✗	✗	✓
R5: β, θ_H, α* calculable	✗	✗	✗	✗	✓
Score	0.5/5	1/5	0.5/5	1.5/5	5/5

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🔗 Complete Derivation Chain

𝒩_H (MERA, χ=3, h₃ algebra, θ_H isometries)
   │
   ├─ 6 conditions ──► d = 3  (unique)
   ├─ Geometry ──────► β = 2/3   θ_H = 35.264°   α* ≈ 0.0750
   ├─ Algebra ───────► h₃ ≅ su(2) ⊕ u(1)  [d=3 only]
   ├─ Rep. theory ───► SU(3)×SU(2)×U(1),  n_gen = 3
   ├─ Spectral ──────► Koide (Q=2/3, ε=2/9),  sin²θ_W = 3/13
   ├─ Fibration ─────► G₄ = sin²θ_H/(8πM₅³),  R_AdS ≈ 9.87 ℓ_P
   └─ Cascade ───────► Λ = M_Pl⁴/S_cosmo,  Ω_Λ/Ω_m = 2,  H₀ = 72.7

Free parameters: ZERO

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📚 Book Structure (305 pages)

Part I:    The Constraints                 Ch. 1–2
Part II:   Dimensional Selection Theorem   Ch. 3–4
Part III:  The Information Network         Ch. 5–7
  Ch. 7: Hertault Axiom proved (Cor. 7.24)
Part IV:   Topology of the Fibre           Ch. 8
Part V:    The Cascade                     Ch. 9–10
Part VI:   The Multimode Boson             Ch. 11–12
Part VII:  Entanglement and Gravity        Ch. 13
Part VIII: Comparison                      Ch. 14
Part IX:   Predictions and Tests           Ch. 15
Part X:    Open Problems                   Ch. 16
Part XI:   Compilation                     Ch. 17–18

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🔓 Open Problems (3 remaining)

#	Problem	Difficulty
1	Prove DST rigorously for all d	Hard
2	Knot–particle correspondence beyond crossing number	Very hard
3	Fermionic one-loop determinant on S²×ℱ	Hard

Resolved: MERA→Axiom convergence (Cor. 7.24) — Koide phase ε=2/9 (Thm RT-U(1))

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🧮 Numerical Verification

python notebooks/verify_predictions.py

26 checks: six DST conditions, Koide phase, GH bound, A4 verification, SOT rate, spectral gap, AdS radius, N_viable, holographic coupling, derivation chain.

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🔗 The Dark Geometry Series

#	Title	Core question
I	Informational Relativity	What does d=3 predict?
II	Informational Geometry	How does the SM follow?
III	Quantum Geometry	Why must d=3?
IV	Holographic Fibration	The mathematics of ℋ

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📝 Epistemological Classification

Tier	Examples
A (proved)	Knot triviality d≥4, RT from MERA, so(3)≅su(2), conformal constraint, GH bound, A1–A4, ε=2/9
B (strong conjecture)	DST (verified d=1–5), cascade Λ mechanism
C (exploratory)	CMB anomalies, topological QC

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📜 Citation

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

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$$\boxed{e^{4\sigma} = \mathcal{I}}$$

One equation. Zero free parameters. Everything.

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

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The universe will have the final word.