Gravity from Description-Length Minimization
General relativity is not a postulate — it is the unique gravitational theory that minimizes the description length of the CMCA tape. Einstein's equations emerge as a theorem about information, and Newton's constant is derived from group theory with zero free parameters.
The Problem: Gravity is Different
The Standard Model describes three forces — electromagnetism, the strong force, and the weak force — using quantum field theory (QFT). Each force has a force-carrying particle (photon, gluons, W/Z bosons), and the framework is extraordinarily accurate. Predictions match measurements to twelve decimal places.
Gravity refuses to fit this pattern. Our best description of gravity is general relativity (GR): massive objects curve spacetime, and objects follow the straightest possible paths through that curved geometry. GR is also highly accurate — it correctly predicted gravitational waves, black hole shadows, and GPS clock corrections.
The incompatibility: When physicists try to combine GR and QFT at extreme energies (the Planck scale, ~1019 GeV), the equations produce uncontrollable infinities. Every other force's infinities can be tamed by renormalization; gravity's cannot. After a century of effort, no quantum theory of gravity agrees with experiment.
String Theory
Replace point particles with tiny vibrating strings. One vibration mode automatically behaves like a graviton — the hypothetical force carrier of gravity. This works mathematically, but only in ten or eleven dimensions: six or seven extra dimensions must be hidden away. There is also a landscape of ~10500 possible universes with no principle selecting ours.
Status: No experimental prediction distinguishable from the Standard Model has been confirmed. The Standard Model particle spectrum is not derived — it is reproduced by tuning the compactification geometry.
Loop Quantum Gravity
Quantize spacetime itself: space is made of discrete chunks of area called spin foam. This avoids extra dimensions, but has difficulty recovering standard QFT at low energies and has not derived the Standard Model. General relativity is taken as input — Einstein's equations are not derived, just quantized from outside.
Status: No particle masses, mixing angles, or coupling constants are predicted. The 25 free parameters of the Standard Model remain unexplained.
The GTE Route: Derive GR from Below
Instead of quantizing GR from outside, the GTE framework derives GR from a deeper discrete structure. Starting from the CMCA tape (a cellular automaton on three tapes sharing a single causal clock), the Einstein field equations emerge as a theorem about information minimization. No quantization from outside is needed — gravity and quantum mechanics share the same substrate.
Status: Einstein's equation derived (CatAD). Geodesic theorem machine-certified (CatAL). Newton's constant predicted to 0.04% accuracy (CatA). Zero free parameters.
The CMCA Three-Tape Geometry
The starting point is the three-tape CMCA (Causal Minimum Complexity Automaton) introduced in Path 2 (L13). The three tapes share a single causal clock τc. This shared clock is the key to understanding how spacetime emerges.
Each tape evolves under the MDL-optimal polynomial
p(L,C,R) = C + R − CR − LCR. The three tapes together carry the
full ΦMDL field — the master scalar field whose kinks are particles
(Path 3). The shared clock τc sets the rate at which information
flows across tapes.
The Dimensional Protocol Principle (DPP): Three 1+1D tapes
sharing a single causal clock τc produce exactly 3+1D spacetime —
three spatial directions from the tapes plus one time direction from τc.
This is not a postulate: it is a theorem of the three-tape geometry, machine-certified
at CatAL (cmca_three_axes_give_31d). The 3+1D signature of spacetime
is forced by the CMCA architecture.
Why a shared clock matters for gravity: When matter is present, the ΦMDL field changes rapidly in space near the matter kink. More MDL bits are required to specify the field configuration — the information density is higher. This elevated information cost is reflected in τc running faster near matter. The gradient of τc is exactly what Newtonian gravity calls a gravitational potential.
The field is slowly varying. Few MDL bits needed per spatial position. τc runs at its background rate. Spacetime is flat.
The field changes rapidly (large gradient). More MDL bits needed. τc is elevated. Spacetime is curved — clocks run faster at higher altitude, slower near massive objects.
The MDL-Lovelock Principle
The motion of every object in GTE is governed by an action — a number assigned to each possible trajectory, with the realized trajectory being the one that minimizes the action. In GTE the relevant action is the PMDL action (Physical Minimum Description Length action).
Why this particular action and not some other? This is where Lovelock's theorem enters. Lovelock (1971) proved that the only gravitational field equations in 3+1 dimensions that are: (a) derived from a covariant action, and (b) second-order in the metric, are Einstein's equations with a cosmological constant.
The MDL-Lovelock correspondence: MDL minimality independently selects the simplest action consistent with symmetry requirements. The remarkable fact is that Lovelock's physical uniqueness requirement and MDL's computational uniqueness requirement identify the same object — the Einstein-Hilbert action. This bridge makes Einstein's equation not a postulate but a theorem.
Step 1: Symmetry requirements
The CMCA tape obeys general covariance — no special direction in spacetime is singled out. Any valid description of the tape dynamics must respect this. This constrains which terms can appear in the action.
Step 2: MDL minimum
Among all covariant actions, MDL selects the one with the shortest description — the minimal number of independent terms consistent with the symmetry. The Ricci scalar R is the simplest covariant scalar built from the metric and its derivatives.
Step 3: Lovelock's theorem
Lovelock (1971) proved: in 3+1 dimensions, the only second-order covariant equations of motion derivable from a scalar action functional of the metric are the Einstein equations with a cosmological constant. MDL selects exactly the functional of this type with minimal complexity.
Step 4: Einstein's equations emerge
Varying the PMDL action with respect to the metric yields Gμν + Λgμν = 8πGN Tμν. This is Einstein's field equation — derived, not postulated. The cosmological constant Λ is also derived (Path 7).
Geodesics as a Theorem
In standard general relativity, there are two independent postulates: (1) the Einstein field equations, and (2) free particles follow geodesics. The GTE framework replaces both with theorems derived from the PMDL action.
Follow timelike geodesics — the paths that maximize proper time.
Follow null geodesics — paths of zero proper time.
No local experiment distinguishes free fall from inertial motion.
Deriving Newton's Constant from Group Theory
Newton's constant GN is the most precisely measured and least understood constant in physics. Its value sets the overall scale of gravity. In the Standard Model and in string theory and loop quantum gravity, it is an unexplained measured input. In GTE it is derived from the orbit combinatorics of a single group.
The F21 group
The Frobenius group F21 = ℤ7 ⋊ ℤ3 acts on the 343 states of the ℤ73 neighbourhood space. By Burnside's lemma, this action partitions those 343 states into 17 orbits: 10 "all-distinct" (generic) and 7 degenerate.
| Quantity | GTE prediction | Measured value | Error |
|---|---|---|---|
| MPl / mτ | 6.8683 × 1018 | 6.871 × 1018 | 0.040% |
| GN | derived from above | 6.674 × 10−11 m³ kg⁻¹ s⁻² | 0.040% |
Every quantity in this formula — 21, 7, 10, 7 — is independently derived from the arithmetic of the GTE polynomial. There are no free parameters and no fitting to experimental data.
The Hierarchy Problem Dissolved
The hierarchy problem asks: why is gravity 1036 times weaker than the electromagnetic force? In the Standard Model this is a deep mystery — quantum corrections to the Higgs mass should drag it up to the Planck scale, requiring a fine-tuning of 1017 decimal places to cancel.
GTE resolution: In GTE there is no hierarchy problem because there is no free parameter to fine-tune. The ratio MPl/mτ = 2110 × 77/2 is a group-theoretic fact about how many independent orbit types exist in the ℤ73 neighbourhood space. Gravity appears weak because GN has a specific arithmetic value from the orbit structure — not because two large numbers happen to cancel. There is no coincidence to explain; the value is derived.
More precisely: the weakness of gravity reflects the richness of the orbit structure. Ten independent orbit types means 2110 ≈ 1.67 × 1013 configurations per group element — a large number that propagates into GN making gravity appear weak relative to forces that don't carry this factor.
Key Results
- Gravity is not the fourth fundamental force in the usual sense — it is emergent from the CMCA tape geometry and the MDL action principle.
- The PMDL action is the unique MDL-minimal covariant action for ΦMDL on a curved background. Lovelock's theorem identifies this with the Einstein-Hilbert action (CatAD).
- Einstein's field equations Gμν = 8πGN Tμν are derived from MDL minimization — "curvature = information density of matter."
- Geodesic motion is a theorem about computational economy, not a postulate. Both timelike and null geodesics are machine-certified (CatAL).
- Newton's constant GN is derived from the F21 orbit combinatorics: MPl/mτ = 2110 × 77/2, agreeing with measurement to 0.04% with zero free parameters.
- The hierarchy problem is dissolved: the weakness of gravity is an arithmetic fact about orbit counting, not a fine-tuning coincidence.
Source material
- Path 6 — Quantum Mechanics: transputation and the Born rule
- Path 7 — Cosmology: the cosmological constant and hierarchy problem in full
Path 5 Complete
You've seen how general relativity emerges from description-length minimization on the CMCA tape — and how Newton's constant is a group-theoretic fact, not a free parameter.