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10, so alpha_stratum(2,5) = 5; a fresh 60k ensemble's calibrated read is consistent with 5, with alpha = 10 (the superseded single-wall count's prediction) excluded at p = 0.000 and demonstrated power ~ 1; and a stratum-conditional paired ladder tracks the exact rank-10 Hessian form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law m_C(2,d) = 2(d-1) is census-exact at d = 3, 4, 5, now including the first point where it diverges (by 2x) from the superseded count.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.
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.
Two walls stand at the edge of the two-ring instrument, and this paper is the map of both. Many qubits first. One setting reads two amplitudes and N qubits carry 2^N, and that wall is the only thing between the instrument and many-body quantum mechanics. Part I is the map of the attempt to scale through it, chains of certified rings in lockstep, each qubit a setting with its digit certificate, entanglement between neighbours as the shared-radius coupling of the companion paper, local turn laws carrying the gates, a geometric form of the tensor networks that are classical physics' own escape from the same wall. What carries over is stated exactly and verified. Across every cut of a chain the two sides share one spectrum, so the cut's radius is its certificate. An eight-link chain built by one layer of levers holds at most one bit per cut and is held by a chain of bond two to a fidelity of 1.000, and a random eight-qubit state has rank 16 at the middle cut and is held to 0.162, the wall exhibited. The move past it is the nesting of the companion repair paper, a ring inside every cell, done here as a setting inside every share. A setting whose two shares are themselves settings, applied down to single qubits, holds any pure state of N qubits exactly, 2^N − 1 settings for 2^N amplitudes, the standard's own count, verified on random states at every size to eight qubits and on the same random state the chain held to 0.162, with every setting on the ladder. At a node the entanglement of that qubit with the rest is sin θ √(1 − |⟨ψ₀|ψ₁⟩|²), the Wootters concurrence for a pair, largest at equal shares over mirror branches. A Z turn on qubit k turns the azimuths at depth k and nothing else, and a measurement reads the root's shares and leaves its branch. On ibm_kingston a random three-qubit state was prepared from its seven settings as the machine's own gate angles and its populations read within 0.048 of the tower's shares. The contraction that turns the chain into a reading is classical arithmetic, and the ladder guard of the companion paper stands on it, 112 fixed-point words carrying their base-60 digit pairs, 200 random corruptions of one to six bits, every one caught from the shadow alone and every word recomputed, while the unguarded contraction returned a wrong energy in 155 of the 200. The five problems of the chain program are each settled where they stand: the tower is the canonical form, a gate on any qubit needs no route through its neighbours, the contraction's cost is measured, the classes the tower compresses are counted, and the width a chain needs to follow the dynamics is the entanglement across its cuts, measured. Three levels second. A pure qutrit carries four real parameters and one setting reads two lengths. The populations are read by two levers, one inside the other, the tower of Part I at its smallest: the first splits level 0 from the rest and the second splits the rest between levels 1 and 2, so every split is reachable, a state fully in one level included. Two certificates ride them, the sum to one and the product law, the coherence between two levels at most the product of their two amplitudes and equal to it for a pure state, and on a thousand random qutrits every one-percent misread of the populations is caught by one of the two, 1,000 of 1,000, 992 of them by the sum alone. The threefold turn a qutrit wants is not a single-digit multiplier, and that is a theorem, not bad luck. The Pisano period over the rank of apparition is 1, 2 or 4 (Vinson 1963), confirmed by enumeration to modulus 250. But the sixty-ring carries the threefold turn as a two-term law, F(n + 20) ≡ 5F(n + 1) + F(n) (mod 10) at all sixty positions, and 178 moduli below 250 carry the same law at a third of their period. The two relative phases are read as well, by three two-level settings, one for each pair of levels, whose azimuths obey the addition law exactly, the third certified by the other two, so that three populations and two certified phases rebuild every pure qutrit, verified on a thousand. Every number is verified by the script in the appendix, and the document is not built unless every check prints True.
CORRECTION NOTICE (v5, 2026-09-06). The central result of v1-v4 is WITHDRAWN. This record has not been peer reviewed. Issued 2026-09-06 by the author (v5 of this record). This notice was added to the archived record; the manuscript itself is unaltered. This record has not been peer reviewed. What is wrong The force exponent reported here as alpha = 3.92 +/- 0.16 is an artifact of the fitting procedure, not a property of the data. The fit was unweighted least squares in linear space applied to a steeply falling profile, which places 95.8 percent of its objective on the single point at the shortest retained separation. Re-fitting the same data with nothing discarded, changing only the error model, gives 2.82 (relative-error weighting) or 2.89 (log-space regression). Across fifteen combinations of standard estimator and fit window the exponent spans 2.53 to 3.92, and the value published here is the largest of them. What the data show instead, and how firmly The retraction is certain. The replacement is not. On the open 24-site chain the same data give alpha = 2.6 +/- 0.2, where the interval is the spread of a plateau in the local logarithmic slope over six consecutive separations, not a statistical error (the bootstrap error on the published fit to the same data is 0.63); the plateau value was reproduced at 2.5921 by a separately written implementation run on the same recorded state (SIM-2026-073). This is a single geometry at a single size. On a 24-site periodic ring, analysed with chord distance as pre-registered, two probes that are both energy-density-like (Delta = 1) give 1.83 and 2.37 against a CFT two-point value of 2 (SIM-2026-074): one size, not converged, disagreeing with each other by 29 percent, and not attributable to a single operator channel. No corrected exponent is established. The exchange channel is an open question. Treat 2.6 +/- 0.2 as the provisional open-chain value, not as a result. Also withdrawn or voided alpha = 3.92 +/- 0.16 (24 sites) and alpha = 3.97 +/- 0.15 (16 sites, exact) from stress-tensor exchange. Was: headline result and its first corroboration. Now: withdrawn. Why: the same estimator artifact: both move in lockstep under it (3.92 to 2.56, 3.97 to 2.75 on dropping the contact point), because they share one observable, one fit window and one fitting routine. scalar exchange (alpha = 2) excluded. Was: stated as excluded at about 12 sigma. Now: withdrawn. Why: the 12 sigma used the covariance error 0.16; at the bootstrap error 0.63 computed before release the exclusion is about 2.2 sigma, and under any window starting at separation 3 the point estimate itself falls to about 2.56. alpha = 4 is universal; alpha = 4.00 at Ising, tricritical Ising and Potts. Was: confirmed at three classes. Now: withdrawn; the operator towers at all three classes stand. Why: the exponent was measured only at Ising and inherited by the other two; the towers are spectra, not forces. third independent evaluation alpha = 4.07 (chi = 3 MERA), and with it the statement that this was the first holographic force measurement from an optimized tensor network state. Was: supporting evidence and a priority claim. Now: both withdrawn. Why: the profile changes sign inside its own fit window and cannot be a fixed-sign power law; the priority claim rests on that evaluation alone and was never gated against prior work. the discriminating test: stress tensor (Z_j Z_j+1) versus spin primary (X_j). Was: registered killer test. Now: void. Why: at the critical point these are one operator, related by exact Kramers-Wannier self-duality. Measured on the periodic ring: identical expectation values to 1e-15 and identical force profiles to 1e-13 at 14 sites, 2.4e-9 at 24 sites (SIM-2026-074). Open boundaries break the duality, which is why the open-chain run of this test (SIM-2026-073) showed a spurious difference between the two arms. The test compared the energy density to itself. Neither label was correct: Z_j Z_j+1 is the energy density, not the stress tensor; X_j is its dual, not the spin primary. transport time T = (pi/2) l_AdS to 1.2 percent as a geometric prediction. Was: confirmed. Now: measurement stands (T = 2.239 at N = 16); geometric reading not established. Why: l_AdS cancels in the cited derivation, which also gives pi rather than the pi/2 the manuscript compares against; the chi = 3 estimate of l_AdS gives a 57 percent miss; the transport driver never applies a MERA. The true spin primary Z_j was never in the design, and the direct replacement test is blocked: Z_j is odd under the Z2 symmetry, so the energy-difference observable used throughout this work subtracts across symmetry sectors and is undefined for it. A replacement test using two Z2-even probes has been pre-registered (PREREGISTRATION-T.md, committed before its driver) and run; its result is deliberately not part of this correction and will appear in a later version. Not previously reported The published fit has a reduced chi-squared of 6.008 on 9 degrees of freedom, and a bootstrap standard deviation of 0.63 rather than the quoted 0.16. Both were computed before release and omitted. What is unaffected Of the claims in this record: the two-phase optimization method, the chi = 3 criticality threshold, the exact-diagonalization gates, the measured OPE coefficient C_sigma-sigma-epsilon = 0.50 +/- 0.03 against the exact 1/2, and the conformal towers at all three classes. Not part of this record, mentioned only because they are sometimes read together with it: the later C_TTT stress-tensor self-coupling measurements (SIM-2026-067/068/069, dated after this record) and the matched off-critical ring control (SIM-2026-074) are unaffected by this correction and make no claim about the force exponent. How the error was found and what this record contains The error was found by the author's own internal adversarial audit, not by a reviewer; the journal submission it was found after is closed, and no review of this work has taken place. Files in this record: the full re-analysis (SIM-2026-072.json, driver, figure), the pre-registered chain test (SIM-2026-073.json) and ring test (SIM-2026-074.json) with both pre-registrations as committed before the runs, the runnable drivers with their shared module and the negative test suite for their validation gates, and the error register (ERRATA.md). Concept DOI 10.5281/zenodo.19660302 always resolves to the current version. The abstract of the original manuscript follows unchanged, for the record. Every claim in it about the force exponent should be read against the correction above. We report results from classical simulation of critical spin chains that establish quantitative physical content in the MERA-AdS correspondence. First, the interaction energy between boundary-primary excitations follows a power law F(dr) ~ dr^(-alpha) with three consistent evaluations yielding alpha = 3.92 +/- 0.16 (24q), 3.97 +/- 0.15 (16q), and 4.07 +/- 0.18 (16q chi=3 MERA), all consistent with stress-tensor exchange (alpha = 2*Delta_T = 4) and inconsistent with scalar exchange (alpha = 2); the exponent is explained by the Ising CFT fusion rule eps x eps = [1]. Because Delta_T = 2 is fixed by conformal symmetry in every unitary 2D CFT, alpha = 4 is universal: we confirm this at two further fixed points - the tricritical Ising model (c = 7/10), where a scale-invariant MERA recovers the full M(4,5) primary spectrum (Delta_sigma = 0.092, Delta_eps = 0.204, Delta_T = 1.999), and the three-state Potts model (c = 4/5), where a Z3-symmetric MERA at chi = 12 recovers Delta_sigma = 0.134, Delta_eps = 0.805, Delta_T = 2.001 - giving alpha = 4.00 at all three universality classes. Second, the holographic transport time matches the AdS_3 null geodesic prediction T = (pi/2) l_AdS to 1.2% at N = 16 sites on a time-grid-converged measurement, with the residual shrinking from 2.8% at N = 12. Third, we introduce a two-phase optimization method (ALM + Z2 projection) for Z2-symmetric MERA ground states at 24 qubits. Fourth, extending the MERA bond dimension from chi=2 to chi=3 overcomes the expressibility barrier, achieving 0.15% energy error at 16 qubits and enabling the first holographic force measurement from an optimized tensor network state. The OPE coefficient is also measured directly from the fixed-point tensors: C_sigma-sigma-eps = 0.50 +/- 0.03 against the exact Ising value 1/2. A hardware path is designed for Quantinuum Helios (57 CNOTs, 0.57% logical error). Version 4 changes (relative to v3): Transport: N=16 residual restated as the time-grid-converged value of -1.21% (dt = 1e-3), with -2.78% at N=12 showing the finite-size trend; the 0.72% figure in v3 was an artifact of the dt = 0.05 integration grid and is now labeled as such. The N=24 transport point carries a new reliability caveat (double-lobe trace structure). The l_AdS comparison table is recomputed against the converged transport time (best fit 1.425, deviation 1.21%). Effective-mass derivation rewritten in a consistent AdS3 convention m^2 l^2 = Delta(Delta-2); m_eff = 0.693 = 1/l_AdS follows from Delta = 1 exactly. NEW: universality section extended to a third fixed point - a Z3-symmetric MERA at chi = 12 recovers the three-state Potts (c = 4/5) tower, giving alpha = 4.00 at all three universality classes (c = 1/2, 7/10, 4/5). NEW: the OPE coefficient C_sigma-sigma-eps = 0.50 +/- 0.03 measured directly from the fixed-point tensors, with C_sigma-sigma-sigma = 0 at machine precision as the Z2 structural check. Supplemental table row SIM-007 annotated (record lost; script preserved); data availability expanded; Zenodo DOI added to the front matter.
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.
Reproducibility note and two wording corrections (28 October 2026). The note’s appendix promises check_dissipation_law.py, which version 1 did not carry. It has been written and run: 109 checks, 109 pass, with 53 derived from the C60 adjacency alone, 45 from the stated premises, 6 internal-consistency, 4 Monte-Carlo and 1 using a reconstructed input. The MEASURED set is empty, and that is itself a check: the note’s own flow-position paragraph says it compares no kernel number with measured data, and that was confirmed rather than assumed. Definitions are tagged as such and are never counted as derivations. Every entry of Table 1 and every number in §3, §4, §5.3, §6.1 and §6.2 reproduces. 1. §3.1’s “fails by at least 19%” is a property of one draw, not a bound. On the setup described, the smallest control deviation is 40.6%, so the floor holds there comfortably. But across 500 random cases (25 draws × 5 β × 4 t) the minimum falls to 10.1%, with 0.6% of cases below 19%. The substantive content is far stronger than the stated floor and should be given instead: every one of the 500 control cases exceeds its own KMS error by a factor of 3.5×10¹³ or more. Suggested wording: “fails by factors of 10¹³ or more above the machine-precision residual.” Relatedly, §3.1’s “relative error 3×10⁻¹⁵” is the typical value, not an attainable bound: the median over 500 cases is 3.23×10⁻¹⁵, the 90th percentile 8.1×10⁻¹⁵, the maximum 2.2×10⁻¹³. 2. §6.2’s “Stratonovich simulation” label does not fit the reported figure. The number itself is right — the theory value 19.44 and the closed form are exact and confirmed — but a direct Stratonovich–Heun integral of the locked case carries a large O(dt) bias, because the phase-difference modes relax at rate ~K = 30: dt = 0.002 gives 14.94 and dt = 0.001 gives 17.16, so reaching the reported 0.15% would need dt ≈ 10⁻⁵, far beyond what “N = 6, D = 0.25, K = 30, 400 replicas” implies. The exact Itō form of the same quantity, D⁻¹(⟨ΣFi²⟩ + D⟨∇·F⟩), carries no stochastic-integral bias and gives 19.4651 at dt = 0.001, converging to theory as dt falls. So the estimator, not the result, is what the label misnames. Separately, the ω list is never given, so the printed simulated triple is not reproducible as such; the printed theory triple is internally consistent (24.42 − 19.44 = 4.98 exactly) and is reproduced exactly from a reconstructed ω with mean 0.9 and population variance 0.2075. The note’s “var” is unambiguously the population variance — the identity fails for the sample variance. 3. §1’s “the C60 spectral values 1, 4−φ and φ” is loose on the third. 1 is an eigenvalue of A with multiplicity 9; 4−φ is an eigenvalue of L = 3I − A with multiplicity 5, since 3 − 1/φ = 4 − φ; but φ is an eigenvalue of neither (nearest level 0.0565 away in A, 0.1796 in L). Only −φ is, with multiplicity 5. The note says so itself two paragraphs later, so this is internal looseness of wording rather than a numerical error, and the product identity (4−φ)·φ = 3φ−1 = 3.854101966249685 is exact. Two results are worth naming because they are stronger than stated. Aut(A) ≅ Ih is confirmed structurally, not merely by order: order 120, 31 involutions, centre of order 2 and no element of order 4, which pins A₅ × C₂ and rules out S₅ (25 involutions, trivial centre). And in §6.1 the garbled line is ΔSsys = −TC₁, the total correlation: the system’s entropy falls by exactly the correlation that forms, verified symbolically, with C/8 = H(p₁) exactly — which is why the printed closed form for (E/S)op holds. Ψop was rebuilt from ledger quantities alone, with no A, B or C in it, and agrees with the printed form to 2.5×10⁻¹⁷⁵. A scope and status note on the coherence dissipation law. Separates what the C60 adjacency forces from what it forbids, and states the open-system form that survives after the bare coefficient was retired in the record. The law in every surviving form is posited, not forced, and the paper says so wherever it matters. Includes the two-ledger theorem under stated premises, a no-go result limited to coherence functionals built from the adjacency, and a verification script.
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.