What exact rules can and cannot do

An exact digit relation is a linear identity among the digits of the points that holds for every point of a prefix (def:exactrel). The constructions of page 2 are built from them:

• a designed geometric digit at a full level: the first digit of coordinate i below the cell is a fixed linear function of the last few cell digits (a design, as in the nets of the companion paper, fixes s of these at once, one per coordinate);
• the staggered relation of the brick wall: y_c⋆,T = Σ k_c y_c,T−1;
• the skeleton relation of the residue sequence: Σ v_c y_c,T = 0, on the prefix one digit before the full level.

Relations as such are not scarce: at every full level each geometric digit is some linear function of all the cell digits (the offset map). What is scarce are relations of bounded depth J, those that use only the digits T − J, …, T of each coordinate. Those are the useful ones, because their effect does not depend on the level.

The count (thm:count). Each relation translates into a polynomial of degree less than s, and the residue theorem at a single place shows that relations at different levels cannot share digit positions: their parts in one coordinate are linearly independent. A range of R + 1 consecutive levels offers only J + R + 1 digit positions in one coordinate, so it carries at most

J + R + 1 independent exact relations of depth at most J,   about one per level,

once the range starts late enough (an admissible range: K₀ ≥ J and K₀ > (s − 1)R + s − 2).

The board is fig:count, made interactive: dimension s = 2, depth J = 1, the six levels T = 6, …, 11 (an admissible range). Each column is a level, each row a digit position n of coordinate 0. A relation is a bar in its column covering the positions it uses in coordinate 0: the skeleton relation uses position T; a designed digit of coordinate 0 uses T − 1 and T; a designed digit of coordinate 1 uses only T − 1 there. The theorem allows at most J + R + 1 = 7 bars. Click a column to cycle through the choices, or load a preset.

Consequences. A full design has s relations, so all s geometric digits can be designed on at most a fraction 1/s of the full levels (cor:layer); the block-periodic sequences attain this. Skeleton relations and designs share the same count: their densities satisfy d_S + (s − 1)d_E ≤ 1 (cor:density), which the residue sequence (d_S = 1) and the sequences of rem:gaussfrontier attain. A self-similar scramble carries one level-independent relation, and it is the skeleton relation exactly for the residue sequence and the staggered relation exactly for the staggered residue sequence (prop:onerel). An exact skeleton relation fixes the row of the generator matrix it lives in, and next to such relations there is no room for a designed geometric digit (lem:rigid, cor:nodesign).

So each construction is certified at its own family of sizes N = p^m, its level family: m ≡ −1 (mod s) for the residue sequence, m ≡ 0 for the staggered residue sequence, m ∈ s²ℕ for the block-periodic sequence. Between certified sizes nothing is controlled, but every family with bounded gaps gives a bound for every N (lem:prefix): the first N points are a subset of the next certified prefix, which costs the factor p^−Δ/s, Δ the largest gap. A dense family can therefore win for all N with a smaller constant.

Off its family a sequence really is uncontrolled. Below, exact sep(P_m) for m = 1, …, 10 in dimension two over 𝔽₅; filled markers are sizes in the family, bars the certified bounds. Hover or tap a column.

The count also limits the exponent of the all-N bound: certificates by exact relations of bounded depth leave gaps of at least s between certified sizes arbitrarily far out, so through lem:prefix they give at best the factor p⁻¹, which the residue sequence already attains (rem:exactgaps; a statement about these certificates, not about the sequences).

The count is about exact relations. A self-similar scramble has one free digit per carry length and coordinate, and choosing these one at a time can keep each carry response inside a fixed set of allowed values, a window, without fixing it (prop:steer). A window is an inequality, not an exact relation, so thm:count does not apply, and windows do certify every full level.

Which windows are cheap? A response read by a coordinate that steps forward is harmless just behind 0 and fatal just in front of it; read by a coordinate that does not move, it must stay away from 0 on both sides. Half a cell is the one offset that costs the same for every kind of step (lem:antipode): move Φ and watch the three gaps meet.

Staggered windows (thm:BS). Stagger one coordinate by half a cell against all the others, keep that half in a window of width 2ε, and pull the others back by a. In the limit of large p the bound reaches 1 as soon as (1 − a − ε)² + (½ − ε)² ≥ 1 and (s − 1)ε < a ≤ 1 − ε. In dimension two, at a = 1 − √3/2, this is the hexagonal brick wall: the brick wall with its columns compressed to the hexagonal spacing √3/2.

The residue sequence certifies every prefix P_sK−1 with constants tending to 1; windows certify every full level with 1 − O(1/p). One sequence with both would have the best bound for every N. No digital sequence with both is known. Two routes are excluded in a stated scope:

• exact designs next to exact skeleton relations: excluded for frame scrambles and relations of bounded depth (cor:nodesign);
• windows next to skeleton relations: every self-similar scramble satisfies a skeleton relation with correction terms (lem:corrS), and the free digit that moves an own response moves its correction by the opposite amount. A selection that keeps both in product windows, covering every line, certifies the full levels at most at the order of the inclusion bound of the residue sequence (prop:joint).

Also open (sec:intro, scope): base 5 in dimension three at every full level; the primes s ≤ p < 2^s for s ≥ 4 at every full level (some are covered at every s-th full level, thm:block(c),(d)); prime-power bases beyond those treated; and integration error, which the paper does not measure.