04 / Factor language
Ten turns.
A global constraint.
Compare a finite piece of motion with the standard highway’s local language. Then watch a sharp mass inequality, a delayed transition, and newly manufactured memory emerge from exact finite seeds.
Read the manuscript1. Account for every turn and every black cell
There are 84 distinct cyclic length-ten factors in P104. For any finite black initial field and interval [s, t), let N₁₀ count violating ten-turn windows lying wholly inside that interval. The mass–defect theorem states
3(t − s) − 26(|Bₜ| − |Bₛ|) ≤ 351N₁₀(s, t) + 322.
The complete field at t; the highlighted path runs from s to t. All cells outside the displayed finite black set are white.
Inspect a sliding window
Click the strip to inspect a window. Copper marks a violation; pale marks a window not fully contained in [s, t). The vertical pointer marks your selection.
This window test uses future turns within the displayed segment. The intrinsic event J₁₀ instead uses the current turn and the actual previous nine turns; no invented past is attached to the initial state.
What makes this an all-input inequality?
The proof enumerates physically feasible finite legal words and checks integer potential inequalities on a length-300 graph with 29,334 vertices and 29,559 edges. It does not assume that a legal finite word can extend to an infinitely legal trajectory. Splitting at violating windows gives the penalty 351. The 90-step witness realizes the exact constant 322 with zero violations; 351 is not asserted optimal.
2. See why the attraction slope must be 12
Under the additional promise that every future ten-factor is legal, a finite field of mass m reaches the full standard trace by τ ≤ 12m + 212. This explicit family spends longer and longer in a legal transient: mₙ = 2n + 13 and τₙ = 24n − 3.
Each rail pair adds two black cells and 24 transient steps. As n grows, τ/m approaches 12. Thus no smaller slope with a fixed additive constant can hold uniformly. This says nothing about optimality of the additive constant 212.
From a local language to eventual standard motion
The physical factor graph has three recurrent possibilities: periods 104, 28 and 24. An all-parameter collision argument excludes arbitrary interleaved folds in its non-simple component. Infinite backgrounds can support the latter patterns; finite black support excludes them. Eventual periodicity is the conclusion, not an assumption.
3. Legal motion can manufacture long-lived memory
A legal stretch can write a black cell that will be consumed much later. This complete seed comes with its actual nine-turn past. Changing n lengthens a ribbon while preserving the exact birth and return formulas.
Silence means no J₁₀ event during the legal prefix. It does not promise a legal future: the returning ant eventually breaks the local language. The theorem covers all n ≥ 1 through affine interval certificates; these controls replay selected finite instances.
Two proof chains, with distinct premises
Infinite-future legality → quantitative attraction
Local language → physically feasible graph → all-parameter fold exclusions → recurrent alternatives → finite-support elimination → τ ≤ 12m + 212. The exact event algorithm can decide the next J₁₀ and return a complete compressed endpoint. Deciding each next event does not prove that the whole event chain terminates.
Arbitrary finite words → mass–defect bound
Physical feasibility → length-300 integer potential → all-length legal finite-word bound → split at violations → the inequality above. This proof is independent of the infinite-future attraction theorem.
Complete finite constructions → sharpness and memory costs
Exact rail cancellation proves the sharp attraction family for every n. Affine read-interval checks prove silent memory manufacture and recovery/reset results. Keeping the entire field makes later readback visible; deleting the history would destroy these conclusions.
Proof and reproducibility
Proof dependency map · Mass certificate · Attraction certificate · Silent memory certificate · Exported data and source hashes
Full replay: python3 factor_language/reproduce.py. The manuscript also contains full-state event macros, folded recovery, and reset costs. They share this paper’s problem boundary; the interactions here expose its most direct witnesses.