Preparing a geometric memory with appreciable probability requires relative entropy when geometry is rare in the reference ensemble. The bound quantifies information resources without automatically identifying them with work or elapsed time.
The rarity of a geometric memory becomes a quantitative preparation question when a reference ensemble and a target success probability are fixed. Relative entropy measures the information required to favor that event. For SMM/DTT, the bound separates the work of an active source from the structured memory and controls it must prepare; converting this information requirement into physical energy or duration needs additional laws.
Let σ be the reference state of the memory, and let A denote the event that the memory has the selected geometric property. Its reference probability is p. A preparation produces another state ρ in which the same event has probability q. Relative entropy D(ρ∥σ) measures how distinguishable that preparation is from the reference. This quantity includes the statistical information favouring geometry; it is not automatically an amount of energy or a count of source ticks. The event must be tested in the same way in both states for the comparison to be meaningful.
Measure only whether A occurred. Data processing says that discarding all other information cannot increase relative entropy. The resulting binary distributions give a lower bound. Expanding the binary divergence leaves a term q log(1/p), minus the binary entropy h₂(q), and a further nonnegative contribution. If geometric reference probability is exponentially small in N log N, preparing order-one geometric occupation therefore requires relative entropy of that same order.
Suppose a toy reference gives p=e^(−1000), and the intended preparation succeeds with probability q=0.9. The simple lower bound is 900−log 2, approximately 899.31 nats. Dividing by log 2 expresses this as approximately 1297.4 bits of distinguishability. These numbers illustrate the bound; they do not estimate the information content of the universe. A controller, fresh ancillas and a retained success flag may carry part of this resource. Omitting them from the state description can make a costly protocol appear falsely free.
For a specified thermal reference σ proportional to exp(−H/k_BT), the excess free energy is k_BT D(ρ∥σ). Only after this Hamiltonian, temperature and reservoir model are supplied does the information bound acquire an energy interpretation. An actual minimum-work statement still needs a preparation protocol and its operational restrictions. In DTT the distinction is particularly useful: generative time, memory capacity, thermodynamic work and observed elapsed time are different quantities. Deriving relations between them would be new physical content; relabelling one as another would not explain the preparation of space.
Relative entropy is always measured against a reference. If a proposed formation process changes the reference distribution, the original contraction argument cannot simply be reused. The draft accounts for a changing reference with a sum of operator-norm differences between consecutive logarithms of reference states. This is an information budget, not automatically work. It makes a physical escape from the obstruction explicit: a source may supply structured control that changes which memories are favoured, but that control then enters the accounting. This prevents an apparent solution that hides the required resource in an unreported change of ensemble or in a success-conditioned branch.
The six inner stages in this research programme are nested roles of a single generation–completion process. The seventh level publishes a completed configuration; it is not another independent time coordinate. The locality paper supplies concrete finite implementations of this architecture, but their schedules, memory and permitted transport remain specified inputs. A stronger SMM/DTT account would select those ingredients with one physical update law and explain why they persist under perturbation. It would also calibrate graph steps and completion counts against actual rulers and clocks. A proof about a finite labelled memory is valuable even before that programme is complete, provided its conclusion is kept at the level established by its hypotheses.
There is also a metric distinction inside the lattice example. Counting elementary signed operations produces a word metric, which on the standard lattice is the Manhattan metric. A Euclidean-looking embedding or cubic volume growth does not automatically replace that metric by an isotropic Euclidean norm. A continuum limit must specify the probe, its allowed directions and any averaging or refinement that makes its comparison law isotropic. This is why adjacency, volume growth, diffusion, calibrated distance and signal propagation should be checked separately even after an exact lattice neighbourhood theorem has been proved.
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Scope of the argument. Information-theoretic necessity; sufficiency and minimum physical work are not established.
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