Relativity
Relativity as Fisher geometry of the sieve
PT does not paste Einstein onto the sieve. It attempts to show that the spacetime metric is the natural information geometry of the sieve, and that Lorentzian signature, SO(3,1), and Einstein equations follow from that reading.
Fisher → Bianchi I → SO(3,1) → Einstein
Plain language
Einstein from an information map
Classical general relativity begins with a spacetime metric: a rule saying how to measure distances and durations. Only then does it write how this metric curves under matter and energy.
PT reverses the question. It does not first ask “which space does the sieve live in?”. It asks: “which notion of distance is forced by what the sieve can distinguish?”. That natural distance is the Fisher metric.
Fisher can be understood as a statistical microscope rule. If two distributions look almost the same, many observations are needed to distinguish them: they are close. If a small displacement strongly changes what is observed, they are far. The Fisher metric assigns a number to that sensitivity.
A simple metaphor: imagine a city whose map is unknown, but whose most stable routes between districts are known. By comparing those routes, one can reconstruct a map. PT does something analogous: it reconstructs geometry from persistent distinctions of the sieve.
Fisher therefore measures an information distance. Two states are close if their distributions are hard to distinguish; they are far if a small change makes the distribution clearly different. This notion of distance is not decorative: Cencov uniqueness forces it for statistical families.
Bianchi I is then the geometric name for a very simple map: three directions of space may stretch with three different scale factors, without mixing the axes. That is exactly the right language for reading the active primes 3, 5, and 7 as three directions before isotropic averaging.
The move to Einstein happens once this map carries curvature and conservation. In relativity, curvature is not decoration: it says how rules of measurement change. In PT, those rules come from Fisher; in Einstein, they become gravitational dynamics.
The message is this: spacetime is not a primitive container added to the sieve. It is the continuous reading of what the sieve makes distinguishable. Einstein then appears as the geometric grammar of that map, not as a theory glued on afterward.
Distance
Before meters or seconds, PT measures distinguishability between internal states.
Curvature
If the rule of distance changes from point to point, the map curves: this is the entry into gravity.
Time
When the μ direction receives the sign opposite to spatial directions, it reads as proper duration.
Standard
Physical architecture: Fisher, Bianchi, Einstein
The monograph follows the chain: s = 1/2 → sin²θₚ, γₚ → gᶠμν → Bianchi I → SO(3,1) → Gμν. Each step turns sieve information into geometric structure.
The active primes 3, 5, and 7 provide three spatial scale factors. The depth coordinate μ provides the time candidate, but only after the sign switch of g₀₀.
One obtains a 3+1 form without placing it as an initial backdrop: three spatial directions from the active sector, and one temporal direction from persistence curvature.
PT relativity is therefore not merely “compatible with Einstein”. It tries to explain why Lorentzian geometry and Einstein-type equations are the right structures when persistent information is read as distance, curvature, and energy.
s = 1/2 → Fisher → Bianchi I → SO(3,1) → Gμν
Standard
From Fisher to Bianchi I
The graph below replaces a purely illustrative drawing. It plots the Bianchi I scale factors computed from the chapter 13 formula: aₚ(μ)=γₚ(μ)/μ for p=3,5,7.
This graph does not by itself prove the Fisher → Bianchi I theorem. The proof comes from Fisher uniqueness, CRT factorisation, and diagonality of the active sector. The graph shows the computed form produced by that proof once projected onto the three active primes.
Depth μ is not automatically time. It becomes timelike when the g₀₀ component changes sign. This is the link with the Time page: the three spatial curves below must be read together with the g₀₀(μ) curve that fixes the signature.
In this reading, special relativity appears locally through SO(3,1), while general relativity appears through the Einstein tensor associated with the Fisher-Bianchi metric.
3 directions
The active primes 3, 5, and 7 play the role of the three scale directions of the Bianchi I metric.
1 duration
The μ direction becomes proper duration via g₀₀ < 0 and dτ = √|g₀₀| dμ.
Einstein
Gμν appears through the Hessian/cumulant route and through the Lovelock lock.
Technical
Technical demonstration: Fisher, Einstein, and Lovelock routes
Chapter 13 presents Lorentzian signature as an algebraic result, SO(3,1) through structural tests, and Einstein tensor components checked on the Bianchi I metric. The global status is mixed: THM/DER/VAL by step, not one undifferentiated block.
The sensitive step is the bridge “Fisher metric = spacetime metric”. The monograph supports it through Lemma F, Cencov uniqueness, and informational monotonicity: an admissible metric must not artificially create distinguishability.
Two routes overlap. The Hessian route reads gab = ∂a∂b ln Z and turns Einstein equations into a cumulant identity. The Lovelock route locks the form: in 3+1 dimensions, with covariance and conservation, the admissible tensor is essentially Einstein’s tensor.
statistical family
The sieve defines distributions over residue classes, indexed by depth μ.
Cencov
The natural monotone metric on these families is Fisher: gᶠ measures distinguishability.
CRT
CRT factorisation separates prime factors and diagonalises the active sector.
Bianchi I
On p∈{3,5,7}, scale factors take the form aₚ = γₚ/μ.
signature
The g₀₀ component becomes negative in the physical regime: the metric is Lorentzian.
SO(3,1)
The isometry algebra of the active sector gives local relativistic symmetry.
Einstein
For ds² = -N²dμ² + Σ aₚ²dxₚ², one obtains in particular G₀₀ = H₃H₅ + H₃H₇ + H₅H₇.
Lovelock
Covariance and conservation in 3+1 lock the form of the dynamical tensor.
THM
Lorentzian signature and SO(3,1) structure belong to the strong reconstructions of chapter 13.
BRIDGE/DER
The Fisher-spacetime identification is the conceptual bridge that must remain explicit.
VAL
Component computations and companion scripts support numerical coherence of the sector.
Status
Epistemic status
This page presents a reconstruction, not a loose analogy. The mathematical core is Fisher geometry; the identification with spacetime is a strong physical bridge; relativistic equations are derived and checked in this frame, with statuses kept separate.