03 / Wake rigidity
A highway, reconstructed
from its wake.
A repeating ant leaves a trail of translation classes. Explore how a small amount of geometric information forces the standard 104-step highway — and how parity excludes other proposed wakes.
Read the manuscript1. Collapse an infinite highway into one period
Full drift is (2, −2). Two read locations belong to the same class exactly when their difference is an integer multiple of this full drift. The labels t = x + y and r = x mod 2 retain both lanes; replacing the drift by (1, −1) would lose information.
Black cells: complete synthesized finite seed after the selected phase. Path: the next 104 reads. Coordinates use north as positive y.
Moving the period boundary changes the local view. The full displacement and number of classes visited an odd number of times remain fixed.
Why count odd visits?
Every visit flips a cell. An even number of visits cancels; an odd number changes its final color. The exact ordered reads within each translation class must alternate white and black. The exported seed satisfies the all-epoch color criterion, including the infinite tail.
2. Edit the proposed wake
Click a circle to add or remove an odd class. The standard profile has counts (2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2), from layer −10 to 0. The strongest theorem needs less: growth 12, width 10, and at most one odd class at each strict interior layer.
The necessary parity screen
Normalize σ to +1 by translating layers by one when σ = −1. Let V count even-layer classes and J count ordered even-below-odd pairs.
J + V/2 ≡ 1 (mod 2)
Normalized (Q₀, Q₁, C₀, C₁):
The uniqueness theorem
Why does a screen survivor remain a question?
Parity and signed residue counts are necessary. They do not construct a consistent read ordering or prove that a proposed wake is realizable. The growth-eight survivor illustrates this distinction. The thin-wake classification requires the additional growth, width, and thinness assumptions shown above, with full drift fixed throughout.
3. Follow the finite proof of an all-period statement
Geometric hypotheses → finite count cases
Full drift (2, −2), growth 12 and transverse width 10, with at most one odd class at each strict interior layer, constrain the layer occupancies. Integral crossing-flow equations reduce the possible counts to 76 cases. The standard profile and its residues emerge from classification.
Integer clock → a certified period cap
A finite window graph and 90,158 checked integer edge inequalities bound the period separately for each surviving count case. The bounds are proved before enumeration; the conclusion does not assume a period limit.
Ordered reads → 83 exhaustive jobs
The search enforces physical read alternation and exact full-drift classes. All 83 jobs are exhausted, visiting prefixes. Only equivalent standard highway representations remain. This page displays the resulting certificate data; it does not rerun that search in your browser.
Surviving word → an infinite highway
An independent all-epoch color verifier checks the survivor and synthesizes a complete finite seed. The period explorer above replays this exact seed. Together the bound, exhaustive exclusions and infinite-tail check establish the uniqueness theorem.
Proof and reproducibility
The general inverse-synthesis theorem also covers non-axis drift: a clock bounds candidate periods, ordered class constraints produce finite candidates, and the infinite-tail criterion accepts complete seeds.
Proof dependency map · Full classification checker · Mixed-wake parity proof and checks · Exported data and source hashes
Reproduce the full chain: python3 wake_synthesis/reproduce.py --jobs 4. The --quick option omits the large exclusions. Mixed-wake parity builds on the attributed periodic-growth results cited in the manuscript.