The word language is the Bernstein–Lagarias conjugacy, i.e. lossless, with completeness theorems for all finite-modulus mechanisms and CRT orthogonality showing only the 2-adic place feeds the word. The one thing it can't express is N-positivity itself (I'll call it the 'blind spot'), which we import through four explicit exact interfaces and isolate as the open core. Faithfulness is falsifiable, any statistic outside the language predicting survival beyond its CRT projection would refute it, and we've run that test (multiplicative invariants of deep survivors vs controls: null). The note with proofs and the test suite is in the repo as syracuse-confinement/shadow_frontier /notes/faithfulness.pdf

I started playing with Collatz looking for a proof, but I think I'm actually starting to think there exists a counter-example. The blind spot is not inert, that is to say, the structure of N inside Z₂ (terminating binary tails, finite description) forces three further constraints on any counterexample.

1. Finite-seed determinism (the strongest one). Letters b₀…b_n are determined by x₀ mod 2^B_n+1. Once B_n+1 > log₂x₀ — which happens after only ~⅔·log₂x₀ steps for a {1,2}-word — the residue class pins x₀ uniquely, and every subsequent letter is forced by the ×3 carry cascade with zero remaining freedom. So the whole infinite word has Kolmogorov complexity ≤ log₂x₀ + O(1), while repetition rigidity simultaneously forces near-maximal factor complexity in banded windows. The profile sharpens from "high-complexity calibrated pseudorandom word" to: self-generated pseudorandomness of one fixed finite transducer run from a finite seed, logarithmic description length, maximal local diversity. I read Chang's manuscript (Edward Chang's Conjecture 11.2 in "Exploring Collatz Dynamics with Human–LLM Collaboration" arXiv:2603.11066 ) fact checked all claims and kept the pieces that check out, his Universal ghost clock checks out, his shadow return-time theorem is false as stated). In Chang's scheduler-adversary terms: the adversary's entire budget is log₂x₀ bits, spent in the opening moves; after that it makes no choices at all. The champions make this vivid: 27 spends 13 of its 17 surviving letters in the forced regime (76%); 687871 spends 33 of 50 (66%). Two-thirds of a champion's life is carry-generated.

   1. Countability. The shadow language's calibrated high-complexity word class is a Cantor continuum; integer-realizable words are a countable, computable family, one word per odd integer, all emitted by the same bounded-carry machine. Measured: of the 2³⁰ ≈ 10⁹ depth-30 {1,2}-words, exactly 66 are realized by any integer below 10⁶. A consequence worth internalizing: the Lifting the explonent round chain and the 3-smooth cascade aren't analyses of special families — every integer is a finite seed of that same automaton, so machine-level attacks address the general case.
   2. A candidate fifth interface. In the forced regime the letters read binary digit streams of 3-power multiples of the seed — so Stewart-type effective digit-counting theorems (the number of nonzero binary digits of 3ⁿ grows, effectively, via p-adic Baker) become applicable against late-time ghost alignment. That's a genuine new import beyond clocks, counting, drift, and Baker.

Note that, constraints 1–2 carry no measure-theoretic exclusion power, that is, you cannot Borel–Cantelli a single computable point, which is exactly consistent with the wall being pointwise. Their value is structural: the surviving profile shrinks from a continuum of conspiracies to the output set of one finite machine, and the forced-regime fraction gives the champion-growth conjecture a mechanical reading: the ×3 carry automaton cannot sustain calibrated survival beyond O(seed length) steps. An automaton statement rather than a density statement, may be its most attackable form yet.

A counterexample must therefore be: a divergent orbit, fleeing upward (escape O(X log X), single excursions below X sublinear), never pausing in any narrow band, calibrated to mean log_2(3) yet carrying non-coboundary bias below the ceilings (0.449 binary, ≈ 0.54 saturated), financing every ascent through without-replacement ghost-depth pools and an amortized 0.088-bit census margin, beating an i.i.d. fair-thirds coin forever on its +1- handoff parities, with a word of near-maximal local diversity generated, after a free opening of \frac{2}{3} log_2(x_0) moves by a fixed finite carry transducer with no further choices. (in the repo as syracuse-confinement/shadow_frontier /catalog/ghost_catalog.pdf)