Mechanism-Diagnostic Geometric Constraint on Spin-Fluctuation Superconductors: A Y-Space Analysis of Coherence, Energy Scale, and Fermi Velocity

Author: Adam Murphy, Independent Researcher, Latah, Washington
Contact: impactme.ai | theoryofeverything.ai
Date: March 27, 2026
Version: 2.1 (revised after multi-agent AI review + GPT-4 secondary review)

AI Assistance Acknowledgment: This work was developed in collaboration with Claude (Anthropic), GPT-4 (OpenAI), and Gemini (Google). AI systems contributed to data analysis, statistical computation, circularity auditing, and manuscript preparation. All scientific judgments and interpretations are the author's.

Revision Note: Version 1.0 was reviewed by a 10-specialist, 6-model AI panel (theoryofeverything.ai review system, March 27 2026). Three of four math specialists independently identified a dimensional inconsistency in the definition of Y; one specialist identified a slope inconsistency between Sections 5 and 6. This revision corrects all identified issues. A complete list of changes appears in Appendix C.

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Abstract

We report a mechanism-diagnostic geometric constraint among independently measured properties of superconductors: transition temperature Tc, bosonic glue energy scale θ_glue, and bare Fermi velocity vF*. The quantity Y ≡ ln(Tc/θ_glue) − ln(vF*/v₀), where v₀ = 10⁶ m/s is a fixed reference velocity, separates spin-fluctuation (SF) from phonon-mediated (PH) superconductors with perfect rank separation (Mann-Whitney U = 55, p = 0.0005, rank-biserial r = −1.0, Cohen's d = 2.6, N = 16). SF materials cluster near Y ≈ 0 while PH materials have Y ≈ −3.4 in this convention. We emphasize that Y is defined relative to a chosen velocity normalization; the SF/PH separation, ranking, and all relative comparisons are invariant under rescaling of v₀, but the absolute location of the "balance point" is convention-dependent. Within SF materials, a suggestive positive trend between Y and ln(ξ/a) (OLS p = 0.017, N = 11) does not survive removal of a single high-leverage point (CeCoIn₅) and is not confirmed by robust regression or rank-based tests. We document a shared-variable inflation in an earlier R-space formulation and present Y-space as the corrected primary result.

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1. Introduction

1.1 The Question

Superconductors span an extraordinary range of energy scales, crystal structures, and pairing mechanisms. Despite this diversity, the basic phenomenon — zero-resistance current flow via Cooper pairing — is universal. A natural question is whether any constraint connects the fundamental parameters of superconductivity across mechanism families.

1.2 Our Approach

We compile a dataset of 16 superconductors spanning five mechanism families (heavy fermion, cuprate, pnictide, ruthenate, and phonon-mediated) with independently measured values for Tc, θ_glue, coherence length ξ, lattice constant a, and bare Fermi velocity vF*. We define a diagnostic quantity Y and show that it acts as a mechanism separator: spin-fluctuation superconductors cluster in a distinct region of Y-space from phonon-mediated superconductors.

1.3 Transparency Note

This analysis evolved through multiple iterations. An earlier formulation in "R-space" showed a strong linear correlation that was subsequently found to be inflated by a shared variable. A first draft of the Y-space paper (v1.0) contained a dimensional inconsistency in the definition of Y and an arithmetic inconsistency between reported regression slopes. Both were caught by AI multi-agent review and are corrected here. We document these issues in Sections 6 and Appendix C for methodological transparency.

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2. Dataset and Methods

2.1 Material Selection

We include materials meeting three criteria: (1) Tc well-established, (2) θ_glue identified with a specific bosonic mode (spin resonance for SF, phonon spectrum for PH), and (3) bare Fermi velocity measured by a technique that does not derive vF from ξ or Tc (ARPES, de Haas–van Alphen, or DFT band calculations with independent validation). This last criterion is essential to avoid circularity.

Two materials carry provenance flags: UBe₁₃ (vF from dHvA estimate) and LaFeAsO (vF from DFT). We verify that excluding these does not change the qualitative findings (Section 4.3).

Note on mechanism labels: The SF and PH classifications used throughout this paper are externally assigned benchmark categories imported from the prior literature, not outputs inferred by our analysis. The present study tests whether Y tracks accepted mechanism categories; it does not independently establish mechanism assignment.

2.2 Parameter Conventions

• Coherence length: ξ_ab from upper critical field Bc2∥c for layered materials, via ξ = √(Φ₀ / 2πBc2), where Bc2 is in tesla and Φ₀ is the flux quantum.
• Lattice constant: In-plane a for layered systems.
• Operational Fermi velocity (vF):* In this paper, vF* denotes the highest-confidence proxy for the least-renormalized Fermi velocity available for a material, selected from ARPES, dHvA, or validated band-structure sources, subject to the rule that it must not be derived from Tc or ξ. It is therefore an operational comparator across materials, not a claim of identical microscopic meaning across all probes. We test sensitivity to probe type in Section 3.5.
• θ_glue: Spin resonance energy for SF materials; θ_D or ω_log for PH materials.

2.3 Source Citations

†vF provenance flagged. Full bibliographic references in preparation.

2.4 Dataset

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3. The Y-Space Separation

3.1 Definition and Dimensional Convention

We define:

Y ≡ ln(Tc/θ_glue) − ln(vF/v₀)*

where v₀ = 10⁶ m/s is a fixed reference velocity. Both arguments of the logarithms are dimensionless: Tc/θ_glue is a ratio of temperatures, and vF*/v₀ is a ratio of velocities. Y is therefore a well-defined dimensionless quantity.

The choice of v₀ = 10⁶ m/s places the SF population near Y ≈ 0 in this convention. A different choice of v₀ would shift all Y values by the same additive constant ln(v₀'/v₀), preserving all relative comparisons, rank orderings, and statistical tests, but moving the absolute location of the "balance point." We adopt v₀ = 10⁶ m/s throughout and note that the physical content lies in the relative structure of Y across materials, not in the absolute numerical value of any single Y.

Equivalently, Y = ln[Tc/(θ_glue × vF*/v₀)]. The condition Y = 0 in our convention corresponds to Tc/θ_glue = vF*/v₀, or Tc = θ_glue × (vF*/v₀).

3.2 SF/PH Separation

The 11 SF materials have mean Y = 0.07 ± 0.87 (SD). The 5 PH materials have mean Y = −3.38 ± 2.03 (SD).

Statistical details: Mann-Whitney U = 55 (maximum possible for N₁=11, N₂=5), two-sided exact p = 0.0005. Rank-biserial correlation r = −1.0, meaning every SF material has higher Y than every PH material — perfect rank separation. Cohen's d = 2.6 (using pooled SD = 1.31). We note that the pooled-variance calculation is sensitive to the extreme PH outlier Al (Y = −6.59); excluding Al gives pooled SD = 1.10 and Cohen's d = 3.1.

This separation is invariant under change of reference velocity v₀, since all Y values shift by the same constant.

3.3 Physical Interpretation

The ratio e^Y = Tc/(θ_glue × vF*/v₀) compares the transition temperature to a composite scale set by the glue energy and electron speed. In our convention (v₀ = 10⁶ m/s), SF materials have e^Y ≈ 1 while PH materials have e^Y ≈ 0.03. This can be described loosely as a difference in how much of the available energy budget materializes as Tc, but we emphasize that e^Y is an empirical characterization, not a derived thermodynamic efficiency. Its relationship to established coupling parameters (e.g., the dimensionless coupling constant λ, or the Eliashberg function) is not established by the present analysis. The key empirical result is the separation between mechanism classes, not the absolute numerical location of either class on the Y axis.

3.4 Y Values

| Material | Y | |Y| | e^Y | Family | Rank by |Y| | |----------|---|-----|-----|--------|---------| | YBCO | −0.016 | 0.016 | 0.984 | Cuprate | 1 | | UPt₃ | +0.018 | 0.018 | 1.018 | Heavy Fermion | 2 | | BaKFe₂As₂ | −0.051 | 0.051 | 0.950 | Pnictide | 3 | | BSCCO | +0.262 | 0.262 | 1.300 | Cuprate | 4 | | Tl-2201 | −0.288 | 0.288 | 0.750 | Cuprate | 5 | | UBe₁₃ | +0.405 | 0.405 | 1.500 | Heavy Fermion | 6 | | CeCu₂Si₂ | +0.560 | 0.560 | 1.750 | Heavy Fermion | 7 | | LSCO | −0.611 | 0.611 | 0.543 | Cuprate | 8 | | Sr₂RuO₄ | −0.693 | 0.693 | 0.500 | Ruthenate | 9 | | LaFeAsO | −1.059 | 1.059 | 0.347 | Pnictide | 10 | | CeCoIn₅ | +2.239 | 2.239 | 9.388 | Heavy Fermion | 11 | | — SF mean — | 0.070 | — | — | — | — | | Nb | −3.705 | 3.705 | 0.025 | Phonon | — | | Pb | −3.284 | 3.284 | 0.037 | Phonon | — | | Al | −6.585 | 6.585 | 0.001 | Phonon | — | | MgB₂ | −2.040 | 2.040 | 0.130 | Phonon | — | | H₃S | −1.307 | 1.307 | 0.271 | Phonon | — | | — PH mean — | −3.384 | — | — | — | — |

3.5 Probe Homogeneity Tests

To test whether the SF/PH separation is sensitive to the mixture of velocity measurement techniques, we repeat the analysis on restricted subsets:

Test A (ARPES-only SF): Restricting to the 7 SF materials with ARPES-derived vF*, the separation from all 5 PH materials is preserved: Mann-Whitney U = 35, p = 0.0025, with perfect rank separation maintained.

Test B (Experimental-only): Excluding all DFT-derived and estimate-level vF* sources (removing UBe₁₃, LaFeAsO from SF; removing Nb, Al, MgB₂, H₃S from PH) leaves 9 SF vs 1 PH (Pb). The separation is preserved but the PH sample is too small for a meaningful test. This underscores that expanding the PH dataset with experimentally measured vF* values is a priority.

The class separation is not an artifact of mixing measurement probes.

3.6 Permutation and Bootstrap Analysis

Exact permutation test: Of the 4,368 distinct ways to assign 11 SF and 5 PH labels to our 16 materials, only 21 permutations produce a median difference as large as or larger than observed. Exact two-sided permutation p = 0.0048. The separation is not driven by distributional assumptions.

Bootstrap confidence interval: The 95% bootstrap CI for the difference in medians (10,000 resamples) is [1.29, 6.57]. Zero is far outside this interval.

Leave-one-out class stability: Removing any single material from the dataset preserves perfect rank separation between SF and PH (all LOO Mann-Whitney p < 0.002). The separation is not driven by any individual material.

3.7 Baseline Comparison: Y vs Simpler Diagnostics

A natural question is whether the composite Y separates mechanism classes better than its individual components.

The bare Fermi velocity alone achieves perfect rank separation (all SF materials have vF* < 0.30, all PH materials have vF* ≥ 0.50). The energy ratio Tc/θ_glue alone does not separate the classes: MgB₂ (Tc/θ = 0.065) and H₃S (Tc/θ = 0.135) overlap with SF materials like Sr₂RuO₄ (Tc/θ = 0.030) and LaFeAsO (Tc/θ = 0.104).

Y achieves the strongest statistical separation (highest U, lowest p) because it combines the partial information from both components. However, the honest statement is that vF* does most of the separating work. The added value of the energy ratio component is to tighten the separation and provide physical interpretability linking Tc, glue scale, and electronic structure in a single coordinate. Whether this added interpretability justifies the composite over vF* alone is a judgment call that larger datasets would help resolve.

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4. The SF Y-Region (Convention-Dependent)

4.1 Which Materials Cluster Near the SF Center?

In our convention (v₀ = 10⁶ m/s), the three materials closest to the SF cluster center (Y ≈ 0.07) are YBCO (Y = −0.016), UPt₃ (Y = +0.018), and BaKFe₂As₂ (Y = −0.051). These happen to be canonical representatives of three distinct SF families. The empirically striking proximity of materials spanning two orders of magnitude in Tc (0.56 K to 93 K) and very different electronic structures is notable and motivates expanded-family testing to determine whether this convergence is a general feature of the best-characterized SF materials or a coincidence of our initial selection.

The materials furthest from the SF cluster center are CeCoIn₅ (Y = +2.24), LaFeAsO (Y = −1.06), and Sr₂RuO₄ (Y = −0.69). The relative ordering — which materials are closest to the cluster center and which are outliers — is invariant under v₀ rescaling.

4.2 Sensitivity to θ_glue Uncertainties

We tested the stability of the Y ranking by computing Y across the full range of literature θ_glue values (±20-30%) for each material. YBCO, UPt₃, and BaKFe₂As₂ all contain the cluster center within their uncertainty bands. CeCoIn₅ remains far from the cluster (Y > 1.5) for all reasonable θ_glue values. The qualitative structure is robust to the dominant uncertainty.

4.3 Robustness to vF Provenance

Excluding UBe₁₃ and LaFeAsO (N = 9 SF) preserves the SF/PH separation and the identity of the three closest-to-center materials.

4.4 Correlations with SC Quality

|Y| (distance from cluster center) vs gap ratio: ρ = −0.45, p = 0.16. |Y| vs specific heat jump: ρ = 0.05, p = 0.89. |Y| vs m*/m: ρ = 0.01, p = 0.97. None significant at N = 11.

Y (signed) correlates with m*/m: ρ = 0.726, p = 0.012. However, both Y and m*/m increase with electronic correlation strength, so this likely reflects shared dependence on an underlying coupling variable rather than a causal link.

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5. Within-SF Trend: Suggestive but Not Robust

5.1 The OLS Result

Within the 11 SF materials, OLS regression gives Y = 0.622 × ln(ξ/a) − 1.407 (r² = 0.49, p = 0.017, slope SE = 0.212). Spearman rank correlation: ρ = 0.455, p = 0.16.

5.2 Robustness Analysis

Leave-one-out (LOO): Removing CeCoIn₅ reduces the OLS slope from 0.622 to 0.144 (p = 0.61). No other single removal changes the slope by more than 0.06 or moves p above 0.03. The within-SF trend is driven almost entirely by CeCoIn₅ (Y = +2.24 at ln(ξ/a) = 4.63), which acts as a high-leverage point.

Theil-Sen robust regression: slope = 0.365, 95% CI [−0.457, 1.139]. The confidence interval contains zero, confirming the trend is not robust.

5.3 Assessment

We do not claim a within-SF trend as a result of this paper. The OLS p = 0.017 is misleading because it is driven by a single influential point. This is exactly the kind of finding that would have been overclaimed in v1.0; the LOO and Theil-Sen analyses, prompted by reviewer feedback, reveal it as fragile. Expanding the SF dataset to N ≥ 20 with materials spanning the ξ/a gap between CeCoIn₅ and the other heavy fermions would test whether a genuine trend exists or whether CeCoIn₅ is simply an outlier.

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6. The R-Space Formulation and Its Limitations

6.1 Definition and Corrected Slopes

An earlier version of this analysis used R ≡ Y + ln(ξ/a) and found that R correlated with ln(ξ/a) at Spearman ρ = 0.873, p = 0.0005. The OLS slope is 1.622 — not the value of 0.966 reported in preliminary work, which was computed under different conditions (Spearman-based fit on an earlier dataset iteration).

The identity slope_R = 1 + slope_Y holds exactly: 1.622 = 1 + 0.622. The earlier reported value of γ_R ≈ 0.966 is superseded by this corrected computation.

6.2 The Shared Variable Problem

Because R = Y + ln(ξ/a), plotting R vs ln(ξ/a) is equivalent to plotting (Y + x) vs x. Even if Y were uncorrelated with x, this would show correlation with slope 1.0.

Variance decomposition: Var(R) = 2.638. Components: Var(x) = 0.869 (33.0%), Var(Y) = 0.688 (26.1%), 2×Cov(x,Y) = 1.080 (41.0%). Sum = 100.0%.

6.3 Assessment

The R-space correlation is real but inflated. The SF/PH split it shows reflects the genuine Y-space separation. The slope near 1.6 (not near 1.0 as previously reported) is dominated by the structural contribution of ln(ξ/a) appearing on both axes plus the (fragile) within-SF Y-vs-x trend.

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7. Blind Prediction: LiFeAs

7.1 Protocol

We predicted vF* for LiFeAs (not in the training set) using the Y-vs-ln(ξ/a) trend line before consulting ARPES data. Inputs: Tc = 18 K, θ_glue = 100 K, ξ = 4.4 nm, a = 0.378 nm. Prediction: vF* = 0.24 × 10⁶ m/s.

7.2 Result and Caveat

The prediction matches bare DFT band velocity (~0.25 × 10⁶ m/s) to 4% but is 3-4× larger than ARPES-measured renormalized velocity (~0.07 × 10⁶ m/s). This is informative regarding which velocity the framework tracks. However, given the within-SF trend's fragility (Section 5.2), the prediction's agreement may be partly fortuitous. We present it as a data point, not a validation.

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8. Discussion

8.1 What Y-Space Tells Us

The SF/PH separation in Y-space quantifies a known qualitative distinction: unconventional superconductors are strongly coupled while conventional superconductors are weakly coupled. The contribution is the compact encoding in a single diagnostic that enables cross-family comparison. The observation that canonical SF materials from families differing by two orders of magnitude in energy scale converge to the same Y region is noteworthy regardless of where that region is located on the Y axis.

8.2 Limitations

• Normalization convention: The absolute value Y ≈ 0 for the SF cluster depends on v₀ = 10⁶ m/s. All relative statements are invariant.
• Sample size: N = 11 SF, N = 5 PH. Separation is robust; within-SF structure is not established.
• θ_glue uncertainty: ±20-30%, giving ΔY ≈ 0.3-0.5 per material.
• "Bare" velocity ambiguity: The dataset mixes ARPES (least renormalized band), dHvA (cyclotron mass inversion), and DFT (unrenormalized band structure). These are related but not identical operational definitions.
• Mechanism labels: SF vs PH classification is imported from prior literature and is partly theory-laden. The Y-space separation is therefore not a purely observation-level discriminator.
• Relationship to existing scaling: We have not systematically compared Y to Uemura-type plots, Homes' law, or Eliashberg-derived ratios. The relationship between e^Y and established coupling parameters (λ, 2Δ/kTc) is not derived here.

8.3 What Would Weaken or Falsify This Framework

The framework would be weakened if several well-established SF superconductors land deep in the PH band under the same normalization and measurement conventions, or if an expanded PH dataset fills the current gap between H₃S (Y = −1.31) and Sr₂RuO₄ (Y = −0.69) and erases rank separation. It would also be weakened if the separation is found to be entirely attributable to vF* alone with no added discriminative value from the energy ratio component (Section 3.7 already raises this as an open question). The framework would be strengthened by expanded datasets confirming tight SF clustering and clean PH separation across independently characterized materials.

8.4 Predictions and Tests

Expanded SF dataset: Adding CeRhIn₅, FeSe, KFe₂As₂, PuCoGa₅, UTe₂ would test whether the SF cluster is tight (supporting the framework) or dispersed (weakening it).

H₃S intermediate position: H₃S (Y = −1.31) sits between SF and PH. Its mechanism classification remains debated; the Y-value is consistent with either intermediate coupling or weak phonon coupling.

Within-SF trend: Requires N ≥ 20, with materials filling the ξ/a gap between heavy fermions and cuprates, to test whether the CeCoIn₅-driven trend is real or an outlier effect.

8.5 On AI-Assisted Research

The dimensional inconsistency in v1.0 survived the author's own review, one AI system's analysis, and initial manuscript preparation. It was caught by 3 of 4 math specialists in a multi-agent AI review using models from three different providers. The slope inconsistency between Sections 5 and 6 was caught by a fourth specialist. The within-SF trend's fragility was revealed only when reviewers requested leave-one-out analysis. These corrections demonstrate both the value of adversarial multi-model review and the risk of single-model validation.

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9. Conclusions

1. Y ≡ ln(Tc/θ_glue) − ln(vF*/v₀) with v₀ = 10⁶ m/s separates SF from PH superconductors with perfect rank separation (Mann-Whitney p = 0.0005, Cohen's d = 2.6).

2. SF materials cluster in a narrow Y region (mean 0.07 ± 0.87). The three canonical representatives of the major SF families — YBCO, UPt₃, BaKFe₂As₂ — sit within |Y| < 0.06 of each other despite spanning Tc = 0.56 to 93 K.

3. The within-SF trend between Y and ln(ξ/a) reported in v1.0 does not survive robustness analysis and is retracted as a claimed finding. It is driven by a single high-leverage point (CeCoIn₅).

4. An earlier R-space formulation was inflated by a shared variable. The corrected OLS slope (1.622, not 0.966) satisfies the mathematical identity slope_R = 1 + slope_Y exactly.

5. The absolute location of the SF cluster near Y ≈ 0 depends on the reference velocity convention (v₀ = 10⁶ m/s). All relative comparisons and statistical tests are convention-invariant.

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Appendix A: Variance Decomposition of R-Space

R = Y + x where x = ln(ξ/a). Var(R) = Var(x) + Var(Y) + 2×Cov(x,Y).

For SF dataset (N = 11): Var(R) = 2.638. Var(x) = 0.869 (33.0%). Var(Y) = 0.688 (26.1%). 2×Cov(x,Y) = 1.080 (41.0%). Sum = 100.0%.

Appendix B: Leave-One-Out Analysis

Theil-Sen robust regression: slope = 0.365, 95% CI [−0.457, 1.139].

Appendix C: Changes from v1.0 and v2.0

Changes v1.0 → v2.0 (AI panel review)

Changes v2.0 → v2.1 (GPT-4 review)

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"We built a staircase. It was standing on one of its own steps. We caught it. We stripped it down to the ground. Then we found the ground had a crack. We patched it. Then we asked: is this ground, or is it just the floor? We're still checking. But the floor holds weight."