Build a record.
Examine its geometry.

Move from nested roles to spatial addresses, then compare readout rank, a refinement exponent and calibrated clock ratios without identifying them by definition.

Conditional constructions with independently defined observables

Six nested roles and outer completion

The six inner roles organize a proposed construction; outer completion publishes its record. The nesting diagram does not represent seven independent observable time axes.

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Selected role 7

Counter radix q 2

Use the sliders with a pointer, or focus a slider and use the arrow keys. Home and End select its limits. All values are model inputs.

Role 7: 6 inner roles included; affine readout rank 3; 64 addresses in this radix-2 block. The seventh role publishes the completed six-role archive.

Exact rank and address counts for a supplied six-digit paired counter. The widget does not derive a physical spatial frame, clock duration, or necessity of six roles.

Try this investigationMove from the sixth role to outer completion. Separate the completed record from the operations that constructed it.

Many addresses, fewer spatial sites

The declared paired binary readout maps 64 addresses to 27 sites. Equal address weights give a nonuniform distribution over those sites.

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Source address n 0

Use the sliders with a pointer, or focus a slider and use the arrow keys. Home and End select its limits. All values are model inputs.

Address 0, digits d₁…d₆ = 0 0 0 0 0 0 → (0, 0, 0). This site has 8 of the 64 addresses: probability 12.5% under uniform address sampling. Total: 64 addresses, 27 distinct sites.

Exact multiplicity classes
Site class	Sites	Addresses each
Centre	1	8
Face centres	6	4
Edge centres	12	2
Corners	8	1

Exact binary enumeration and multiplicities for the stated paired map. Probabilities assume uniform address sampling, and the counter scan is not identified with physical transport.

Try this investigationCompare the centre, a face centre and a corner. Count how many addresses map to each before assigning a probability.

Compare geometry and clock readings

Vary the branching, contraction, resource usage and alternative clock. The comparison makes the extra conditions behind numerical agreement visible.

Readout and refinement

Inner role aOne of six paired-channel prefixes; outer completion is a separate role.

Counter radix pValues per internal digit. p is separate from scale contraction q.

Children per record BDistinguishable children at every refinement step.

Scale contraction qEach step changes length by 1/q. Dimensionless and greater than one.

Refinement steps nCompleted refinements; zero gives no measured clock ratio.

Explicit resource and clock assumptions

Relative branch calibration rBranch/resolution calibration ratio; r=1 means the same encoder.

Branch-resource efficiency ηη=1 saturates the stipulated log-cost relation; smaller η adds branch work.

Extra completion overhead hAdditional clock activity hτ; h=0 assumes no extra serial completion cost.

Inputs are dimensionless model choices. Activities use a fixed common reference unit with a*=1. Use Tab, type values and press Enter or choose Update results. Changing a field also updates after you leave it.

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.

Same archive and matter activity, different clock readings
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

Paired readout progression at the selected radix
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.

Try this investigationKeep the same spatial archive but change the alternative clock. Check whether numerical agreement with the refinement exponent survives.

The joint construction remains the target

A useful next model must produce the spatial records, clock readings and propagation law through the same specified dynamics. Agreement at one calibration point is not a universal derivation.

Six nested roles and outer completion

Many addresses, fewer spatial sites

Compare geometry and clock readings

The joint construction remains the target

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