Paired readout rank, refinement dimension and clock conversion are separate measurements. Their values can agree in a calibrated example and differ when the archive, resource cost or material clock changes.
Compare the same archive with two clocks
At role 6, paired rank is 3. B=8 and q=2 give refinement exponent D=3. Four steps produce 4,096 records at relative scale 1/16. The calibrated activities give u/τ=3. The protected clock gives u/Tₚ=3, while the ungated clock gives u/Tₐ=3/4 on exactly the same archive. Agreement with the protected clock assumes r=η=1 and h=0.
- Paired readout rank
- 3independent linear directions
- Refinement exponent D
- 3log B / log q; dimensionless
- Protected conversion u/Tₚ
- 3dimensionless activity ratio
Selected counter prefix: 64 addresses; 27 distinct sites.
Refinement archive: 4,096 records; ℓₙ/L = 0.0625.
| Quantity | Value | Units |
|---|---|---|
| Branch activity u | 8.317766 | reference activity units |
| Resolution activity τ | 2.772589 | reference activity units |
| Extra completion activity hτ | 0 | reference activity units |
| Protected reading Tₚ=(1+h)τ | 2.772589 | reference activity units |
| Ungated reading Tₐ=u+(1+h)τ | 11.09035 | reference activity units |
| u/τ | 3 | dimensionless |
| u/Tₐ | 0.75 | dimensionless |
| Inner role a | Rank | Addresses pᵃ | Sites |
|---|---|---|---|
| 1 | 1 | 2 | 2 |
| 2 | 1 | 4 | 3 |
| 3 | 2 | 8 | 6 |
| 4 | 2 | 16 | 9 |
| 5 | 3 | 32 | 18 |
| 6 (selected) | 3 | 64 | 27 |
Neither a rank-three readout nor a refinement exponent of three proves a spatial manifold or a universal clock law. All controls preserve this distinction.
1. A paired prefix gains rank every other role
Choose independent frame vectors e₁,e₂,e₃ and a positive length ℓ. The paired map Q=ℓ[(d₁−d₂)e₁+(d₃−d₄)e₂+(d₅−d₆)e₃] turns six radix-p digits into three coordinates. In the all-zero-initialized prefix, the first a digits vary and the others stay zero. Its affine rank is ceiling(a/2): 1,1,2,2,3,3.
sites=(2p−1)ᵐ for a=2m;
sites=p(2p−1)ᵐ for a=2m+1.
The seventh completion role is not another coordinate or an extra digit. Increasing p changes the address and site counts without changing the rank. The chosen frame and pairing remain inputs.
2. Refine counts and scale independently
At each completed step, take B children per record and contract a comparison scale by 1/q. After n steps, Nₙ=Bⁿ and ℓₙ=Lq⁻ⁿ. Eliminating positive n yields D=log B/log q. This q is a contraction factor; it is unrelated to the continuation count in the response workbench.
For the infinite address space with the specified common-prefix ultrametric, matching cover and measure bounds make D a Hausdorff dimension. A finite archive alone has Hausdorff dimension zero; its displayed D refers to the refinement family or a declared scaling window. At n=0 there is one record and scale L, and no finite-depth ratio can be read from 0/0.
Keeping rank three while choosing B=4,q=2 gives D=2. At B=6,q=2, D≈2.584963. The readout rank and refinement exponent are independent unless a further theorem connects them.
3. Make the common-resource premises visible
The calibrated encoder in the derivation supplies u=a*n log B and τ=a*n log q. Its contacts and common activity unit must be independently justified. The controls r, η and h introduce a deliberately simple resource-cost variant to expose those premises:
Tₚ=(1+h)τ; Tₐ=u+(1+h)τ.
Here a*=1 reference activity unit. Common calibration means r=1. Saturated branch-resource use means η=1: no additional cost beyond the stipulated logarithmic branch work. The overhead h measures separately counted serial completion work in units of τ. These variations are explicit bookkeeping assumptions, not additional source-derived physical laws.
For positive n, u/τ=(r/η)D and u/Tₚ=rD/[η(1+h)]. Only with the declared common calibration, saturated cost, zero overhead and protected material response does the model establish the intended equality with D. Other parameter choices can accidentally cancel; agreement at one setting is not a universality theorem.
4. Keep the geometry and change the clock
At the baseline B=8,q=2,n=4, there are 4,096 refined records at L/16. The branch activity is 4 log 8 and resolution activity is 4 log 2. The protected clock reads τ, so u/Tₚ=3. The ungated clock reads u+τ, so u/Tₐ=3/4. Both use the same archive and the same branch evolution.
With common resource assumptions at arbitrary D, the ungated factor is D/(D+1), not D. Adding h=1 while preserving the baseline archive changes the protected factor to 3/2 and the ungated factor to 3/5. The six nested roles alone select neither clock coupling nor completion cost.
A material band evolving with branch activity acquires the appropriate u/T conversion only after its physical clock is specified. Its group velocity also need not equal a universal causal-cone slope. Test geometry, activity, clock and propagation separately while keeping calibration fixed across settings.
Check the exact arguments
- The paired spatial rank sequence
Exact rank and site counts, restricted channel bounds and the assumptions they need.
- Refinement dimension and the clock ratio
The address-space dimension proof and independently calibrated encoder.
- Calibrate which operations a clock ignores
Protected material readouts and delayed tests for hidden coupling.
- Why one clock model does not prove universality
The ungated-clock countermodel and working-draft source references.
- Refinement, dimension and clock comparison
Why equal numerical diagnostics need a common-source explanation.