Module Dependency Structure
Interactive Flow Diagram
Master Equation
δℐtot
Total Coherence Cost: the single functional from which all physics emerges. Stationarity of this functional generates all physical laws.
[
DA,ℛ
Dirac operator encoding geometry and matter: A = gauge fields (determining particle interactions), ℛ = resolution field (determining spacetime geometry).
,
ϱ
Quantum state (density matrix): encodes all information about the system. The coherence kernel reconstructs the full Hilbert space structure.
]
=
0
This single master equation encodes
Gravity
Gravity emerges from the geometric term: the KL divergence between spectral measures yields Einstein's equations via maximum entropy.
,
Matter
Matter content emerges from octonion structure: the exceptional geometry of 𝕆 uniquely determines gauge groups, generations, and mass ratios.
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Quantum
Quantum state (density matrix): encodes all information about the system. The coherence kernel reconstructs the full Hilbert space structure.
, and
Thermo
Thermodynamic arrow of time: Landauer erasure cost at the vacuum temperature drives irreversibility and determines dark energy density.
.
Free Parameter
Physical Postulate
Derived
In Progress
Foundations
Octonion Algebra
The octonions are the largest "number system" where division works. They're 8-dimensional and uniquely non-associative. TCR shows that the forces of nature (electromagnetism, weak, strong) aren't arbitrary—they're the only possibilities allowed by octonion mathematics.
Coherence Kernel
A mathematical function that measures how "similar" two quantum states are. From this single object, TCR reconstructs all of quantum mechanics—Hilbert spaces, wave functions, and probabilities all emerge automatically.
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Derived Structures
Exceptional Geometry
G₂ = Aut(O)
dim = 14
G₂ is the symmetry group of the octonions—the set of all transformations that preserve octonion multiplication. It has exactly 14 dimensions and is one of the five "exceptional" Lie groups. All gauge symmetries trace back to G₂.
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S⁶ Geometry
The 6-dimensional sphere formed by unit imaginary octonions. The geometry of this sphere encodes particle mixing angles. The Cabibbo angle, for instance, comes from distances on S⁶.
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Stab(i₀) ≅ SU(3)
If you pick any point on S⁶ and ask "which G₂ transformations leave it fixed?", the answer is SU(3)—exactly the symmetry group of the strong nuclear force (QCD). The strong force isn't assumed; it's derived.
Jordan Algebra
H₃(O)
dim = 27
The exceptional Jordan algebra: 3×3 matrices of octonions with special multiplication rules. Its 27 dimensions match the degrees of freedom of one generation of fermions. This structure determines mass ratios and Yukawa couplings.
Quantum Theory
RKHS Construction
A theorem (Moore-Aronszajn) guarantees that any coherence kernel automatically generates a Hilbert space. This is how TCR derives quantum mechanics: the Hilbert space of QM isn't assumed, it's constructed.
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Gleason's Theorem
A mathematical theorem proving that in 3+ dimensions, there's only ONE consistent way to assign probabilities to quantum measurements. This forces the Born rule to be true—it's not a postulate but a theorem.
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Born Rule
derived
The rule that probabilities equal |amplitude|². Standard QM assumes this; TCR derives it. Gleason's theorem shows it's the unique probability rule consistent with quantum logic.
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Uncertainty Principle
Heisenberg's ΔxΔp ≥ ℏ/2 follows from the coherence kernel structure. Position and momentum can't both be sharp because the kernel encodes their fundamental incompatibility.
Gauge Structure & Gravity
Gauge Theory
U(1) × SU(2) × SU(3)
The Standard Model gauge group describing electromagnetism, weak, and strong forces. In TCR, this isn't chosen—it emerges from octonion geometry. SU(3) is Stab(i₀), and U(1)×SU(2) comes from its embedding.
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Anomaly Cancellation
κ₁/κ₂ = 5/3
Quantum field theories can become inconsistent ("anomalous") unless particle charges satisfy specific constraints. TCR derives κ₁/κ₂ = 5/3, matching the value needed for consistency—and the GUT prediction.
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Three Generations
Why are there exactly 3 copies of each particle (electron/muon/tau, etc.)? TCR says it's because of "triality"—a unique property of 8-dimensional geometry that relates three equivalent representations.
Gravitational Sector
H₀
single parameter
The Hubble constant—the expansion rate of the universe. This is TCR's single free parameter. Once H₀ is specified, all other scales (Planck mass, particle masses, cosmological constant) are determined.
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Maximum Entropy
Physical postulate: nature selects the configuration with maximum uncertainty (entropy) compatible with constraints. This information-theoretic principle replaces Einstein's geometric assumptions about spacetime.
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Heat Kernel
a₁ = R/6
A mathematical tool connecting geometry to physics. The heat kernel's expansion coefficients encode curvature. The a₁ = R/6 term directly links spectral data to the Ricci scalar, enabling gravity to emerge.
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Einstein Equations
derived
General relativity's field equations Gμν = 8πTμν are derived, not assumed. Maximizing entropy on the coherence space yields Einstein's equations—gravity is an entropic force, not fundamental.
Cosmology
Fermionic Suppression
Physical postulate: at cosmological scales, fermions (quarks, electrons) don't contribute to vacuum energy because their masses far exceed the Hubble scale. Only massless or very light bosons contribute.
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Dark Energy
Ω_Λ = 0.690
TCR predicts Ω_Λ = 13/(6π) = 0.690 from counting 13 Standard Model bosons. The observed value is 0.68±0.01. This is exact agreement within experimental error—no fine-tuning required.
Quantitative Predictions
Cabibbo Angle
θ = π/14
The mixing angle between quark generations. TCR predicts θ = π/14 from the geometry of G₂/SU(3). This gives sin(θ) = 0.2225 vs. observed 0.2253—agreement to 1.3%.
Top Quark Mass
m = v/√2
The top quark is predicted to have Yukawa coupling exactly 1, giving mass m = v/√2 ≈ 174 GeV. This follows from the top saturating a bound in the Jordan algebra. Observed: 173 GeV (0.6% agreement).
Spacetime Dimension
d = 4
Why 4 dimensions? TCR shows d=4 is the unique choice where both stable atoms can exist AND gravity propagates as waves. Other dimensions fail one or both requirements.
Mass Hierarchy
m_p/m_e ~ e^7.5
Why is the proton 1836× heavier than the electron? TCR derives m_p/m_e ≈ e^7.5 ≈ 1808 from the exponential structure of Yukawa couplings. Observed: 1836 (1.5% agreement).
Discrete Spacetime
A_min = (π−3)ℏG
Spacetime is not continuous—area is quantized in units of (π−3)ℏG ≈ 0.14 ℓ_P². This "quantum foam" emerges from hexagonal tiling geometry. TCR predicts 50× finer quantization than Loop Quantum Gravity.