Arithmetic dynamics and finite evidence

Reverse-and-add through folds, carries, and reflection

Reverse-and-add looks globally nonlocal because every digit is paired with its reflection. Folding those pairs together exposes a simpler description: a symmetric word of column sums followed by one directional carry bit. That presentation yields two exact elementary results and several useful finite observations. It does not prove that 196—or any decimal integer—is a Lychrel number.

A worked picture

Reflection chooses the columns; the carry chooses the direction

In decimal, start with $47$ . Reversal gives $74$ , so both mirrored column pairs have raw sum $4+7=11$ . The fold therefore forgets which 11 came from which side: it is the symmetric word $(11,11)$ .

The units column writes 1 and sends carry 1. The tens column then sees $11+1=12$ , writes 2, and sends a final 1. Hence $47+74=121$ .

This separates three operations: the input is the digit word, pairing mirrored positions gives the list of column sums, and the moving carry bit completes the addition. The pair sums are symmetric, but the carry enters each column from only one side.

One step is therefore simple. Iteration is not: the output digits become the next digit word, so the next symmetric pairing depends on where the previous carry wrote its digits. The exact results below isolate this coupling without pretending to solve the complete orbit.

FPRD-D39 · definitions

Fold the reversal before following the carry

Fix a base $b\ge2$ and write $A_b=\{0,\ldots,b-1\}$ . Let $\operatorname{rev}_b(n)$ reverse the canonical base- $b$ digit word, and put

$T_b(n)=n+\operatorname{rev}_b(n).$

For $n\in\mathbb N_0$ , write its canonical digits least-significant first as $a_0,\ldots,a_{L-1}$ , using $L=1,a_0=0$ for zero and $a_{L-1}\ne0$ otherwise. The folded column-sum word $c_i=a_i+a_{L-1-i}$ is symmetric. Starting with $\gamma_0=0$ , define

$s_i=(c_i+\gamma_i)\bmod b\in A_b, \qquad \gamma_{i+1}=\left\lfloor\frac{c_i+\gamma_i}{b}\right\rfloor.$

Retain the terminal carry $\gamma_L$ . Reversal-created leading zeroes do not change the integer. For the canonical base- $b$ digits $d_0,\ldots,d_{M-1}$ of $m$ , define its reflection distance by $D_b(m)=\#\{i:d_i\ne d_{M-1-i}\}$ . It counts ordered positions and vanishes exactly at palindromes.

FPRD-T144 · exact representation theorem

One step is a symmetric fold plus a two-state sweep

For each fixed base $b\ge2$ , after the symmetric fold supplies $c_0,\ldots,c_{L-1}$ in increasing digit index, the carry stage is a deterministic subsequential transducer with states $\{0,1\}$ . It computes $T_b(n)=\sum_{i=0}^{L-1}s_i b^i+\gamma_Lb^L$ exactly and emits a terminal 1 exactly when $\gamma_L=1$ .

For a direct proof, sweep from the least-significant column. After columns $0,\ldots,i-1$ have been processed, their written digits are the corresponding low digits of the partial sum, and $\gamma_i$ is exactly the amount carried into column $i$ . Division of $c_i+\gamma_i$ by $b$ writes its remainder and passes its quotient, preserving the invariant. At the final column the remaining carry is the possible leading digit. Starting from $\gamma_0=0$ , induction using $0\le c_i\le2b-2$ and $\gamma_i\in\{0,1\}$ gives $\gamma_{i+1}\in\{0,1\}$ .

The transducer reads the fixed-base alphabet $C_b=\{0,\ldots,2b-2\}$ ; its transition is $\lfloor(c+\gamma)/b\rfloor$ and its output is $(c+\gamma)\bmod b$ . The fold is reflection-symmetric; only the carry travels directionally within this folded presentation. This is a family indexed by a fixed base, not one finite transducer over unbounded bases, and the fold is supplied first: no ordinary one-way transduction of the original digit stream is claimed. The theorem makes one step transparent but does not control a complete orbit.

FPRD-T145 · exact digit theorem

Without length growth, mirrored digits differ by one carry modulo b

Suppose $\gamma_L=0$ , and let $j=L-1-i$ . Symmetry gives $c_i=c_j$ , so

$s_i-s_j\equiv\gamma_i-\gamma_j\pmod b, \qquad \gamma_i-\gamma_j\in\{-1,0,1\}.$

This hypothesis is equivalent to no digit-length growth. For $n>0$ , the outer raw sum obeys $c_{L-1}=a_{L-1}+a_0\ge1$ . If $\gamma_L=0$ , then $s_{L-1}\ge1$ and the output has length $L$ ; if $\gamma_L=1$ , the retained terminal digit makes its length $L+1$ . Zero is the canonical one-digit singleton.

This is modular: digits $0$ and $b-1$ differ by $-1\pmod b$ . The law also permits zero difference everywhere, and its residue set is the full residue set in bases 2 and 3. It therefore gives no exclusion in those bases and does not exclude a future palindrome.

For a non-palindromic example, decimal $150+051=201$ has no length growth. Its folded sums, read from the units side, are $(1,10,1)$ . The middle column creates the only carry, so the mirrored output digits are 1 and 2: they differ by $-1\pmod{10}$ exactly as the theorem predicts. A saved finite replay checks both exact theorems for bases 2 through 16 and inputs through 200,000. That finite replay corroborates the written proof; it is not the proof.

FPRD-T146 · theorem and reproduced finite census

Exactly two decimal column-sum patterns produce a palindrome in one step

A positive decimal input has a palindromic next value exactly in one of two cases. Either every column is carry-free, $c_i\le9$ , or every column sum belongs to $\{0,11\}$ and the outer sum is 11. In the second case the output gains a leading digit.

Proof idea. Include the final carry $\gamma_L$ as a possible leading output digit. If the output has length $L$ , equality of mirrored output digits and symmetry of $c_i$ force the incoming carry word to be symmetric. Its first bit is zero; taking the first hypothetical 1 and reflecting the column that created it gives a contradiction. Hence every carry is zero and all $c_i\le9$ .

If the output has length $L+1$ , its first and last digits are both 1. Starting at the two outer columns and moving inward, the same palindrome equations and carry recurrence force each raw sum to be 0 or 11; the outer sum must be 11. Conversely, these conditions make each carry-out the indicator of an 11-column, and the resulting output digits are symmetric. This is the carry classification recorded by Vaughn Suite in 2003; the proof here is reconstructed directly from the recurrence above.

The independent exhaustive replay over $1\le n\le10^6$ gives

$152{,}058=151{,}250+808$

one-step palindromes, with zero uncovered cases in that interval. The large first class is carry-free: its symmetric folded digits are already a palindrome. The 808 saturated cases show a second way a carry pattern can still finish palindromically. The theorem supplies the unbounded classification; the displayed counts are only for the stated finite interval.

Vaughn Suite's 16 September 2003 carry note states the two cases and gives separate length-dependent counting formulas. The classification is therefore not presented as an FPRD novelty.

How to read the rest

The theorem, the census, and the orbit evidence answer different questions

FPRD-T144 is exact for every input in each fixed base, but describes only one update.
FPRD-T145 is also universal, but only when that update does not add a digit.
FPRD-T146's written carry-recurrence proof gives the universal decimal classification. The exhaustive census covers only $1\le n\le10^6$ ; it supplies the displayed finite counts and checks the classification only on that interval.
FPRD-T147, FPRD-T148, andFPRD-T149 concern bounded portions of one orbit and cannot establish its infinite future.

FPRD-T147 · reproduced bounded finding

The sampled 196 orbit remained visibly far from reflection

The published computation fixes the window to outputs $T^k(196)$ for $1\le k\le4{,}000$ . The minimum reflection distance is 2. The minimum normalized distance is $4/33\approx0.121212$ at step 68. Among the 3,771 outputs longer than 100 digits, the mean of $D/L$ is 0.659803985542. This reproduces the source's qualitative $0.66L$ description with an explicit estimator and finite window.

A finite orbit segment staying away from the palindrome set does not imply that every later iterate does so.

FPRD-T148 · reproduced bounded finding

Growth appeared to move the reflection axis rather than destroy it

The published computation uses the first 3,000 transitions and averages each output's normalized mismatch fraction. On 1,735 non-growing outputs the shift-0 mean is 0.496558797244. On 1,265 growing outputs the shift-+1 mean is 0.497066641824. The mismatched shifts from −2 through +2 range from 0.884 to 0.897. The finite table therefore reproduces the observed one-position relocation.

The exact shifted-index identity was not derived. This is a correction to an experiment, not a universal theorem.

FPRD-T149 · reproduced bounded finding

Raw blocks looked diverse while mirrored digits remained coupled

Across states $T^k(196)$ for $0\le k<4{,}000$ , every within-state decimal digit block through length 4 and every incoming-carry block through length 12 occurs. Incoming-carry density is 0.499366103113. Mirrored digits retain mutual information 0.563591475633 bits, while mirrored incoming carries have only 0.000013795714 bits. Raw block diversity therefore did not erase the conditional structure exposed by FPRD-T145.

The archived source reported carry density 0.5009 for unspecified later iterates. That exact number is not treated as reproduced: the public result uses the pinned window and convention above.

Finite maximal block diversity is not normality, entropy, or a probability model of the infinite deterministic orbit.

FPRD-FAIL-LYCHREL-01 · stopped route

The raw substitution-hierarchy analogy did not provide an invariant

The excursion folded histories into digit-pair tiles and carry colors, hoping for a low-complexity hierarchy like those used in substitution constructions. The bounded raw data showed no such substrate, so that specific route was stopped.

The initial conclusion was too broad. Raw subword complexity is blind to conditional structure, as the mirrored ±1 law shows. This failure says nothing decisive about finite-state invariants, backward preimages, anti-concentration, or every tiling presentation.

FPRD-C04 · open problem

The decimal Lychrel question remains open

Does some positive decimal integer avoid palindromes under every iterate of $T_{10}$ ? Does 196? The external records checked on 28 August 2026 still treat 196 and the entries of OEIS A023108 as apparent candidates rather than proved decimal Lychrel numbers.

What this batch establishes. One step has an exact two-part presentation: symmetric folded columns and a one-bit directional carry. That yields a universal mirrored digit law and a reproducible finite census. The longer-orbit data remain bounded evidence, and no FPRD result here settles 196.

Sources and evidence status

FPRD Lab, Folded reverse-and-add and carry analysis, 6 June 2026; this is the cited historical source record. The corresponding durable packet is not present in tracked source.
Vaughn Suite, Reverse-and-add carry note, email to Wade VanLandingham, 16 September 2003. This is the primary source for the two one-step decimal classes and their counting formulas; it has no DOI or archival venue.
John Walker, Three Years of Computing: Final Report on the Palindrome Quest (1990), for the primary computational history of 196; and OEIS A023108 for the continuing “apparent candidate” status and bibliography.
Reproducible bounded audit: integer checker, independent digit-list replay, certificate, and estimator notes.
Result-review audit: checker · output.

The original June experiment code and raw run directories remain unavailable. The new orbit audit recreates the finite experiment under fully pinned windows and estimators; it does not claim byte identity with an unavailable historical run.