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. 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. 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.
Version 4 (2026-08-25). Two additions from the correction campaign, carried as dated in-body correction records. The (2,5) entry, printed as a prediction in v3, is now census-exact -- and it was the wall law's first divergence test. At the constructed maximal-wall stratum point the stiff transverse count is exactly 10, so alpha_stratum(2,5) = 5; a fresh 60k ensemble's calibrated read is consistent with 5, with alpha = 10 (the superseded single-wall count's prediction) excluded at p = 0.000 and demonstrated power ~ 1; and a stratum-conditional paired ladder tracks the exact rank-10 Hessian form to <= 0.002 while a rank-20 comparator separates by >= 2.7 on identical draws. The law m_C(2,d) = 2(d-1) is census-exact at d = 3, 4, 5, now including the first point where it diverges (by 2x) from the superseded count. The (3,4) entry, printed open in v3, is now a PAIR of statements, and neither alone is alpha(3,4). The posted product-wall counting landed census-exact (alpha_stratum(3,4) = 5, so the asymptotic exponent satisfies alpha(3,4) <= 5), while the accessible window at n = 3e4 is steeper (calibrated 90 percent set [5.4, 6.9], with the previously read 4.0-4.1 excluded at power ~ 1): the codim-10 stratum's prefactor has not engaged at reachable depth, and window and asymptote have visibly decoupled. This is the scope lesson of the (d >= 3) x (d' > d) regime -- census plus conditional ladder measure the law, global windows measure the pre-asymptotic shoulder -- and it retro-explains the earlier fresh-seed ladders' instability as shoulder reads. Version 3 (2026-08-20). One SUPERSESSION: the qutrit-qutrit exponent closes. Versions 1 and 2 left alpha(3,3) open, v1 by a small-sample fit reading 3.25 and v2 by the bracket [2.3, 3.0] read off a descending window ladder. Both readings were artifacts of the estimator problem for which the pair-type table was already corrected in v2. The descent is attenuation tracking sample count rather than tail depth: an ensemble drawn from an EXACT alpha = 3.0 law descends by 0.62 across the ladder against the real data's 0.34, so a descending ladder is evidence FOR a constant exponent, not against one. Replacing the window fit with a calibrated Monte-Carlo test and inverting it over candidate exponents gives alpha(3,3) = 3.0 at p = 0.517 with a 90 percent confidence set [2.75, 3.10]; v1's 3.25 is excluded at p = 0.017. Power against alternatives is 0.990 / 0.590 / 0.120 / 0.575 / 0.990 / 1.000 at true alpha = 2.50 / 2.75 / 3.00 / 3.25 / 3.50 / 4.00. An independent re-audit at n = 10^4 reads flat near three on two seeds. Disclosed defect, carried in the paper: the test's own calibration row missed at the low edge (rejection rate 0.207 at a nominal 0.10 when the truth is alpha = 2.50, 4.4 standard errors high), so the lower end of the confidence set is softer than printed. This ensemble separates 3.0 from 3.25 with about 58 percent power and has none to separate 3.0 from 2.95. The (3,3) CARRIER family remains unattributed: the exponent is settled, the stratum that carries the accessible tail there is not. Version 2 (2026-08). Changes from v1: the pair-type exponent table is corrected (v1's alpha(2,4) = 2.08 was a 50-sample order-statistics fluctuation, placed at the 0.15th percentile of its own estimator's sampling distribution; three independent codebases now agree on the wall law alpha(2,d) = d, census-exact at d = 3 and 4); the transverse constant 1/12 is derived in closed form (diagonal cost 1/6, halved by the congruent partner's equal-cost correction); the (2,3) tail's carrier stratum is fully attributed (an isotropic-intertwiner family sharing the exponent 3); the qutrit-qutrit exponent is honestly bracketed open at [2.3, 3.0]; and the dispersion rule (tail-exponent fits print bootstrap dispersion or do not print) is adopted. The v2 ledger appendix carries the correction anatomy. Why is the multipartite certification constant heavy-tailed, where does the tail live, and what kills it? Localization: the credential pencil is exactly block-diagonal over party-subset supports; the tail is the worst pair; geometry statistics reconstruct from pair statistics (14/14 factor-band comparisons). The law: a frame-congruence theorem: whenever the two parties' operation frames subtend equal Bloch angles, the rotation intertwining them is an exact common kernel direction; the zero set is codimension one at generic angles, transverse growth is universally quadratic (mu = (Delta theta)^2/12), and the qubit-pair tail exponent is exactly 1/2 (the previously measured 0.39 is the finite-window slope of this law). The dichotomy: pairs containing a d >= 3 party measure exponents 2.08 to 3.69 because higher-dimensional operation pairs carry many local invariants that must all match: freedom helps the certifier. Engineering: heterogeneous dimension profiles tame the tail; qubit-pair presence splits all measured geometries into two clean classes; screening is a five-minute two-party calculation. Pre-registered; two entry gates that caught wrong conventions and all deaths ledgered.
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.
Licensing: paper, figures and data are under CC BY 4.0; the source code is under the Apache License, Version 2.0. The paper text names the code licence "Apache License 2.0" — this is the same licence. Mutual information between distant regions of a topologically ordered state can witness long-range nonstabilizerness ("long-range magic"). For anyon states on a torus it takes an operational form: the Shannon entropy of the flux (anyon-type) statistics along a loop, evaluated in every frame generated by the modular transformations S and T. The witness is one-sided: a non-integer entropy certifies magic, an integer entropy certifies nothing. We ask whether it can certify the states that were actually prepared in the trapped-ion demonstration of non-Abelian D(D_4) topological order. In exact arithmetic, under a model of the prepared states (below), we find that it cannot. Modeling the 22 ground states of that experiment as the joint eigenbasis of the abelian Wilson loops, we find all of them blind to the witness in every frame, because S and T permute them up to phases and each is uniform over two or eight anyon types, so the entropy is 1 or 3 bits throughout. We then give a protocol that leaves the blind set: one logical string operator, controlled by an ancilla and applied at a rotation angle t≠π/4. In simulation, and in an exploratory robustness analysis, at least 42.5 % of all angles give a distance of at least 0.25 bit from the nearest integer; the budget is 200 to 300 flux samples (about 300 with a conservative decision rule). Because the target states are blind, even 1 % depolarizing noise produces false certificates at a few thousand samples unless the decision rule is protected; in our model the protected rule tolerates up to 6 % in the best case. One pre-registered prediction was refuted and one proved ill-posed; both are reported as such. A short chapter derives requirements for a photonic realization with qudits of dimension 8: announced state-independent loss is harmless, and state-dependent loss shifts the tested state only at second order. Scripts, result files, and the pre-registration are deposited with this paper. This record contains the manuscript, the 13 scripts (verification, analysis, and the figure), their result files, and the pre-registration with its deviations. The README explains how to re-run the scripts and which result files are reproduced bit for bit after normalizing line endings to LF (Python on Windows writes CRLF). Use of AI tools: In the research, processing, and writing of this paper and its results I worked together with generative AI tools, in practice a system of multiple coordinated AI instances that I set up and orchestrate (large language models, mainly Claude, by Anthropic, inside Claude Code). At their current context sizes I found it far more effective to work with several specialized instances, each with its own role and its own harness of rules and parameters that I designed and refined through feedback, than to load a single instance with all of the material; for my workflow that would have been inefficient, though this depends on the individual implementation. I lead this collaboration: I choose the research directions, set the goals, and make the final decisions in open exchange with the AI, learning actively as the work proceeds. The AI carries out the drafting, including the mathematical and technical parts, the numerical computation, and the literature search, under my direction. The AI works autonomously only task by task, within the structure I develop through feedback: it completes a task, and at open questions that need me it stops until the point is settled before the next step. Along the way I witness and take many of the decisions that shape the path, and it is common for me to spot things that need improvement. The work spans many separate runs, and a single simulation or build task alone can take up to an hour, so it could not happen all together in one autonomous run; and had I let the AI do all of it together alone, even if it is possible, it would no longer be my work but the AI's. I run multiple verifications at the different stages of the work and one before release, and all references are checked against the original sources. In the end what matters are human eyes, a principle that is itself written into the parameters of my system: I reach out to experts after publishing for review and feedback, so I learn what is solid and what must be corrected or falsified. My scripts for reproduction and review are released with this record. These tools are not authors; I am the author, and I take full responsibility for all scientific content and decisions leading to these results and their publication.