The phrase “six inner levels of time” is easy to picture incorrectly. In the SMM/DTT research programme, the levels describe nested roles in forming an observable configuration. A later role depends on work performed within earlier ones. The outer role makes the completed result available to observation. A small counter model shows how this organization can be made explicit—and where its physical explanation still needs to be developed.
What makes a level nested?
Start with an ordinary distinction: a process can perform substantial internal work before it produces a new accessible record. The order of that work and the order of completed observations answer different questions. DTT uses inner generation and outer completion to organize this distinction within a proposed account of physical formation.
A nested level adds a dependency. Completing one small cycle enables an operation at a wider level; completing that wider cycle enables the next. Calling six unrelated procedures from the same controller would give a sequence, but common scheduling would not explain why their physical outputs agree.
The levels also need not be separate coordinates. A state can contain several internal variables without providing that many observable directions. A readout may combine variables, discard distinctions or retain information only after a completion condition is satisfied. To see what the terms mean, the update and readout must be specified separately.
One update rule, six dependent cycles
Consider six binary digits d₁, …, d₆, where each digit is zero or one. Encode them in a single integer:
n = d₁ + 2d₂ + 4d₃ + 8d₄ + 16d₅ + 32d₆.
Let one autonomous update advance n by one, modulo 64. The first digit changes at every update. The second changes when the first carries; the third changes when the first two carry together. Each higher digit depends on the completed lower cycles. The individual digit periods are 2, 4, 8, 16, 32 and 64 elementary updates.
This is one rule acting on one state. The six periods are consequences of its carry structure. They are not six frequencies that must be adjusted independently. Starting from zero, the first 2a addresses exhaust every assignment of the first a digits while the remaining digits stay zero.
The construction illustrates nesting exactly. It does not establish that nature implements a binary counter, or that its elementary advances are physical clock readings. Binary digits, six places and the carry rule are declared ingredients. Their value is that the dependency can be inspected directly.
How six internal variables can yield three directions
Now choose three linearly independent readout vectors e₁, e₂, e₃ and a positive scale ℓ. Pair the digits by the rule:
Q(d) = ℓ[(d₁ − d₂)e₁ + (d₃ − d₄)e₂ + (d₅ − d₆)e₃].
Varying the first digit opens one direction. Varying its partner changes the same direction. The third digit opens a second independent direction; the fourth remains in it. The fifth opens a third; the sixth again remains in the existing span. Across the six address prefixes, the affine ranks are therefore 1, 1, 2, 2, 3, 3.
The proof has two parts. At each prefix, all outputs lie in the span of the directions whose pairs have begun. Conversely, the prefix contains the zero address and an address with a single available odd-position digit equal to one. Their differences supply each of those independent directions. The upper and lower rank bounds agree.
There are 64 internal addresses, but each paired difference takes only three values: −1, 0 and 1. The full image therefore contains 27 distinct sites. Eight addresses map to the origin, while each corner has one representative. Uniformly sampling the internal addresses gives probability 1/8 at the origin and 1/64 at a given corner.
Address count, site count, rank and sampling probability are different quantities. The pair rule establishes the rank sequence within this architecture. It does not select orthogonality, equal physical lengths or a universal three-dimensional space. The detailed spatial-readout chapter examines these distinctions and the readout's sensitivity to errors.
What the outer record must retain
A completion rule could expose an archive only after the full 64-address sweep. That archive might retain the site set, multiplicities, a completion flag and any state needed for subsequent evolution. Its public availability then marks an outer event. In this example, the completion role is an interface on the completed work, rather than a seventh independent direction.
The archive needs enough information for its intended predictions. The current spatial coordinate alone can be insufficient. Addresses zero and three both map to the origin. Their successors are one and four, which map to ℓe₁ and ℓe₂ respectively. A deterministic next-coordinate rule cannot assign both successors to the same current coordinate unless additional state or a restricted preparation is supplied.
A complete sweep also returns the counter to its initial state. Repeating an identical sweep does not, by itself, distinguish one outer event from another. An event counter, changing retained data or some other history record is needed to make successive completions distinguishable. Such memory is part of the model's resources.
This is why completion must be operationally defined. A statement that the work is finished should specify what has been certified and what information survives. The completion paper guide introduces the related requirement that equivalent records support consistent future tests.
From an exact example to a physical explanation
The counter and paired readout provide a useful realization of six nested roles and three observable directions. A stronger SMM/DTT model must explain why its source admits this organization, what physically implements the detector, and why the resulting records support persistent local propagation.
The clock question is equally specific. Counting completed sweeps defines an event index. A physical duration requires a clock observable and an interaction that makes its readings meaningful. If two models have identical paired images but different clock couplings, their spatial ranks agree while their measured propagation rates may differ.
I want to develop source restrictions that constrain these choices together. A promising result would identify one admissible-operation principle that forces the pairing, protects the retained record and limits its clock response. The six-level proposal would then carry a stronger physical obligation than an organizational analogy.
What source operation could select the paired readout while preserving the memory and clock information required at completion? Researchers in mathematical physics, dynamics and the philosophy of time are invited to propose a concrete model or counterexample.
Sources and further reading
- Start here: a guide to SMM and DTT — the distinction between inner generation and outer completion.
- From six nested roles to a stable three-dimensional spatial readout — the counter, paired readout, rank sequence and multiplicity calculations developed in this post.
- One source, retained memory and the construction of predictive records — why completed records may need information beyond their present display.
- Completion principles: a guide from histories to phase — an explanatory route into the published completion framework, with its primary paper linked.