Book
pages exist together, but reading gives them an internal order.
Time
Time as the measurable signature of persistence
In PT, the sieve does not unfold inside an already-given time. It first carries a hierarchy of distinctions; proper time appears when Fisher geometry gives the depth μ a signature opposite to spatial directions.
Visual reading of chapter 13
logical order 2 → 3 · 5 · 7 hierarchy, not chronology
metric switch μc ≈ 6.97 g₀₀ : + → −
proper time τ = ∫√|g₀₀|dμ physical invariant
Plain language
How to read this page
Our intuition imagines time as a stage: the universe is placed on it, then events arrive one after another. PT reverses the image. It begins with a structure of constraints, then asks when that structure becomes readable as duration.
A useful metaphor is a closed book. The pages exist together, but reading orders them. Reading order does not create the pages; it only makes an already-given structure readable.
PT cosmogony works in that way: sieve thresholds are not dates. They are dependencies. Time appears only when this hierarchy receives a geometry able to turn reading depth into proper duration.
Score
notes are written; performance turns structure into lived duration.
Map
a path becomes measurable only once a scale converts steps.
Plain language
Time as reading, not as scenery
The first difficulty is to leave the familiar image of scenery. We tend to imagine time as a large horizontal ruler on which the universe is placed. In that image, there is a “before”, then a “during”, then an “after”. PT instead asks how that ruler can appear.
In the PT reading, the sieve is first a structure given by constraints. One can read it layer after layer, but that reading is not yet physical chronology. It is like a musical score: all notes are on the page, yet music exists as duration only when an oriented performance crosses the score.
PT cosmogony is understood in that way: the thresholds 2, 3, 5, 7, then the echoes, are not instants lined up on an external clock. They are conditions of readability. Some distinctions must be available before others can make sense.
The crucial point is therefore: an order is not yet a clock. A cooking recipe has an order, a mathematical proof has an order, a sentence has an order; but that order alone does not produce measurable duration. For duration to appear, an internal rule of measurement is needed.
That is where the metric enters. It does not merely say “in what order to read”; it says how much a small reading step is worth. When this metric gives the depth μ the sign opposite to spatial distances, that step no longer reads as one more length. It reads as duration.
In simple terms: the sieve does not unfold in time. Time appears when the internal reading of the sieve receives a geometry that allows an internal clock to exist.
Logical order
dependencies are already oriented, but no physical duration is measured yet.
Internal scale
the metric gives weight to small reading steps in depth μ.
Timelike sign
the sign opposite to spatial lengths prevents μ from being merely a fourth distance.
Standard
From logical order to proper time
A logical order is not enough to make a clock. To obtain physical time, one needs a metric: a rule saying how to measure a small difference between two internal states.
The Fisher metric measures exactly this distinguishability. If a small change in μ strongly changes the sieve distribution, Fisher distance is large; if it changes it only slightly, distance is small.
When the g₀₀ component becomes negative, μ stops being a bare depth. It receives the timelike sign: it no longer adds like a spatial length; it defines proper duration.
In other words, PT separates three levels: dependency order, Fisher measurement of that order, then the signature that allows one direction to be called “time”. The diagram summarises that promotion, not a cosmic chronology.
dependency order → Fisher measures → g₀₀ < 0 → dτ = √|g₀₀| dμ
Standard
Where the time direction appears
The curve below is no longer a metaphorical sketch: it is computed from the chapter 13 formula on the active sector {3,5,7}.
The first thing to read is the sign. Before μc ≈ 6.97, the μ direction does not yet carry the timelike sign. After that threshold, g₀₀ < 0: depth can be converted into proper time.
μ* = 15 already lies in that Lorentzian regime. This is why cosmogony can say measurable time appears just before physical stabilisation of the cascade.
Standard
Why the opposite sign changes everything
If μ had the same sign as spatial directions, it would merely be a fourth length. It could be added to other distances, like one more direction of travel.
With the opposite sign, the situation changes. The temporal contribution can compensate a spatial contribution in the total interval. This is the minimal relativistic logic: the direction no longer describes another place, but the internal rhythm separating two states.
The small demonstrator does not simulate PT. It only shows this mechanism: I = dx² − |g₀₀|dμ². When the temporal term dominates, separation reads as duration.
Spatial term 1.00
Temporal term 1.00
Interval 0.00 null boundary
I = dx² − |g₀₀|dμ²
Technical
Technical demonstration
Chapter 13 does not merely say “let μ be time”. It treats the identification μ ↔ τ structurally: the spacetime metric is read as a spectral invariant of sieve transfer matrices, and μ is the unique coordinate producing the temporal component.
By Cencov uniqueness, Fisher is the natural monotone metric on statistical families: it measures distinguishability without artificially creating information. In PT, two nearby depths μ are close or far according to the actual deformation of the sieve distribution.
On the active sector p∈{3,5,7}, CRT decomposition provides the three spatial blocks. A depth coordinate μ remains, whose curvature is carried by the persistence potential S(μ) = −ln αEM(μ).
The sign of that curvature fixes the signature. When g₀₀ < 0, the line element becomes Lorentzian, ds² = −|g₀₀|dμ² + Σp ap²dxp². The dμ² term is no longer a fourth length: it defines proper duration.
The essential distinction is therefore μ versus τ. μ is a reading coordinate, analogous to sieve depth; τ is the scalar measurable by an internal clock: τ = ∫√|g₀₀|dμ.
Time Rigidity then shows that physical content does not depend on the chosen coordinate name. For every strictly monotone reparametrisation μ̃=f(μ), the product √|g₀₀|dμ remains invariant, so observables written in terms of τ are unchanged.
statistical family
Start from the family of sieve distributions indexed by depth μ.
Cencov / Fisher
The monotone metric compatible with information-non-creating maps is Fisher.
active sector
On p∈{3,5,7}, CRT diagonalises spatial blocks and leaves μ as the scale coordinate.
potential
S(μ)=−ln αEM(μ) encodes persistence sensitivity along depth.
curvature
The time component is read as g₀₀(μ)=−d²lnαEM/dμ² on the active sector.
switch
The sign switch occurs at μc≈6.97; for μ>μc, signature is Lorentzian.
proper time
Measurable time is τ=∫√|g₀₀|dμ, not the bare parameter μ.
rigidity
Under μ̃=f(μ), √|g₀₀|dμ remains invariant: H₀, ΩΛ, q₀ and derived equations are read in τ.
Invariants not to lose
Fisher measures local distinguishability of sieve distributions.
The curvature of S carries the temporal component.
The sign of g₀₀ decides whether μ remains a scale or becomes timelike.
The opposite sign separates duration from distance.
τ is the observable scalar, not the bare graduation μ.
The product √|g₀₀|dμ is invariant.
Status
Epistemic status
The page does not claim to directly measure a primordial clock. It lays out the PT chain: sieve order → Fisher geometry → sign change of g₀₀ → invariant proper time. The strong status concerns the metric structure; the cosmological interpretation remains a physical reading of that structure.