Forged Frames: Snapshot and Persistence Criteria Underdetermine Tensor Product Structures, and the Undercut Persists to d = 128

Here's what this paper means for actual humans: WHAT THIS IS ABOUT. Quantum mechanics describes a system as a whole, but physics, and every quantum computer, depends on carving that whole into parts: this qubit versus that one, the system versus its environment. Mathematically, the carving is called a tensor product structure (TPS), and here is the uncomfortable fact this paper starts from: the carving is not written into the theory. It is extra structure someone has to supply. Several research programs have proposed that the dynamics alone (the Hamiltonian, sometimes plus a quantum state) singles out the "right" parts automatically, so that subsystems emerge rather than being chosen. WHAT WE DID. We tested that claim the direct way. We built quantum systems where the true division into parts is known by construction, because we engineered it in ourselves. Then we hid it, handed an optimizer only the ingredients the proposals say suffice (the Hamiltonian, or the Hamiltonian plus one state, or how entanglement grows in time), and let it search over every possible alternative carving to see whether the criteria recover the truth. WHAT WE FOUND. They do not. At every size we tested (4- up to 128-dimensional systems), the optimizer finds "fake" carvings that score better than the engineered truth on all of the simple criteria, often 40 to 90 times better, and the true carving is not even a local optimum: a search started exactly at the truth walks away from it. A single state's snapshot can be forged too. But the story has a second half. When we strengthened the test to require agreement on a whole family of unentangled states (the "product manifold") rather than a single snapshot, the surviving impostors turned out not to be alien worlds at all. They are the true carving rewritten in "dressed" coordinates, the same quasiparticle-style change of variables condensed-matter physics has used since the 1960s (Schrieffer-Wolff transformations), and in this final version we construct that transformation explicitly and show the optimizer's answer sits on it. Genuinely alien carvings that mimic the whole state family do not appear, and we measure quantitatively how impossible their job is (a "rigidity frontier"), anchored by an exact classical theorem. WHY IT MATTERS PRACTICALLY. (1) Any algorithm, simulation method, or interpretation-of-quantum-mechanics program that takes "the least-interacting factorization" or "the factorization where my state looks simple" as input inherits the failures documented here; we provide the counterexamples and the code to generate more. (2) The positive result sets the bar for what does work: agreement on a family of states identifies the physical parts up to a harmless, physically meaningful change of coordinates. (3) The whole test suite is released with fixed seeds, so every number reproduces, and the corrections we made along the way, including an inference we got wrong and later withdrew, are recorded in the paper rather than erased. WHAT IT OPENS UP. The paper ends with a sharpened question: how LITTLE experimental access suffices to pin down the parts? Not the full state family, but a handful of implemented operations known only through their measurement statistics? That question gets a quantitative answer (two operations, sampled at two times) in the companion paper archived alongside this one, which turns the boundary measured here into exact laws and laboratory protocols. WHO AND HOW. A human-AI collaboration: research direction and standards of evidence by Ryan Cardwell (Crystalline Labs); experimental design, code, numerics, and drafting by Claude (Anthropic), with adversarial reviews by other AI models incorporated and credited. All results reproduce from the archived scripts in the deposit.