Repeated rule application: stable output or restored input?

Decide which two entries you want to compare

Repeated rule application can leave the first result unchanged or bring back the starting input. Those are two different checks. In this original paper exercise, one rule sends 1 to 0 and then keeps 0 at 0. Another sends 0 to 1 and then returns 1 to 0. Both paths finish at 0, yet the relationships between their entries differ.

Use four fictional states named 0, 1, 2 and 3. The numbers are labels, not quantities, prices or measurements. Each rule below assigns exactly one output to each of those four states, and every output is another state in the same set. That makes a second application possible without inventing an extra table entry.

Keep three columns on the page: start, first output and second output. To obtain the second output, look up the first output in the same identified rule. Looking up the original start again would merely repeat the first calculation. It would not follow the path through two applications.

Read the complete P rule before tracing a path

Rule P has four entries: 0 goes to 0, 1 goes to 0, 2 goes to 2 and 3 goes to 2. These are the complete assignments for P. No pattern outside the four listed states needs to be guessed, and no actual pin or other object is being assigned a state.

Start at 1. The entry for 1 gives the first output 0. Now use the entry for 0, which gives the second output 0. The complete trace is 1 to 0 to 0. Its second output matches its first output, but it does not match the starting 1.

Starting at 3 gives 3 to 2 to 2. Again the second output equals the first, while the start differs. Starting at 0 gives 0 to 0 to 0, and starting at 2 gives 2 to 2 to 2. In those last two rows all three entries agree.

Read all four traces together. The second output matches the first for every permitted start. Call that property stable on repeat for this page. The property does not require the first application to leave every starting state unchanged; the rows starting at 1 and 3 explicitly show changes.

Keep the first result separate from the original input

A note saying that P returns the same result needs a named comparison. For the start 1, it returns the same result on the second application as on the first. It has not returned the original 1. Write second equals first beside that trace if stability is the question you are checking.

The two starting states 0 and 1 both reach 0 under P. Seeing the output 0 alone therefore does not tell you which of those two starts was used. The complete trace preserves the supplied start, whereas a final output by itself loses that distinction in this table.

Do not create a reverse entry saying that 0 always returns to 1. Starting at 0 would then be reconstructed incorrectly. Nor does applying P again recover the missing choice: its entry for 0 remains 0. These observations concern the printed P assignments, without making a claim about a real deletion command or software system.

Read the complete S rule for a different second-pass result

Rule S has these four entries: 0 goes to 1, 1 goes to 0, 2 goes to 3 and 3 goes to 2. This is a separate table. Its four outputs remain within the same state set, so the second application again uses an existing entry of the same rule.

Starting at 0 gives 0 to 1 to 0. Starting at 1 gives 1 to 0 to 1. Starting at 2 gives 2 to 3 to 2, and starting at 3 gives 3 to 2 to 3. In every row, the second output matches the start.

Call this property restores after two for the exercise. S meets it across all four rows. Its first output differs from its second output in every row, so S does not meet the stable-on-repeat check used for P. Keep both observations beside the complete traces.

The particular path 0 to 1 to 0 ends at the same state as P's path 1 to 0 to 0. Reading only their final zero would conceal their different starts and first outputs. The three-column record gives you the entries needed to check the stated property.

Check every row before making a table-wide statement

If you examined only P's row starting at 0, all three entries would match. That row alone would be consistent with both properties. The row starting at 1 shows why P fails restoration after two, despite passing stability across the complete table.

A single failing row is enough to reject a claim that all four rows pass. A single passing row does not establish that claim. Here you can check every permitted start directly because the complete state set has only four members and the assignments are fully supplied.

Retain the exact failing row when recording a failed property. For P's restoration claim, 1 to 0 to 0 is sufficient. For S's stability claim, 0 to 1 to 0 is sufficient. Someone reviewing the finding can locate the two entries used without reconstructing a guessed rule.

A four-step return does not establish a two-step return

Use another complete rule, T: 0 goes to 1, 1 goes to 2, 2 goes to 3 and 3 goes to 0. Starting at 0 and applying T repeatedly gives 0 to 1 to 2 to 3 to 0. The fourth output returns to the supplied start.

After only two applications, that same trace is 0 to 1 to 2. The second output 2 matches neither the first output 1 nor the start 0. T therefore fails both of the two-application properties. A later return has not changed those earlier entries.

The other two-application traces are 1 to 2 to 3, 2 to 3 to 0 and 3 to 0 to 1. Each also has three different entries. Continuing each trace through four applications returns its own start, but the number of applications remains part of the statement.

Save T's four-step observation separately from the two-step checks. A note saying that a rule eventually returns can omit the very count needed to answer the question. In this table the return after four is demonstrated by the assignments; it is not a reason to report restoration after two.

The identity rule passes both checks

For rule I, let each state go to itself: 0 to 0, 1 to 1, 2 to 2 and 3 to 3. Every complete trace then has the same start, first output and second output. I passes both stability and restoration after two.

You can also see why both properties together force an unchanged first output for every start in this exercise. Stability says that the second output equals the first. Restoration says that the same second output equals the start. Those two equalities make the first output equal the start as well.

This conclusion requires both properties across the complete state set. It cannot be inferred from P's single unchanged row or from the fact that S eventually returns its input. The identity table supplies an explicit control where both comparisons hold throughout.

Keep the rule version and direction with the trace

Each second application above uses the same table as the first. Applying P first and S second would ask about a mixed sequence. For example, starting at 1 would then give 1 to 0 under P and 0 to 1 under S. Label that mixed path separately rather than treating it as evidence about P repeated.

Likewise, changing one assignment creates a different table that needs its own check. Keep the original table beside the revised version. These small authored examples establish no operating rule for real software, safe retry procedure, transaction reversal, encryption method or product function.

Keep nearby mapping and grouping decisions in their own scope

The category mapping guide checks definitions and evidence before accepting label replacements. Our four-state tables supply their assignments completely. They compare outputs after repeated use, without granting authority to merge actual categories or resolve an undefined label.

The near-matches chain guide keeps numerical label values fixed while testing relationships between pairs. Here a unary rule produces a new state for the next application. Following one of these traces is a different operation from following links between unchanged labels.

Read the actual pin accounts separately

The native JAM Paper 222419057WM account describes eight hundred pins. Its single image shows eight labeled retail containers with different colors. Those containers are recorded with the pins; the image authenticates no delivered count, material, surface compatibility or performance.

The native Unbranded map-pin account describes twenty thousand pins. Its single composite image shows twenty filled compartment containers. Neither account supplies P, S, T or I, and no pin color represents a state in this paper exercise.

Keep a purchasing question outside the rule table

If you separately consider a pin order on republeq.com and want to ask about USDT, identify the actual offer and intended quantity before discussing payment. The four states and their printed rules are invented paper inputs, not product identifiers, checkout prices or cryptocurrency amounts. This article does not establish that a payment option is currently available for an account, location or basket. Read the choices and conditions actually presented for the intended transaction, and ask the responsible support contact to clarify essential information that remains missing. If a cryptocurrency option is offered, establish the specified asset and network, the requested amount, any stated validity period and the reference associating payment with that order. Do not infer a transfer destination from a pin photograph, a container label, an earlier transaction or an arrow between fictional states. Keep the JAM account separate from the map-pin account when asking about supplied contents. An answer about one does not automatically describe the other, and a changed quantity may require reviewing the resulting order information again. Ask separately about applicable delivery, cancellation, return and refund conditions, including how an approved amount would be determined, its denomination and any deductions. No present USDT eligibility, discount, conversion result, fee level, refund amount, processing time or delivery outcome is promised here. You can inspect every printed rule using an available sheet without ordering either item. Preserve any eventual order acknowledgement and payment record separately from the state traces. Applying a paper rule twice neither authorizes repeating an actual payment nor establishes that an uncertain transaction has been reversed.

Retain the comparison that supports the result

P has second equal to first across the table, while S has second equal to start. T returns after four applications and fails both two-step checks. I passes both. Keep the complete assignments, starting state and successive outputs beside those findings so another reader can check exactly which comparison passed.