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Definition

FPRD-D38

Guarded-affine traces at a fixed Collatz layer

Exact statement

Fix the accelerated map T(n)=n/2T(n)=n/2 for even nn and T(n)=(3n+1)/2T(n)=(3n+1)/2 for odd nn. A guarded-affine trace records a chronological word w∈{0,1}kw\in\{0,1\}^k, its affine action Tk(n)=(3sn+cw)/2kT^k(n)=(3^s n+c_w)/2^k, its guard rw≡−cw3−s(mod2k)r_w\equiv-c_w3^{-s}\pmod{2^k}, and—at a fixed layer (K,S)(K,S)—D=2K−3SD=2^K-3^S and Π(w)=cw−Drw\Pi(w)=c_w-Dr_w.

StatusFixed and self-contained; no Collatz-conjecture implication
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The accelerated Collatz system is used as a demanding calibration for exact finite-trace composition. The definition separates the presented parity word, its affine dynamics, finite modular observations, and the acceptance/resource readout.

Definitions

  • s=∣w∣1s=|w|_1 is the number of odd steps in ww, and cwc_w is determined by Tk(n)=(3sn+cw)/2kT^k(n)=(3^s n+c_w)/2^k.
  • rwr_w is the unique residue in [0,2k)[0,2^k) whose cylinder follows ww.
  • At a contracting layer, D=2K−3S>0D=2^K-3^S>0; accepted-trace readout uses the least representative of the guard cylinder greater than two.

Hypotheses and scope

  • Chronological parity words use the accelerated two-branch map exactly as displayed.
  • The fixed-layer score and continuation potentials are used only when the layer parameters and admissible-representative convention have been specified.

Proof or evidence

The public proof guide derives the affine append and concatenation laws, the residue cylinder, the fixed-layer score, and the four distinct presentation/observation/readout layers.

Guarded-affine presentation and worked trace →

Verification notes

This run checked the definition against repository commit 3a264faa09 and the frozen claim ledger. The symbols, least-representative convention, and nonconclusions agree. Definition maturity stops at L3 because no proof is asserted.

Limitations

  • This is a definition and calibration, not progress toward proving Collatz.
  • Finite traces are not identified with complete orbits.
  • No novelty claim is made for classical parity-vector or residue-cylinder formulas.

Open work

Compare the presentation vocabulary with the closest parity-vector and 2-adic literature before any novelty language.

Notes

This definition makes the Collatz system a test presentation for exact cut actions, observations, and resource potentials. It does not define a general affine-system theorem, identify finite traces with complete orbits, or claim progress toward the Collatz conjecture by itself.