This work extends the theorem-level connection established in the Sigma Transition/Noether Same-Section Reviewer Package (doi:10.5281/zenodo.21639274) from the unweighted transition-incidence structure to the additional anisotropic rate metric measured on the same source-locked ⁸⁸Sr quantum-walk section. For a positive opposite-channel rate assignment (Jₓ, Jᵧ, Jₓ, Jᵧ), compression to the canonical rank-three Sigma augmentation carrier produces the exact spectrum Jₓ, Jᵧ, and (Jₓ + Jᵧ)/2. The prescribed source quarter-turn is an isometry if and only if Jₓ = Jᵧ, whereas the corresponding half-turn remains an isometry for every positive pair. Consequently, unequal measured axis rates do not invalidate the previously established unweighted C₄ same-section registration. Instead, they reduce the symmetry of the weighted carrier metric canonically from C₄ to C₂. An explicit rank-three metric realization, its edge-length classes, and a scale-independent anisotropy certificate are derived without fitting spatial directions, angles, auxiliary points, or laboratory coordinates. For the published Young et al. central tunnelling rates, Jₓ/ℏ = 2π × 178.1 Hz, Jᵧ/ℏ = 2π × 148.3 Hz, the resulting residual is Ωᵣₐₜₑ = 22201/1331712 ≈ 0.01667102196270665. Using only the reported marginal standard errors, the exact weighted-C₄ condition Jₓ = Jᵧ is incompatible with the central estimates by at least 149/11 standard-error units for every admissible correlation. The already registered finite transition-complex equivalence transports the supplied rate-induced Dirichlet form to the Sigma-labelled transition star. This closes the weighted application-form route once the experimental rate assignment has been registered. It also provides an exact falsification interface: the unweighted source symmetry, the weighted metric symmetry, and their respective residuals are mathematically distinct and can be tested independently. The accompanying reproducibility layer reconstructs the published central rates from the public analysis material and evaluates a disjoint even/odd repetition split of the official experimental record. The split supports conditional same-experiment reproducibility of the anisotropic rate shape. It is not an independent experimental confirmation, because it uses the same experiment and retains the published nearest-neighbour opposite-axis model. The principal significance is that the Sigma-to-experiment correspondence now survives a nontrivial weighted deformation: the experimentally registered transition section retains its exact source-generated combinatorial structure while its measured rate metric acquires a rigorously determined anisotropic normal form. The tunnelling rates Jₓ and Jᵧ enter this application as externally supplied effective-Hamiltonian characterization data; predicting their numerical values independently of the chosen experimental preparation is neither required nor intended here. The result does not identify the carrier with laboratory space, establish microscopic identity with the apparatus, exclude longer-range interactions, or constitute an independent experimental validation of the Sigma framework.
This work extends the theorem-level connection established in the Sigma Transition/Noether Same-Section Reviewer Package (doi:10.5281/zenodo.21639274) from the unweighted transition-incidence structure to the additional anisotropic rate metric measured on the same source-locked ⁸⁸Sr quantum-walk section. For a positive opposite-channel rate assignment (Jₓ, Jᵧ, Jₓ, Jᵧ), compression to the canonical rank-three Sigma augmentation carrier produces the exact spectrum Jₓ, Jᵧ, and (Jₓ + Jᵧ)/2. The prescribed source quarter-turn is an isometry if and only if Jₓ = Jᵧ, whereas the corresponding half-turn remains an isometry for every positive pair. Consequently, unequal measured axis rates do not invalidate the previously established unweighted C₄ same-section registration. Instead, they reduce the symmetry of the weighted carrier metric canonically from C₄ to C₂. An explicit rank-three metric realization, its edge-length classes, and a scale-independent anisotropy certificate are derived without fitting spatial directions, angles, auxiliary points, or laboratory coordinates. For the published Young et al. central tunnelling rates, Jₓ/ℏ = 2π × 178.1 Hz, Jᵧ/ℏ = 2π × 148.3 Hz, the resulting residual is Ωᵣₐₜₑ = 22201/1331712 ≈ 0.01667102196270665. Using only the reported marginal standard errors, the exact weighted-C₄ condition Jₓ = Jᵧ is incompatible with the central estimates by at least 149/11 standard-error units for every admissible correlation. The already registered finite transition-complex equivalence transports the supplied rate-induced Dirichlet form to the Sigma-labelled transition star. This closes the weighted application-form route once the experimental rate assignment has been registered. It also provides an exact falsification interface: the unweighted source symmetry, the weighted metric symmetry, and their respective residuals are mathematically distinct and can be tested independently. The accompanying reproducibility layer reconstructs the published central rates from the public analysis material and evaluates a disjoint even/odd repetition split of the official experimental record. The split supports conditional same-experiment reproducibility of the anisotropic rate shape. It is not an independent experimental confirmation, because it uses the same experiment and retains the published nearest-neighbour opposite-axis model. The principal significance is that the Sigma-to-experiment correspondence now survives a nontrivial weighted deformation: the experimentally registered transition section retains its exact source-generated combinatorial structure while its measured rate metric acquires a rigorously determined anisotropic normal form. The tunnelling rates Jₓ and Jᵧ enter this application as externally supplied effective-Hamiltonian characterization data; predicting their numerical values independently of the chosen experimental preparation is neither required nor intended here. The result does not identify the carrier with laboratory space, establish microscopic identity with the apparatus, exclude longer-range interactions, or constitute an independent experimental validation of the Sigma framework.
This work establishes the first theorem-level connection between a structure derived internally by the Sigma calculus and a published experimental quantum system. The Sigma calculus predicts a canonical finite transition structure generated by four equivalent source directions. Until now, this structure had been established only within the internal mathematical framework. Here it is shown that the complete structure is realized, at the level of unweighted nearest-neighbour transition incidence, in the published site-resolved (⁸⁸Sr) quantum-walk experiment of Young et al. The result is stronger than a visual or geometric similarity. An exact finite equivalence is constructed between the Sigma source complex and the experimentally documented transition section. This equivalence preserves the full outgoing transition star, admissible paths, face relations, discrete derivatives, and the associated Noether data. It can be verified by explicit finite matrix identities and therefore provides both a reproducible certificate and a clear criterion by which the proposed registration could be falsified. After the experimental transition structure is registered, the canonical Sigma projection produces a three-dimensional carrier whose normalized Gram matrix is exactly that of a regular tetrahedral simplex: its diagonal entries are (1), all off-diagonal entries are (-1/3), and the resulting simplex residual vanishes exactly. No spatial angles, geometric coordinates, external point cloud, or fitted reconstruction are introduced to obtain this result. The significance of this package is therefore that Sigma has crossed an important boundary: a nontrivial structure derived from its own finite transition and Noether principles is no longer confined to the abstract calculus but has been identified exactly within a source-locked section of real experimental data. The scope of the result is deliberately precise. It establishes equivalence for the published unweighted nearest-neighbour effective-transition section. It does not claim microscopic identity with the experimental apparatus, exact equality of tunnelling rates, reconstruction of laboratory space, or exclusion of additional longer-range interactions.
The slowly varying envelope (SVE) reduction of a gapped classical electromagnetic carrier predicts the single-particle Schrödinger equation as an exact identity plus a closed-form residual: the constrained vector MacCullagh carrier admits an exact per-branch balance whose retained part is the Schrödinger operator and whose sole marked approximation leaves a closed-form remainder, quadratic in the detuning. Under the single identification below, the carrier branch ℏω(k) is the relativistic energy function E(p) = m2 c4 + c2 p2 and the envelope generator ℏ[ω(k) − ωC ] is p its kinetic part E(p) − mc2 , recovered consistently in three cross-checking registers—the Foldy– Wouthuysen operator, the dispersion Taylor series, and direct numerics—with leading correction the fine-structure term −p4 /8m3 c2 . A parameter-free model reproduces the archived residual slopes to 10−4 , the fitted exponent converging to the predicted 2. One action-scale identification, ℏωC = mc2 , enters exactly once as an input; by Buckingham-π the reduction fixes the coefficient c2 /ωC = ω ′′ (0), matched to ℏ/m under the identification, and leaves one free scale, so ℏ is supplied, not derived. The result is a dynamical correspondence at the envelope stratum, not a derivation of quantum mechanics: |ψ|2 is normalized wave action, never a Born probability, and an explicit ledger fences what the correspondence forces, permits, and does not produce—the Born rule, state reduction, quantized charge, and entanglement—each with its standing theorem or structural fact. Companion no-go paper: "Constraining Classical Field Models of Self-Generated Particle Structure: Successive No-Go Theorems", DOI 10.5281/zenodo.21531712 (all versions). The two papers are co-posted and cross-cite by concept DOI. This record additionally carries the complete Track S/N/P code-and-data archive (envelope-code-v3.35.zip): symbolic-audit, numerical, and magnitude pipelines with their test suites (106, 20, and 13 tests), all figure-generating scripts, the campaign manifests (track_n_campaign_summary.json and companions), and dependency floors (code/requirements.txt). Version 2 (September 2026). Retitled ("a Gapped Electromagnetic Carrier") and substantially revised; 49 → 31 pages. Principal changes since v1: (i) the reduction theorem now carries explicit hypotheses — ψ₀ ∈ H^{s+4} via a bounded-multiplier argument, with a band-limit alternative and the H^s-weighted packet functional K_{4,s} (eighth-moment form) replacing the earlier spectral-scale shorthand; (ii) the residual analysis adds the finite-sweep bias law with a parameter-free model reproducing all archived window slopes to 1e-4, guarded by a committed test file; (iii) Sections V and VI are merged and the four scope tables consolidated into a single honesty-ledger table with explicit class codes; (iv) the Wolff placement is rebuilt as a three-horn trilemma mirroring the companion paper; (v) the conclusion is rewritten, situating the result against the independent kinematic (Zhang) and intrinsic-periodicity (Puddu) routes; (vi) the effective-medium reading of the gapped carrier is stated as a constant-coefficient effective field theory with no substrate assumed or determined; (vii) the carrier branch and the envelope generator are distinguished throughout, the companion dependence is stated results-based, and the long-time figure marks the Gaussian-packet breakdown estimate; (viii) all printed record DOIs now use the Zenodo concept (all-versions) DOIs. A complete change ledger (paper1_delta_v3.13_to_v3.35.pdf) accompany this version, together with the complete Track S/N/P code and data archive (envelope-code-v3.35.zip).
A partial-transpose image of the positive-semidefinite cone supplies an established alternative ordered presentation on a fixed composite carrier. Transporting both the cone and the processes preserves an abstract equivalence, whereas fixing the process representation can discriminate the two orders. We study the iteration of one specified right-argument transpose rule on supplied finite qubit carriers. With three standard leaves, its two bracketings have different cones, so no linear identification fixing every product operator pointwise can make them agree. Subset-orientation labels give an explicit repair. Relative to a separately supplied tensor-closed standard process category, we construct the raw tensor of arrows, associator, symmetry, asymmetric unit maps, and quasi-inverse tensorator, and prove symmetric monoidal equivalence by faithful transport. Complementary subsets define the same isolated cone, but their identification is not a congruence for the raw tensor product. At fixed rank, a precisely defined two-lift groupoid is a split central Z₂ extension; its odd lift is not an ordinary completely positive endomorphism at fixed representatives. A five-qubit example then separates matched reference transport, a fixed calibrated anchor, and an effect-based quotient. The latter loses a linear subspace rather than only a two-point transpose orbit and is preserved only by histories satisfying the standard kernel-invariance condition. A named real local unitary connects the anchor pair, while a real controlled-NOT (CNOT) gate exposes the limitation of an enlarged single-anchor history class. The contribution is an assumption-controlled analysis of one finite composition rule and its interfaces, not a new general coherence, orientation, or resource theorem, and not a derivation of quantum theory from primitive complementarity. Selected supporting statements and finite witnesses are accompanied by a partial Lean 4.19.0 formalization. A recorded local build and type/axiom audit accepted 117 theorem declarations and 57 definitions or abbreviations. The full symmetric monoidal category and equivalence in Theorems 4.5 and 4.6, and other explicitly listed manuscript links, remain incompletely formalized. The public source supplement also includes three preserved exact Python implementations, expected results, pinned dependencies and claim-to-code maps. Packaging and source/log audits are not another Lean compiler run or external peer review. The public evidence selection preserves the successful type/axiom output and the failure/success chronology. Machine-local account paths are omitted from derived run summaries, and one failed-build diagnostic has an explicitly normalized home-directory prefix. Accepted Lean proof sources remain unchanged; private review archives and original machine-local receipts are not public upload files.
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.
Physics may have mistaken a quotient for an origin. Quantum theory represents a preparation by a trace-one positive operator ρ. This is exact for relative outcome weights, but it is produced by the many-to-one map N(σ)=σ/Tr(σ), for which N(λσ)=N(σ) for every λ>0. If total response scale is inert under every admissible continuation, the identification loses nothing relevant. If one declared continuation can distinguish σ from λσ, however, normalisation has erased a future-relevant distinction. The trace-one state remains an exact terminal quantum reading, but it cannot by itself be a faithful future-sufficient carrier. The reason this order is easy to miss is structural: once the quotient is taken as the complete input, every downstream construction that factors through it is necessarily blind to the positive-ray scale already identified away. The mathematical ingredients used below are largely established and are not presented as isolated novelties. Jordan–von Neumann, Frobenius, and finite-dimensional Gleason–Busch-type results serve instead as consecutive links in a conditional dependency-order theorem. Equality of complete future-response profiles first fixes the minimal predictive quotient. Under explicit locks, positive refinement fixes square-root access; quadratic recomposition fixes a real inner-product carrier; and the stated return-algebra, positive-involution, and central exhaustive-phase locks then select its complex Hilbert representation. Positive additive effect weights acquire the unique pre-normalised trace form Wω(E)=Tr(σωE); and only after the non-zero total sω=Tr(σω) is closed does one obtain ρω=σω/sω and the Born reading. The familiar finite-dimensional Hilbert–trace–Born structure therefore reappears, but in a different dependency position. Two no-go results make this dependency claim exact within their stated domains: canonical ray normalisation cannot by itself faithfully carry active scale, and, in dimension at least two, a normalised invertible congruence is affine on the full state space only when its transport is scale-uniform. A common-profile theorem states when independently reconstructed readings are uniquely profile-equivalent over one fixed common future-profile image, rather than merely compatible descriptions. Under locks L0–L9 on one typed finite-dimensional assembly datum, these results form a single conditional genealogy; the paper neither derives quantum theory from NIR alone nor asserts that nature realises the locks. Its central result is not a new quantum rule. If two realised or admissibly generated representatives σ and λσ on the same positive ray, with σ≠0 and λ≠1, have different complete declared future-response profiles, no faithful representation of those profiles can factor solely through canonical ray normalisation N. The trace-one state is not wrong: it is the exact answer to a later, conditional question.
We construct an effective carrier model within the Worldline–Residue–Resource–Action framework. Two conditionally independent registers encode prime-factor multiplicities. A label-symmetric equality projector controls a common source coupled to two supplied particle-pair channels. One pulse generates the earlier collision-statistic probabilities without address-dependent interaction strength. A restricted single-register obstruction and a classification of diagonal permutation-invariant effects specify the construction’s resource assumptions. The implementation reproduces species probabilities, but not the preceding interface’s coherent output channel. Bare-energy conservation holds at the calibrated reference volume; detuning away from resonance requires interaction energy and switching work. A response identity relates the reference pressure-volume moment to the logarithmic-volume derivative of mean pulse work. Composition with the implemented muon-decay kernel recovers energy and pressure histories at full conversion. This is an executable effective interaction candidate and compatibility analysis; lepton identities, weak interactions, and the full fifteen-channel carrier are not derived from arithmetic alone. Version 0.2, editorial revision of 9 October 2026. Includes the English manuscript, Korean guide, and reproducibility archive. The 123 carrier-level checks are internal consistency checks, not experimental validation or external journal peer review. The manuscript was prepared before this deposit; its preparation-time statement that no public DOI had been assigned is superseded by this record. Preceding WRRA records: r11, https://doi.org/10.5281/zenodo.23237629; particle interface v0.2, https://doi.org/10.5281/zenodo.23241511; dynamic decays v0.2, https://doi.org/10.5281/zenodo.23246308. Correction release 0.2-r1 (9 October 2026): Table 3 now states that the hot electron branch has a positive pressure-volume moment; the antielectron-neutrino overbar in Section 6.2 is correctly rendered in the PDF. The scientific model, reproduction code and numerical results are unchanged. The manuscript retains its v0.2 designation and filenames; 0.2-r1 identifies this corrected deposit. A third Astra medium AI internal review and post-correction check examined the final files. Both required corrections were resolved, with no further mandatory technical corrections identified. The review report is included. This is not external journal peer review or acceptance. Earlier statements in the guide/review about two review rounds describe the historical pre-correction state.