Verification code and documentation for Book Zero of the Dark Geometry series (806 pages, thirteen Parts): the canonical reference volume of the programme, gathering the eleven founding papers of Dark Geometry into a single source — with the connective essays and the opening/closing vision chapters that frame them — around one axiom and one number.
The axiom. The conformal mode of the spacetime metric equals the local informational saturation ratio:
e^(4σ(x)) = I(x) = S_ent(x) / S_Bek(x) (equivalently e^(Dσ′) = I with D = d+1 in the spacetime base; the two conventions coincide at d = 3)
The only empirical input is d = 3 (space has three dimensions). From it, every constant below follows by trigonometric projection through the Hertault angle θ_H.
The axiom is not postulated but derived (established in Book I, consolidated in Book IV): it is the unique minimum of an entropic free energy — vacuum volume cost against the Casini energy of the content — taken covariantly and pointwise,
F[σ] = ρ_Λ · V̂ · e^((D−1)σ) + (S_ent / 2π R̂) · e^(−σ) ,
with the Casini saturation itself a theorem of the Gibbs variational principle.
Book Zero is the canonical reference volume of the Dark Geometry programme. Where Books I–IV develop one face each in full technical depth, Book Zero is the single place where the eleven founding papers of the framework are gathered, connected by linking essays and framed by the opening and closing vision chapters. It is the map of the whole territory: read it first to see how the pieces fit, then go to the individual volumes for the proofs.
Dark Geometry rests on one idea — that dark matter and dark energy are not substances but the conformal mode of spacetime (the Dark Boson), and that everything follows from one axiom and one integer, d = 3. That single idea has eleven complementary faces, each a founding paper:
| # | Face (founding paper) | What it contributes |
|---|---|---|
| 1 | DG — Dark Geometry | The overview and synthesis: the axiom, the conformal mode, and the unified picture across all faces. |
| 2 | IDG — Informational Dark Geometry | The thermodynamic / information face: gravity as the dynamics of information; the four informational laws. |
| 3 | HDG — Holographic Dark Geometry | The holographic face: the fibration H = M³ ×_σ F as the geometric stage; the axiom as its section. |
| 4 | QGU — Quantum Gravity Unification | The quantum-gravity face: five QG programmes (Asymptotic Safety, LQG, AdS/CFT, CDT, celestial holography) as projections of θ_H. |
| 5 | HF-DG — The Holographic Fibration | The rigorous mathematical treatment of the foundational object and the uniqueness of the axiom. |
| 6 | MERA-DG — The Hertault Network | The pre-geometric substrate: a tensor-network (MERA) realisation whose RG flow is the depth-parameterised dynamics. |
| 7 | mDG — Microscopic Dark Geometry | The particle-physics face: the Standard Model spectrum — gauge group, three generations, masses, mixings (~170 predictions). |
| 8 | spDG — Sub-Planckian Dark Geometry | The vibratory origin of mass: masses as nodes of the Dark Boson; black holes as condensates; cosmological inheritance. |
| 9 | wzDG — Equation-of-State Profiles | The dark-energy face: the w(ρ) and w(z) profiles, and why the two equations of state must not be conflated. |
| 10 | dg-tensions — The Cosmological Tensions | The H₀ and σ₈ tensions resolved from first principles, locked by the Coupling Identity Δn/ξ = 20/(27π²). |
| 11 | Condensate Dynamics — The Hertault Condensate | From the equilibrium constraint to an equation of motion for the conformal mode; gravitational-wave echoes. |
All eleven are logically equivalent presentations of the same structure — different charts of one manifold. Their convergence on the same parameter values (θ_H, β = 2/3, α∗, ξ) is not a coincidence but a theorem. Book Zero is where that convergence is laid out as a single argument.
Everything is a projection of the Hertault angle
θ_H = arccos√((d−1)/d) = arcsin(1/√d) ⇒ for d = 3, θ_H ≈ 35.264°,
with the holographic exponent β = (d−1)/d = 2/3 and sinθ_H = 1/√3, cosθ_H = √(2/3) = 0.8165, sin2θ_H = 2√2/3.
| Quantity | Formula | Value |
|---|---|---|
| Dark-energy fraction | Ω_Λ = cos²θ_H = β | 2/3 ≈ 0.667 |
| Matter fraction | Ω_m = sin²θ_H = 1−β | 1/3 ≈ 0.333 |
| Coupling | α∗ = √2/(6π) | 0.0750264 |
| Non-minimal coupling | ξ = β/[4(1+β)] | 1/10 |
| Holographic UV fixed point | g∗^UV = √6/3 = β·√(3/2) | 0.8165 |
| Coupling-identity shift | Δn = 2β·α∗² = 2/(27π²) | 7.5053×10⁻³ |
| Conformal weight | s(s−1) = β | s ≈ 1.4574 |
| Field range | u_max = π/α∗ | 41.873 |
| Hubble ratio (tension) | H₀^local / H₀^Planck = 1+ξ | 11/10 |
| Barbero–Immirzi parameter | γ_BI = (ln2/π)·sinθ_H = ln2/(π√3) | — |
| Log BH-entropy correction | c_log = (dim SU(2)+cos²θ_H)/2 = (3+β)/2 | 11/6 |
| Quantity | Formula | Value | Accuracy |
|---|---|---|---|
| Proton/electron mass ratio | m_p/m_e = 2d·π^(d+2) = 6π⁵ | 1836.118 (obs. 1836.152) | 19 ppm |
| Electroweak vev | v_H (NLO, from α∗, β) | 246.225 GeV (obs. 246.220) | 23 ppm |
| Weak mixing angle (M_Z) | sin²θ_W = d/Φ₃(d) = d/(d²+d+1) = 3/13 | 0.23077 | 0.19 % |
| Proton mass | m_p = d·µ (anchor µ ≈ 299 MeV) | — | — |
| Koide anchor | µ_Koide = 3·α∗³·v_H | ≈ 312 MeV | — |
| Critical density | ρ_c = Ω_Λ·ρ_crit ⇒ ρ_c^(1/4) | 2.30 meV (Planck 2.28) | 0.85 % |
| Structure growth | σ₈^DG = 0.766 ⇒ S₈ = σ₈√(Ω_m/0.3) | S₈ = 0.779 | ~1 % |
| Early-time Hubble | H₀^(DG, early) | ≈ 70.6 km/s/Mpc | ~2.3σ resid. |
| Inflation | n_s = 0.9654, r = 3.3×10⁻³ (Starobinsky–DG, N_e=60) | — | — |
| Quantity | Formula | Value |
|---|---|---|
| Neutrino mass-squared ratio | Δm²₂₁/Δm²₃₁ → 1/F₉ = 1/34 (Fibonacci gap, F₉ from d²=9 config. space) | 0.02941 (obs. 0.0307; JUNO test by 2030) |
The horizon acts as a 2×2 unitary beam splitter U(θ_H): the primordial state is partitioned into a transmitted trunk of weight cos²θ_H = 2/3 (dark energy) and a reflected branch of weight sin²θ_H = 1/3 (matter). The echo amplitude cascades as (sin²θ_H)ⁿ = (1/3)ⁿ.
Stacking the cascade over 256 layers yields the cosmological-constant hierarchy and an exact rational identity:
Λ/M_Pl⁴ = β·(1/3)²⁵⁶ ≈ 10⁻¹²² and Λ × S_cosmo = M_Pl⁴
(with S_cosmo ~ 10¹²², L ≃ 255 in the book's microscopic counting). The same beam-splitter geometry predicts a gravitational-wave echo structure: Δt ~ 36 ms for stellar-mass mergers (LIGO band) versus ~mHz for the parent echo (LISA band).
| Script | What it reproduces |
|---|---|
dg_constants.py |
Every constant of the volume from d = 3 (θ_H, β, α∗, ξ, Δn = 2/(27π²) exactly, s, u_max) and the honest observational scoreboard stated like-for-like: Ω_Λ at 2.6σ from Planck (stated as such), Hubble ratio 11/10 at 0.9σ, σ₈→S₈ = 0.779 at 0.6σ of KiDS (compared in S₈, not σ₈), m_p/m_e at 19 ppm, v_H(NLO) at 23 ppm. |
dg_beam_splitter.py |
The (1/3)ⁿ echo cascade; the 256-layer attenuation Λ/M_Pl⁴ = β(1/3)²⁵⁶ ~ 10⁻¹²²; the exact rational identity Λ × S_cosmo = M_Pl⁴; the LIGO-band (~36 ms) vs LISA-band (mHz) echo distinction. |
DARK GEOMETRY -- the constants from d = 3
theta_H = arccos sqrt(2/3) = 35.264 deg
beta = (d-1)/d = 0.666667
alpha* = sqrt(2)/(6 pi) = 0.0750264
xi = beta/[4(1+beta)] = 0.100000
Delta n = 2 beta alpha*^2 = 7.505273e-03 ( = 2/(27 pi^2) exactly )
s : s(s-1) = beta = 1.457427
u_max = pi/alpha* = 41.8732
Honest observational scoreboard (like-for-like):
Omega_L = cos^2 = 0.6667 vs Planck 0.685 +/- 0.007 -> 2.6 sigma (stated as such)
H ratio 11/10 vs measured 1.0843 +/- 0.0177 -> 0.9 sigma
sigma8 = 0.766 -> S8 = 0.7787 vs KiDS S8 0.766 +/- 0.020 -> 0.6 sigma (S8, not sigma8!)
m_p/m_e = 6 pi^5 = 1836.1181 vs 1836.1527 -> 19 ppm
v_H(NLO) = 246.225 GeV vs 246.220 -> 23 ppm
pip install -r requirements.txt
cd code
for f in *.py; do python "$f"; doneRequires Python 3.9+, numpy>=1.24, scipy>=1.10.
Book 0 — Dark Geometry: Behind the Horizon is archived on Zenodo: doi:10.5281/zenodo.19673186 (print/ebook editions are listed there). This repository contains the verification code, not the book text.
| Volume | Repository | Book (Zenodo) |
|---|---|---|
| Book 0 — Dark Geometry: Behind the Horizon | dark-geometry |
10.5281/zenodo.19673186 |
| Book I — Informational Relativity | informational-relativity |
10.5281/zenodo.18132261 |
| Book II — Informational Geometry | DG-Book2-Informational-Geometry |
10.5281/zenodo.18870211 |
| Book III — Quantum Geometry | DG-Book3-Quantum-Geometry |
10.5281/zenodo.18929646 |
| Book IV — The Holographic Fibration | DG-Book4-Holographic-Fibration |
10.5281/zenodo.19546658 |
Companion code repositories:
DG-S8H0-simulations (CLASS-DG cosmology pipeline),
DG-condensate-dynamics,
decollapse_repo.
Every quantitative claim of the book 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 — the scripts in code/ are that discipline made public. Observational
comparisons are made like for like (in particular, weak-lensing comparisons are
made in S₈, the observable the surveys report). The framework has zero free
parameters: ξ = 1/10 and α∗ = √2/(6π) are algebraically derived (Tier A), not fitted.
See CITATION.cff. If you use this code, cite the corresponding book and this repository.
Code: MIT (see LICENSE). The books are separate copyrighted works.