Finite memory & remote feedback
A black cell can remember.
The ant writes a black cell, walks away, and eventually reads it again. Follow one such lifetime, then stretch the journey while keeping the seed and the intervening event count fixed.
01 · The physical event
When does an old black cell return?
An \(S_D\) event reads a black cell that was absent from the preceding \(D\) read positions. The true past is recovered by reversing the complete state. A newly generated event additionally consumes a lifetime written after the source checkpoint, outside a chosen fixed region \(K\).
The whole black field is retained. The line shows the most recent 208 read positions; changing the view changes only the camera. Arrow keys move the time slider by one update.
0 matching events in 800 replayed updates; 0 initially black cells. Counts describe this finite window.
Try a causal test. On the blank plane, jump between birth 21 and readback 197. Raise D to 176: this 176-update-old read stops being an \(S_D\) event. Then choose “One black cell at the origin”: an initially black coordinate can qualify again after its old lifetime is erased and a new one is written.
02 · A fixed seed size, a growing delay
Twenty black cells. Thirty-one intervening events. No uniform age limit.
For every even \(n\ge20\), an explicit complete twenty-cell input writes cell \((0,2)\) at update 60. That cell survives until update \(624n-1516\). Exactly 31 \(S_{128}\) events occur strictly between these two updates.
The small inputs above are replayed literally. This calculator evaluates the proven affine formulas using exact integers for any entered even n; it does not simulate the enormous examples.
First readback
Black-cell lifetime
First permanent P104 entry
Increasing n moves the seed cells farther apart. Thus memory length D, intervening event count and initial black-cell count alone cannot bound the feedback lifetime. The full input coordinates retain the missing spatial information.
- Place five ports at time zero. Their twenty cells depend affinely on n; no later intervention is used.
- Separate all five travel stages. Affine inequalities hold for every n ≥ 20, protecting previously written history.
- Return to the same physical cell. The complete field at collision has a protected recovery region. Exact recovery takes 774 more updates and reaches a permanent standard highway.
03 · Predicting without discarding history
An exact event after 10⁶⁰ highway epochs.
The birth-filtered algorithm tracks both read times and complete field changes using finite arithmetic segments combined by XOR. In this checked input, a remote initial obstacle is first encountered at \(104N\), with \(N=10^{60}\). Its initial lifetime is skipped by the birth filter; the first qualifying new lifetime is born at \(104N-102\) and consumed at \(104N+46\).
Exact birth update
Exact consumption update
Age: 148 · memory length D: 128
Inspect the exact endpoint representation
Read cell:
XOR segments retain the complete post-state after reads, including untouched off-path cells.
What makes this a proof-producing query?
- Recover the true D-turn suffix from the source state.
- Use the finite suffix dynamics to locate the next stale read.
- Carry every skipped write as a timed arithmetic segment.
- Check the consumed lifetime’s birth and its position outside fixed K.
The browser displays a fixed certificate regenerated by the original exact algorithm. It evaluates its decimal integers exactly; it does not offer a general browser implementation of the event decider.
04 · What persistent nonperiodicity would require
More time forces feedback at larger scales.
Under the premise that a finite-support trajectory never becomes a nonzero translating periodic orbit, the growing-scale theorem forces many distinct newly generated black lifetimes to be read back far away and late. This calculator assumes that premise; it does not classify any input as nonperiodic.
For source mass 20 and source coordinate width 100:
At least this many distinct qualifying lifetimes are consumed by H:
Each counted lifetime is born after the displayed threshold, lives for more than D updates, and is consumed outside the displayed fixed square. The same lifetimes satisfy all three properties.
Show the exact horizon
From interactive witnesses to the proof
The lifetime inspector executes the classical complete-field rule. Exact exported fixtures cross-check it against the publication-owned Python implementations. The parameter-family theorem uses all-parameter affine guards; the event algorithms and growing-scale theorem use analytic bounds and exact finite arithmetic.
Memory preprintProof and dependency mapTwenty-cell constructionExact feedback and growing scalesChecked page data
The published mathematical owner remains the memory paper. Its SD events are defined separately from the factor paper’s J10 events; their counts are not interchangeable.