An orbit can be presented as a trace

An arithmetic orbit is usually displayed as a list of integers. FPRD also records the finite word of branch choices that produced the list, the modular cylinder of starting values with that same word, and the affine action of the whole segment. These are presentations of one finite dynamical event, but they expose different structure.

For the accelerated Collatz map$T(n)=n/2$ when$n$ is even and$T(n)=(3n+1)/2$ when it is odd, the segment

$$7\longmapsto11\longmapsto17\longmapsto26\longmapsto13$$

has chronological parity word$w=1110$. The same four steps are summarized exactly by

$$T^4(n)=\frac{27n+19}{16}, \qquad n\equiv7\pmod{16}.$$

The congruence says more than the single example: every positive starting value in that residue cylinder follows the same four branch choices. The affine formula says how all of those starts move.

Active program · exact costs and exponential equations

Selfridge's problem asks whether addition can make a power of two cheaper

Integer complexity counts the number of ones in an expression using addition and multiplication. The evident product$2^n=(1+1)^n$ costs$2n$ ones; Selfridge asked whether any formula can do better.

The current program studies a normalized family with three additions. It preserves expression cost while turning each branch into an exponential Diophantine equation. Within that family, the entire$g=0$ positive-type-I branch and the positive-type-II tower for$a\ge1$ are classified. In positive type II, exactly ten controls remain, and each costs at least$2N+4$.

The complementary $a=0<g$orientation has exact 3-adic valuation laws for$f=1$ and$f=2$. In the first two minimum layers, the surviving periodic graphs reduce to seventeen proper presentations, each costing at least$2N+6$. The classical S-unit theorem proves that the full normalized complementary family is finite, but it gives no effective exponent bound.

In the remaining $f=0$ branch, the only exceptional cancellations are$(2+3)(1+8\cdot3)+3=2^7$ and$(4+3)(1+8\cdot3)+3^4=2^8$. Both cost more than twice the output exponent. Every other$f=0$ solution satisfies one exact 3-adic valuation law. A checked next-digit continuation still branches heavily, so the residue-table method has reached a structural stopping boundary.

These are restricted theorems about the normalized formula family. They do not resolve Selfridge's open problem.

Follow the research program → · Read the exact restricted results →

Worked case · reversal as paired digit columns

A global digit reversal becomes local after pairing mirrored positions

For the base-$b$ reverse-and-add map $T_b(n)=n+\operatorname{rev}_b(n)$, mirrored input digits look far apart in the ordinary numeral. Pair them first. If $a_i$ are the digits, the paired column sums$c_i=a_i+a_{L-1-i}$ form a symmetric word. One carry bit then sweeps across it.

$$s_i\equiv c_i+\gamma_i\pmod b, \qquad \gamma_{i+1}=\left\lfloor\frac{c_i+\gamma_i}{b}\right\rfloor.$$

The integer is unchanged, but within this fixed-base folded presentation the directional carry is the only source of broken reflection symmetry. The fold is supplied first; this does not claim an ordinary one-way transduction of the original digit stream. It immediately gives the modular ±1 law of FPRD-T145for non-growing steps. In decimal, the same recurrence proves that a one-step palindrome has either no carries or only raw column sums 0 and 11, with outer sum 11. Neither theorem controls every later carry, so they do not prove that 196 is a Lychrel number.

Read the exact results and bounded evidence →

Worked case · exact finite traces

A parity word determines one modular cylinder

Let $w\in\{0,1\}^k$ be a chronological parity word and let$s=|w|_1$. Its iterated action has the form

$$T^k(n)=\frac{3^s n+c_w}{2^k}.$$

Because $3^s$ is invertible modulo $2^k$, the starts that realize $w$ form the single guard cylinder

$$n\equiv r_w\pmod{2^k}, \qquad r_w\equiv-c_w3^{-s}\pmod{2^k}.$$

The word, affine constant, and guard are therefore not three loose annotations. Together they provide an exact finite presentation of the orbit segment. FPRD-D38fixes the conventions and keeps this finite-trace calculus separate from claims about complete orbits.

Worked case · what crosses a cut

The endpoint alone does not sew two traces together

Suppose a prefix $u$ has weight$p$, guard$r_u$, and endpoint$y_u$ at its least guard representative. Every start in its cylinder can be written$r_u+2^k\mu$, and the prefix sends it to $y_u+3^p\mu$.

This is the information transported by the presentation. A numerical endpoint describes one realized path; the affine seam map describes how the prefix routes every possible continuation guard. The complete horizon profile has linear bit demand in$\ell$ for the exact interface defined in FPRD-T139. That is an interface bound, not a language-recognition lower bound.

Worked case · contraction and recovery

A contracting multiplier can still carry an additive recovery

At a fixed layer with trace length$K$ and odd-step count$S$, put$D=2^K-3^S$. The layer is contracting when $D>0$, but the additive constant $c_w$ still matters. For the least admissible start, descent is governed by the score

$$\Pi(w)=c_w-Dr_w.$$

Cutting $w=uz$ separates the remaining score into a prefix budget and two nonnegative costs:

$$\Pi(uz)=2^j\bigl(B_u-\delta_z-D\mu_z\bigr).$$

Here $\delta_z$ is the suffix’s deficit from the best fixed-weight completion, while$\mu_z$ is the modular lift charged at the seam. FPRD-T140proves the identity. It explains why multiplier contraction alone is not enough: a presentation must account for both local disorder and boundary alignment.

Negative result · a failed counting route

More marginal bands do not create joint information

One tempting strategy divides guard values into progressively finer thresholds and combines each band with an unconditional bound on suffix disorder. If$t_1\ge\cdots\ge t_L\ge0$ are the guard thresholds and $A_i\ge0$are those marginal disorder bounds, the resulting envelope is

$$M_L=t_L+\sum_{i=2}^L\min(t_{i-1}-t_i,A_i)+A_1.$$

This is useful precisely because it is negative: it closes a family of increasingly elaborate presentations that never retained the correlation the problem needs. The lowering-capped potential of FPRD-T143 remains an exact monotone upper bound, but monotonicity does not imply a uniform multiplicative loss.

Integrated program · arithmetic quotients

Finite compatibility and arithmetic realization are different gates

The Quotient Arsenal sharpens this distinction in two arithmetic systems. A Collatz parity tower always determines a 2-adic point, but it represents a nonnegative integer only when its least residues stay bounded, equivalently eventually constant. Periodic integrality is a separate divisibility test: for a word $w\in\{0,1\}^k$, the exact equation is $(2^k-3^{s(w)})n=a_w$. The word $001$already separates the doctrines: its unique 2-adic fixed lift is $4/5$, not an integer.

For reverse-and-add, an output of digit length $M$ can arise only from fixed-length profile charts $M-1$ and $M$, with at most one profile in each. The growth bit $\varepsilon=M-m\in\{0,1\}$is therefore a sharp receipt for the one-step chart seam. It recovers the profile chart, not the original digit word, and the seam matching has no cycle monodromy. Later response towers and source fibres remain separate questions.

These are presentation theorems: they identify the exact response, completion, and admissibility interfaces. They do not settle the Collatz conjecture or any decimal Lychrel question.

Read the reviewed guard-chamber and cross-length results →

Open frontier · the missing correlation

The surviving question is joint, not marginal

The exact seam and recovery calculations leave a focused open problem. A useful all-length estimate must retain the simultaneous conditions

$$r_w\le R, \qquad \delta_w\le B-Dr_w,$$

for the actual forced continuation. Counting available guards and available fixed-weight suffixes separately loses their matching. FPRD-C03 records the two honest outcomes: prove sufficient joint loss, or construct an exact accepted counterfamily showing that forced-suffix disorder can preserve exponential mass.

Working answer to the driving question. Finite arithmetic iteration is governed not only by branch multipliers, but by an exact presentation of guards, affine seam actions, additive recovery, and their correlations. This calculus organizes and prunes finite traces. It does not prove or disprove the Collatz conjecture, and the recorded candidate bounds remain exponential.