Symmetry-Resolved Johnson-Harmonic Profiles of Fixed-Order Quantum Correlator Fields: Two Experimental Reanalyses

Paper-5 develops a reproducible symmetry-resolved method for analyzing the internal structure of fixed-order many-body quantum correlator fields using Johnson-harmonic decomposition. Instead of compressing a large collection of labelled correlators into a single overall number, the method keeps the subset structure of the data and separates the centered correlator field into different harmonic layers. This makes it possible to see how correlations are organized inside a quantum system, not only how large the total correlation is. J742_Paper5_Main_Manuscript_DOI… This work is part of the broader QDL five-paper progression: Paper-1 — Exact Permutation Moments: develops exact finite-sample mean, variance, and covariance formulas for Walsh interaction-order energies under permutation. Paper-2 — Exact Distribution Laws: extends the framework to structured exact distributions, Krawtchouk-based reductions, computational-complexity boundaries, and continuation-horizon classification. Paper-3 — Exact Certified Tail Computation: develops certified deterministic counting methods for difficult exact tail probabilities in a structured finite Walsh/Clebsch problem. Paper-4 — Identifiability, Contrast Closure, and Frozen External Testing: studies restricted four-variable log-linear interaction models, rank, identifiability, contrast closure, and a frozen external statistical test. Paper-5 — Johnson-Harmonic Correlator Profiles: applies a symmetry-resolved framework to complete labelled quantum correlator fields and studies how their internal correlation structure is distributed across Johnson-harmonic layers. The central challenge in Paper-5 is that a many-body experiment can produce hundreds or thousands of labelled correlators. If these are reduced to one aggregate norm, important structural information can be lost. Paper-5 preserves the full labelled subset geometry and resolves the correlator field into interpretable harmonic components. What Paper-5 does: reconstructs complete fixed-order labelled correlator fields from public experimental data; removes the common-mode component and decomposes the remaining structure into Johnson-harmonic layers; computes layer energies and normalized structural statistics; uses exact finite-population projector calibration; preserves subset-label information instead of reducing everything to a single total norm; propagates uncertainty using source-native randomized experimental replicates; applies an exact finite-shot bias correction to quadratic layer energies; compares the physical signal with a source-defined noise-reference cohort; provides a reproducible computational route through QDL Research Suite v1.3.0. The mathematical Johnson/slice decomposition itself is established mathematics; the contribution of Paper-5 is the combined experimental-analysis workflow and its application to two independent public quantum datasets. J742_Paper5_Main_Manuscript_DOI… Main locked results: For the six-qubit tomography dataset, each setting contains 20 three-body correlators with harmonic dimensions 20 = 1 + 5 + 9 + 5. Across 28 source measurement settings, the pooled centered correlation structure is: H1 = 48.4133% H2 = 36.7277% H3 = 14.8590% This shows a distributed harmonic profile rather than concentration in a single layer. J742_Paper5_Main_Manuscript_DOI… For the independent 32-qubit hardware dataset, each seed contains 4,960 three-body correlators, decomposed as 4,960 = 1 + 31 + 464 + 4,464. Across nine frozen seeds, the pooled profile is: H1 = 0.341937% H2 = 0.123369% H3 = 99.534694% showing an extremely strong H3-dominant structure. J742_Paper5_Main_Manuscript_DOI… The H3-dominant result remains stable across 675 source-randomized physical twirls. A seed-blocked bootstrap gives a pooled H3 95% interval of approximately: 99.5301% – 99.5388%. J742_Paper5_Main_Manuscript_DOI… After exact finite-shot correction, the pooled H3 fraction changes only from approximately 99.53469% → 99.53486%, showing that ordinary finite-shot sampling bias is extremely small relative to the observed structure. J742_Paper5_Main_Manuscript_DOI… A source-defined noise-renormalization reference cohort gives the opposite profile: Physical corrected H3: 99.534860% Noise-reference corrected H3: 0.138145% providing a strong diagnostic contrast between the physical cohort and the source-defined reference data. J742_Paper5_Main_Manuscript_DOI… How Paper-5 connects to the earlier papers:Paper-1 provides exact permutation-moment tools. Paper-2 studies structured exact distributions. Paper-3 develops certified exact-tail computation. Paper-4 focuses on identifiability and restricted interaction models. Paper-5 moves to a different but related application layer: it uses structured mathematical tools to analyze the internal organization of many-body quantum correlation fields. Why it is useful:The method is useful when a quantum experiment produces many labelled multi-qubit or many-body correlators and researchers want to understand where the structure is located inside that large correlation field. Potential applications include: quantum-state tomography; randomized-measurement analysis; many-body quantum experiments; multi-qubit correlation analysis; quantum-hardware characterization; comparison of correlation structures across devices or experiments; finite-shot robustness checks; structured analysis of subset-indexed data; reproducibility and independent verification of quantum experiments. Rather than asking only: “How much correlation is present?” Paper-5 also asks: “How is that correlation organized across different symmetry layers?” This makes the method useful as a structure scanner for complex quantum correlation data. Reproducibility:The computational implementation for Paper-5 is included in QDL Research Suite v1.3.0, DOI 10.5281/zenodo.22972830. The software provides the Paper-5 computational reproduction route while preserving the original public datasets as the source authority. J742_Paper5_Main_Manuscript_DOI… Paper-5 Preprint DOI:10.5281/zenodo.22974172 J742_Paper5_Main_Manuscript_DOI… Scope:Paper-5 provides a structural and statistical diagnostic workflow. The Johnson decomposition itself is established mathematics. The paper does not claim a new universal Johnson theorem, Bell nonlocality, neutrino thermalization from the H3 statistic, a new physical mechanism, causation, retrocausality, future-state prediction, or a nonzero physical intervention parameter. J742_Paper5_Main_Manuscript_DOI… In simple terms:Paper-5 takes a large set of quantum correlations and breaks it into meaningful symmetry layers. Instead of giving only one total correlation number, it shows where the correlation structure lives inside the data. In the six-qubit experiment, the structure is spread across several layers. In the 32-qubit dataset, almost all of the centered structure appears in the highest Johnson layer, and this pattern remains stable after randomized-twirl analysis and finite-shot correction. In one sentence:Paper-5 is a reproducible structure-analysis framework for complex quantum correlation data that shows not only how much correlation exists, but how that correlation is organized inside the system.