The chosen logarithmic encoders give D=log B/log(1/r)=u/τ. Replacing the clock τ by T=τ+βu leaves the refinement unchanged but changes the speed to v_gD/(1+βD). A calibration at one D does not remove that difference for a whole family.
Explore the declared model
D = 3; u/τ = 3 by the chosen log encoders. Matched-clock speed = 3; mixed-clock speed = 0.75. After one calibration at D = 2, the speed ratio is 0.75. Geometry alone has not selected either clock.
The dark curve is the calibrated speed ratio. The dashed line is equality; the gold point is the current setting. The reference calibration is fixed at D=2.
| Quantity | Value in model units |
|---|---|
| Address count Nₙ | 512 |
| Cell scale ℓₙ/L | 0.125 |
| Count increment u | 6.238325 |
| Scale increment τ | 2.079442 |
| Mixed clock T = τ + βu | 8.317766 |
| Transported distance / chosen length unit | 6.238325 |
| Matched-clock speed | 3 |
| Mixed-clock speed | 0.75 |
| Mixed-clock speed after calibration at D = 2 | 2.25 |
Exact comparison of supplied refinement, logarithmic encoders and two positive clock laws. It demonstrates a clock-selection freedom; it does not establish a physical spacetime dimension, universal limiting speed or material clock mechanism.
Define the geometric encoder
Choose an integer branching factor B, a contraction 0<r<1 and a positive integer depth n. There are Nₙ=Bⁿ labelled addresses, and every cell has assigned scale ℓₙ=Lrⁿ. This widget uses L=1 chosen length unit for the cell-scale readout. Counting addresses and assigning their scale are separate operations before a ratio is formed.
The depth cancels exactly. D is the scaling exponent of this address-and-scale construction. Identifying it with a Hausdorff dimension needs an appropriate metric realization, separation and regularity assumptions. In a proposed three-dimensional realization, even the usual packing restrictions are additional constraints. The slider explores the mathematical encoder, including settings that are not asserted to describe physical space.
Specify two readable clock observables
Use the dimensionless increments u=n log B and τ=n log(1/r), measured in the same declared logarithmic convention. The matched clock reads τ. A second allowed clock reads T=τ+βu for a nonnegative mixing β. Both increase with depth for fixed B and r, so both consistently order completed refinement episodes.
The first equality follows from the chosen encoder definitions. It does not derive the reading of an independently specified material clock. Adding βu is a concrete alternative clock law: it can be varied without changing Nₙ or ℓₙ. When β=0 both clocks coincide. When β is positive they differ by a factor that depends on the setting through D.
Read propagation with the selected clock
Supply a transport rule assigning displacement Δx=v_g u. The coefficient v_g converts one u unit to a chosen length unit. The displayed displacement is transport along that rule, not the shrinking cell size ℓₙ. These are different observables and should not be identified merely because both involve length.
vmixed=Δx/T=v_gD/(1+βD)
The speed columns use the chosen length and clock units. They do not display metres per second or a measured value of c. Increasing n multiplies displacement and both readings by the same factor, leaving the speed ratios unchanged. Varying v_g rescales both speeds, while changing β alters their comparison. This separates a transport normalization from the clock-selection question.
Test one calibration across multiple settings
At one reference exponent D₀=2, the mixed clock can be calibrated to agree with the matched clock by dividing T by 1+2β. Keep that calibration fixed when moving to another setting. The ratio of the resulting measured speed to the matched-clock speed is:
The graph plots this exact function. It crosses one at D=2, and is identically one only when β=0. For β>0 its derivative is −β(1+2β)/(1+βD)², which is strictly negative. Thus one change of units cannot make the two devices agree over a family with varying D. The mismatch is not removed by recalibrating every individual point, because that would abandon the common calibration being tested.
Compare the default and a reference setting
At B=8, r=1/2 and n=3, the encoder has Nₙ=512, ℓₙ/L=1/8 and D=3. With β=1 and v_g=1, the matched speed is three and the mixed speed is 3/4. After the fixed D₀=2 calibration, the mixed speed is 9/4, so R=3/4. The table also exposes the actual u and τ increments used in those divisions.
Change B to four while retaining r=1/2. Now D=2, and the reference calibration makes the speeds agree: R=1. Restore B=8 and set β=0: the clocks agree again, now for a different reason—their observable laws are identical. These exact comparisons distinguish agreement at one reference point from a restriction that enforces agreement throughout the family.
What a joint SMM/DTT derivation must add
The six nested inner roles may organize the dependencies that generate records, while outer completion makes the result available. To obtain one physical geometry–clock–propagation law, the same source dynamics must also constrain which clock observables are admissible and how material transport couples to them. Simply setting β to zero defines the matched family; deriving that selection requires a separate argument.
This workbench is a precise countermodel and comparison tool. It preserves the supplied geometry while varying an observable that geometry alone has not selected. A stronger source construction should show why the mixed alternative is forbidden, suppressed with a controlled bound, or physically equivalent under one setting-independent calibration. The new source chapter and the clock-validation guide explain how to formulate that question without adding hidden clocks or renaming independent assumptions.
Continue with the articles
- From available records to a temporal arrow
Completed returns that preserve or increase accessible records form a positive monoid. Under additional separation and clock conditions, this supplies an effective forward order without proving thermodynamic irreversibility.
- A phase-sensitive registration experiment
Registering a completion flag before or after further source evolution changes later phase-sensitive acceptance statistics. The proposed protocol measures that order effect and compares it with an ordinary qubit baseline.
- A record that carries its own completion certificate
An autonomous history Hamiltonian can generate both a data record and a flag tied to its completed write. The exact construction retains incomplete outcomes, coherent intermediate stages, and finite observation windows.