Here's what this paper means for actual humans: WHAT THIS IS ABOUT. The two companion papers established, for a system split into TWO parts, that you can certify what a quantum device's subsystems are from remarkably little: two generic operations, their statistics, two sampling times. Real devices have many parts. This paper carries the entire theory from two parts to many, and closes the last open corner of the two-part theory on the way. WHAT WE FOUND, IN PLAIN TERMS. Four things. (1) A LAW FOR EVERY CUT AT ONCE. Any wrong choice of parts distorts the family of unentangled states, and for many parts we derive exactly how much, for every way of cutting the device in two and every pattern of deformation across the parts: one closed formula whose weights land on exact fractions (verified to nine decimal places with a sampling-free method). The two-part law is literally the corner case. A structural surprise falls out: with many parts, rigidity is unavoidably direction-dependent; the meaningful object is a weight function over subsets of parts, not a single constant. (2) THE DIAGNOSTIC TOOLKIT GOES FULLY CLOSED-FORM. The standard "how wrong is this factorization" measures turn out to be exact linear functions of just two matrix norms of the frame (the same matrices behind textbook entanglement tests), with coefficients we give in closed form at every pair of dimensions, validated blind on dimension pairs the fit never saw. The two-part frontier constant decomposes into per-side constants with a clean derived formula, and the worst-case ratio of scrambling to visible distortion is computed exactly at three dimension pairs (28/11, 5/2, 265/138), each achieved at an exact-integer point by an explicit saved transformation. (3) CERTIFICATION IS PER-NODE. For many parts, which structure you pin down depends on which operations you certify, node by node on the lattice of ways to partition the device: a node is pinned exactly when your certified operations generate that node's algebra of local actions. Certify only one cut and the interior of a block stays COMPLETELY free (we exhibit an explicit transformation that passes every certificate while maximally scrambling the interior); add one more operation inside the block and it is still free; add two and everything locks. Counting operations is meaningless; per-node generation is everything. (4) A WEAK SECTOR BLINDS THE WHOLE DEVICE. The sampling-time blind spots found in the two-part theory (times at which certification passes while revealing nothing) survive with a sharper lesson: a forgery built for two resonant qubits, simply tensored with "do nothing" on the rest, passes every certificate of the full device while scrambling every cut that touches those qubits. On all-qubit hardware the dangerous sampling times are therefore a property of the whole machine, set by its calibration scale. The known fix (sample at two times, one generic) carries over unchanged. WHY IT MATTERS. For hardware: certifying a large device decomposes into certifying lattice nodes, each with a concrete requirement (generation-complete operations, non-resonant sampling), and the paper says exactly what an uncertified block costs you: everything inside it. For theory: the two-part identifiability program is now a special case of a lattice theory with closed-form constants throughout. HOW MUCH TO TRUST IT. Every quantitative claim was pre-registered in the companion scripts before its run. The appendix ledger lists every prediction: the hits, five failed predictions with their mechanisms, and one computation we invalidated ourselves before it could ship. Negative results rest on exact constructive witnesses, not on searches failing. All code, data, saved transformations, and working notes are in the deposit, seeded for exact reproduction, and every external reference was verified against a live source before citation. WHO AND HOW. A human-AI collaboration: research direction and standards of evidence by Ryan Cardwell (Crystalline Labs); design, code, derivations, and drafting by Claude (Anthropic), operating autonomously under a written research charter. The full arc, from empty directory to this draft, ran in one day of charter-governed iterations.
Here's what this paper means for actual humans: WHAT THIS IS ABOUT. The two companion papers established, for a system split into TWO parts, that you can certify what a quantum device's subsystems are from remarkably little: two generic operations, their statistics, two sampling times. Real devices have many parts. This paper carries the entire theory from two parts to many, and closes the last open corner of the two-part theory on the way. WHAT WE FOUND, IN PLAIN TERMS. Four things. (1) A LAW FOR EVERY CUT AT ONCE. Any wrong choice of parts distorts the family of unentangled states, and for many parts we derive exactly how much, for every way of cutting the device in two and every pattern of deformation across the parts: one closed formula whose weights land on exact fractions (verified to nine decimal places with a sampling-free method). The two-part law is literally the corner case. A structural surprise falls out: with many parts, rigidity is unavoidably direction-dependent; the meaningful object is a weight function over subsets of parts, not a single constant. (2) THE DIAGNOSTIC TOOLKIT GOES FULLY CLOSED-FORM. The standard "how wrong is this factorization" measures turn out to be exact linear functions of just two matrix norms of the frame (the same matrices behind textbook entanglement tests), with coefficients we give in closed form at every pair of dimensions, validated blind on dimension pairs the fit never saw. The two-part frontier constant decomposes into per-side constants with a clean derived formula, and the worst-case ratio of scrambling to visible distortion is computed exactly at three dimension pairs (28/11, 5/2, 265/138), each achieved at an exact-integer point by an explicit saved transformation. (3) CERTIFICATION IS PER-NODE. For many parts, which structure you pin down depends on which operations you certify, node by node on the lattice of ways to partition the device: a node is pinned exactly when your certified operations generate that node's algebra of local actions. Certify only one cut and the interior of a block stays COMPLETELY free (we exhibit an explicit transformation that passes every certificate while maximally scrambling the interior); add one more operation inside the block and it is still free; add two and everything locks. Counting operations is meaningless; per-node generation is everything. (4) A WEAK SECTOR BLINDS THE WHOLE DEVICE. The sampling-time blind spots found in the two-part theory (times at which certification passes while revealing nothing) survive with a sharper lesson: a forgery built for two resonant qubits, simply tensored with "do nothing" on the rest, passes every certificate of the full device while scrambling every cut that touches those qubits. On all-qubit hardware the dangerous sampling times are therefore a property of the whole machine, set by its calibration scale. The known fix (sample at two times, one generic) carries over unchanged. WHY IT MATTERS. For hardware: certifying a large device decomposes into certifying lattice nodes, each with a concrete requirement (generation-complete operations, non-resonant sampling), and the paper says exactly what an uncertified block costs you: everything inside it. For theory: the two-part identifiability program is now a special case of a lattice theory with closed-form constants throughout. HOW MUCH TO TRUST IT. Every quantitative claim was pre-registered in the companion scripts before its run. The appendix ledger lists every prediction: the hits, five failed predictions with their mechanisms, and one computation we invalidated ourselves before it could ship. Negative results rest on exact constructive witnesses, not on searches failing. All code, data, saved transformations, and working notes are in the deposit, seeded for exact reproduction, and every external reference was verified against a live source before citation. WHO AND HOW. A human-AI collaboration: research direction and standards of evidence by Ryan Cardwell (Crystalline Labs); design, code, derivations, and drafting by Claude (Anthropic), operating autonomously under a written research charter. The full arc, from empty directory to this draft, ran in one day of charter-governed iterations.
Here's what this paper means for actual humans: WHAT THIS IS ABOUT. When you build or use a quantum device, everything rests on knowing what its parts are: which degrees of freedom are "qubit 1," which are "qubit 2." The companion paper ("Forged Frames") showed that simple mathematical criteria cannot certify this; convincing fakes exist. This paper answers the question that result leaves standing: what is the MINIMUM experimental information that genuinely pins down a system's subsystem structure, and when can that information silently fail? WHAT WE FOUND, IN PLAIN TERMS. Four things. (1) EXACT LAWS INSTEAD OF MEASUREMENTS. Any wrong choice of parts must visibly distort the family of unentangled states, and we derive exactly how much, at every system size: closed-form constants (verified to seven decimal places), an explicit error bound, and a surprise. The two natural ways to measure "wrongness" (state distortion versus operator scrambling) turn out to be literally the same measurement, up to a fixed number, whenever the two parts have equal size. Wrongness cannot hide. (2) THE TWO-OPERATION THRESHOLD. You can pin down what your subsystems are by implementing just TWO generic operations and checking their statistics, while a THOUSAND mutually commuting operations tell you nothing at all. Diversity of control, not volume of control, is what fixes what your qubits are. The residual uncertainty is proportional to your measurement tolerance, with a constant we derive exactly and show how to compute in advance, so an experimenter can screen candidate operation pairs and pick a strong one (random pairs are typically several times worse, occasionally catastrophically so). (3) THE ALIASING LIMIT: CERTIFICATION CAN PASS WHILE TELLING YOU NOTHING. If you sample your certification data at special "resonant" times (set by the operations' energy scales), a maximally wrong assignment of subsystems passes the test perfectly. What your qubits are depends on when you looked. We characterize these blind spots exactly (for qubits they are universal and computable in advance; the danger windows shrink with the square root of your tolerance), and the fix costs almost nothing: sample at two different times, at least one away from the resonant set. With that, certification is restored; we prove it exactly and measure it. (4) REAL-LAB BUDGETS. We work out what certification costs in actual measurement shots (the error falls off unusually fast, inversely with the total number of shots rather than with its square root), how hardware noise floors the procedure, and how a simple reference measurement removes that floor to first order. A single seven-row table in the paper is effectively a lab protocol: it shows, condition by condition, which kinds of experimental access identify the subsystem structure (two generic operations: yes; one operation, commuting operations, resonant sampling: no). WHY IT MATTERS. For quantum hardware: this is a concrete recipe for certifying that the qubits you think you have are the qubits you physically have, with explicit tolerances, a sampling-time warning that is a real failure mode on calibrated hardware, and shot budgets. For foundations: the contested question "does physics determine its own subsystems?" gets a quantitative operational answer. Physics determines an operational subsystem CLASS (not one preferred coordinate representation of it), and that class is fixed by two operations sampled at two times, and by nothing less that we tested. WHAT IT OPENS UP. The seven-row identifiability table is a blueprint for a tabletop experiment on any small quantum platform; running it on hardware would move this from simulation to laboratory fact. On the theory side the paper leaves sharply posed open problems: the exact global constants when the two parts differ in size, the number-theoretic structure of long-time sampling (the failure times form a small-divisor hierarchy), and the extension from two parts to many. METHODS NOTE, BECAUSE IT MATTERS FOR TRUST. Every quantitative claim was pre-registered in the companion scripts before its run. The full ledger, including predictions that failed and one published-in-note constant that our own later theory overturned, is an appendix of the paper. All code, data, and working notes are in the deposit, seeded for exact reproduction. WHO AND HOW. A human-AI collaboration: research direction and standards of evidence by Ryan Cardwell (Crystalline Labs); design, code, derivations, and drafting by Claude (Anthropic), including autonomous sessions under the program's pre-registration rules, with adversarial reviews by other AI models incorporated and credited.
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.
Version 1 (first deposit, 2026-08). 19 pages. Every quantitative claim traces to a pre-registered evidence block with an archived log in the companion zip; the ledger of failed predictions is an appendix of the paper. A long-running dispute asks whether a quantum world selects its own decomposition into subsystems from the Hamiltonian and the state alone. Stoica (Adv. Theor. Math. Phys. 26, 3895 (2022)) proves that a structure so constructed cannot be both essentially unique and physically relevant, the non-uniqueness being modulo the unitaries commuting with H; Soulas, Franzmann and Di Biagio (arXiv:2512.07468) object that producing a second structure that way modifies the input pair, and prove a positive uniqueness theorem under two hypotheses. Both are correct, about different groups, and this paper makes the difference arithmetic. Working inside the collision architecture of the companion flagship (doi:10.5281/zenodo.22014601, all versions), where an interior memory writes records of a system and audits itself from them, the two invariance groups an interior audit actually has are measured. FIRST, the audit's Hamiltonian-side content is exactly constant on the orbits of the commutant of H, the fibres of the transport map are exactly those orbits, and the resulting gauge has the closed form sum_i m_i^2 - 1 in the eigenvalue multiplicities: the degeneracy has a computable size, three dimensions inside sixteen for a generic four-level Hamiltonian, five at (2,3), and larger exactly when the symmetry group is NON-ABELIAN (Heisenberg gives nine; abelian symmetry forces no spectral degeneracy). SECOND, the audit's total content is invariant not under that commutant but under the stabiliser of the PAIR (H, psi), which is trivial precisely under the Soulas hypotheses: the orbit holds the Hamiltonian side fixed to 2.7e-15 and pays for the second structure entirely on the state side, 1.14. The stabiliser formula quantifies exactly when their free-action condition holds and by how many dimensions it fails otherwise. THIRD, the record statistics carry their own blindness: the directions invisible to the census, whatever the state and the time, form a group, and for a register read in one basis per cell it is exactly span{I, T} at every fragment count, proved by a symplectic argument over F_2 requiring no induction -- in this deposit, by two algorithmically independent F_2 implementations plus a dense-matrix verification. That scope is the result rather than a caveat: the one-bit blindness is the price of a one-basis readout, and it is the only thing that price buys; reading a second basis on every cell collapses the group to the identity and credentials the full matrix algebra. The twist is the product of the observer's own precession with all of its own readouts, T = P R_1 ... R_k exactly, so the blindness group is the CENTRE of the credentialed algebra: an interior observer cannot see the centre its own operations generate. The credentialed algebra is everything commuting with that one bit, of dimension d^2/2. Blindness under this reading is an identity and not a similarity: for T in the blindness group the census under rho and under T rho T-dagger agrees exactly at trace distance 0.740, so no additional sampling can separate them. The paper prints the response functional every margin and census count is computed from, together with its grid and state. Around the group sit its measured collapse conditions: the price list of restorations (menus that eject the twist restore visibility at costs that double register by register, 10/20/40 contexts on the z-menu and 4/8/16 on the y/z-menu at k = 2, 3, 4, stated as observed at three points, not derived); the sorting condition for which twist configurations a restoration menu revives, named as reach membership of the displacement in the Krylov closure of the menu (63/63 exact at k = 2 in three independent engines, 255/255 at k = 3 in exact rational arithmetic; an all-time reach lemma proves the decision grid decision-faithful and shows it deliberately not span-complete); the genericity of visibility (the blindness locus is a union of alignment strata computed in part: a QND family at the minimal codimension 2 and twist families indexed by subsets of cells at codimension 2k, with a bounded search that surfaced a mixed partial-twist family the hand count had missed); and both branches of the open-world dichotomy reproduced from the group definition alone. The transport side is independently rebuilt: the W-alone computation the framework's kill switch depends on was reconstructed from the printed setting alone and the switch does not fire. A closing section isolates the mechanism shared with an earlier half-space result and shows, against the natural reading, that the two halves are distinct subspaces meeting in a quarter: one lemma, distinct instances. All quantitative claims were pre-registered before their runs. The ledger of failed predictions is an appendix and is kept in full, including a posted mechanism that died, a symmetry law corrected to Schur's criterion, a numerics trap caught only because a posted claim came back wrong, engine deaths from the exact-census campaign convicted by their own gates, and errors printed in earlier drafts and caught in review. The delineation of what is inherited from the literature is stated throughout; the paper asserts no priority claims.
Necessary conditions for classifying m-separability of multipartite entanglements