DOI: [To be assigned]
John Swygert
July 28, 2026
Abstract
The van der Waals heavy-fermion metal CeSiI exhibits a nonmonotonic, approximately V-shaped pressure dependence of the resistively defined coherence temperature T*. Under increasing pressure, T* decreases from approximately 50 K at ambient pressure to a minimum near 20 K around the suppression of antiferromagnetic order at approximately 6 GPa, then rises on the higher-pressure branch. A narrow superconducting dome with a maximum transition temperature of approximately 240 mK appears near the same critical-pressure region, accompanied by non-Fermi-liquid transport and evidence of enhanced quasiparticle effective mass. Subsequent room-temperature high-pressure structural measurements report a smoothly decreasing unit-cell volume but abrupt anisotropic lattice responses, altered Ce–Ce and Ce–Si bond lengths, and flattening of the internal silicon honeycomb layer near approximately 6 GPa. [1,2]
The nonmonotonic form of T*(P) permits a prospective matched-scale comparison. Two pressure conditions on opposite sides of the T* minimum may possess equal or nearly equal values of T* while occupying different pressure branches and potentially different electronic, magnetic, and structural organizations. This paper asks whether the shared scalar marker T* identifies an equivalent measured coherence regime across those branches or whether equal coherence temperature conceals reproducible branch-conditioned differences.
The proposed protocol selects several matched or near-matched T* pairs on the descending and ascending pressure branches. Primary measurements are performed within a prespecified coherence-regime temperature window that remains above antiferromagnetic and superconducting transitions. Raw and reduced-temperature transport, Hall, magnetoresistance, magnetic, spectroscopic, and structural observables are then compared using prospectively defined matching tolerances and equivalence margins.
Broad cross-branch agreement would support T* as a comparatively sufficient organizing coordinate for the tested coherence-regime observables. Partial agreement would indicate that T* captures some, but not all, of the relevant physical organization. Reproducible separation across independent observable classes would demonstrate that equal resistively defined T* is insufficient to identify an equivalent measured coherence state under the tested conditions. Indeterminate outcomes caused by pressure uncertainty, sample degradation, insufficient sensitivity, or failed matching are explicitly preserved.
The proposal is presented both as a prospective scientific protocol and as a formal request for adversarial feasibility review and independent experimental testing.
1. Introduction
Heavy-fermion materials are strongly correlated electronic systems in which interactions between localized magnetic moments and itinerant conduction electrons produce quasiparticles with strongly enhanced effective masses. Competition among Kondo hybridization, magnetic exchange, crystal-field structure, dimensionality, frustration, and lattice geometry can generate antiferromagnetism, unconventional superconductivity, non-Fermi-liquid behavior, Fermi-surface reconstruction, and quantum-critical phenomena. [6–9]
CeSiI is unusual because it combines heavy-fermion behavior with a layered van der Waals structure. Its metallic CeSi layers are separated by weakly coupled iodine layers, allowing the material to be exfoliated toward the two-dimensional limit. Thermodynamic, transport, angle-resolved photoemission, and scanning-tunneling measurements have supported the presence of coherent hybridization and an antiferromagnetically ordered heavy-fermion ground state. [3]
The ambient-pressure system displays a resistively defined coherence feature near T* ≈ 50 K and antiferromagnetic ordering near T_N ≈ 7.5 K. Additional work has shown that the hybridization is highly anisotropic and contains momentum-space nodes, strengthening the possibility that a single scalar temperature may not fully describe the organization of the coherent electronic state. [4,5]
Shi and colleagues used pressure to construct a temperature–pressure phase diagram for CeSiI. They reported that T* initially decreases with pressure, reaches a minimum near the pressure at which antiferromagnetic order is suppressed, and then increases rapidly on the higher-pressure side. A superconducting dome appears near the same region. The reported maximum superconducting transition temperature is approximately 240 mK, and the normal-state transport near the critical region displays non-Fermi-liquid behavior and strong quasiparticle-mass enhancement. [1]
Ma and colleagues subsequently investigated the room-temperature crystal structure of CeSiI under pressures up to 8.3 GPa. They did not observe a conventional bulk structural phase transition within that range. The unit-cell volume decreased smoothly, but the in-plane and out-of-plane lattice parameters responded abruptly and differently near approximately 6 GPa. The internal silicon honeycomb layer flattened, and Ce–Ce and Ce–Si bond lengths changed near the same pressure region. [2]
These findings create a narrowly defined experimental opportunity. Because T*(P) is nonmonotonic, two different pressures may satisfy:
P_L < P_min < P_R
and
T*(P_L) ≈ T*(P_R),
where P_L lies on the descending lower-pressure branch, P_R lies on the ascending higher-pressure branch, and P_min denotes the pressure near the minimum of T*(P).
The equality of one measured scalar does not establish that the corresponding physical organizations are equivalent. This paper formalizes a prospective test of that distinction.
2. Research Question
The central research question is:
When two CeSiI conditions on opposite pressure branches possess equal or nearly equal resistively defined coherence temperatures, do they exhibit equivalent measured coherence-regime behavior across independent observables?
The test distinguishes three different propositions.
Marker equality means that the two pressure conditions have matched or near-matched values of T*.
Measured coherence-regime equivalence means that prespecified electronic, transport, magnetic, spectroscopic, or structural observables agree within prospectively defined equivalence margins over a shared reduced-temperature range.
Complete thermodynamic-state equivalence would require equivalence across all state variables, including pressure itself.
The proposed experiment tests the second proposition. It does not test complete thermodynamic-state equivalence because the applied pressures are different by design.
The central question is therefore not whether the two complete states are identical. They are not identical in every respect because P_L ≠ P_R. The question is whether T* functions as a sufficient organizing coordinate for the selected coherence-regime observables or whether it maps multiple distinguishable physical organizations onto the same scalar value.
3. Why the Comparison Is Not Trivial
A simple comparison of low- and high-pressure CeSiI at the lowest accessible temperatures would not provide a strong test. The published phase diagram already indicates that the two pressure regions may approach different magnetic, superconducting, or quantum-critical ground states.
A critic could reasonably state:
Of course the low-temperature states differ. One condition may remain antiferromagnetic while another lies near or inside the superconducting dome.
That difference would not establish that equal T* conceals distinct coherence-regime organizations. It would establish only that T* does not uniquely determine the entire low-temperature phase diagram.
The primary comparison must therefore be conducted above the ordered-state transitions, within the temperature region in which T* is intended to characterize the development of electronic coherence.
The lower temperature boundary for each matched pair shall be:
T_lower = max[0.3T*, T_N(P_L) + δ_N, T_N(P_R) + δ_N, T_c(P_L) + δ_c, T_c(P_R) + δ_c]
The terms δ_N and δ_c are prospectively defined safety margins above the antiferromagnetic and superconducting transitions. Their numerical values must be determined by the executing laboratory from transition width, temperature stability, and measurement resolution before the branch comparison is examined.
The proposed upper boundary is:
T_upper = 1.2T*
A matched pair is ineligible for the primary analysis if these constraints leave no usable common temperature interval.
Repeated normally:
The lower measurement temperature must be whichever is highest: 30 percent of the matched coherence temperature, the antiferromagnetic transition plus a safety margin, or the superconducting transition plus a safety margin. The primary comparison must remain outside the ordered phases.
Secondary measurements may extend below this boundary to determine how the matched-T* conditions evolve toward their respective low-temperature ground states.
4. Equal Marker Versus Equivalent Organization
Let the scalar matching marker be:
M(P) = T*(P)
Let the experimentally sampled coherence-regime organization be represented by a state-observation vector:
S(P,T) = [ρ, dρ/dT, R_H, MR, χ, H, F, L]
The entries may include:
ρ: electrical resistivity
dρ/dT: temperature derivative of resistivity
R_H: Hall response
MR: magnetoresistance
χ: magnetic susceptibility or another magnetic observable
H: hybridization-sensitive spectroscopic observables
F: Fermi-surface-sensitive observables
L: lattice or local-structure observables
The nonmonotonic pressure dependence of T* makes the mapping from pressure to T* non-injective. More than one pressure may correspond to the same T*:
M(P_L) = M(P_R)
while:
P_L ≠ P_R
The experimental question is whether the selected observable vectors also satisfy an operationally defined equivalence relation:
S(P_L,T) ≈ S(P_R,T)
Equal T* accompanied by cross-branch agreement across prespecified observables would support the use of T* as an organizing coordinate for those observables.
Equal T* accompanied by reproducible separation across independent observables would establish that T* is a many-to-one marker within the tested regime: the same measured coherence temperature would correspond to more than one distinguishable physical organization.
This conclusion would apply only to the observables, pressure range, samples, and experimental resolution included in the test.
5. Prospective Propositions
5.1 Broad-equivalence proposition
Matched-T* conditions on the descending and ascending pressure branches exhibit equivalent behavior across the predefined primary coherence-regime observables within experimentally justified equivalence margins.
This result would support T* as a comparatively sufficient organizing coordinate for the measured coherence regime.
It would not imply identical pressure, identical lattice structure, identical low-temperature phase, or complete thermodynamic-state equivalence.
5.2 Partial-equivalence proposition
The matched conditions agree in some observables but remain reproducibly distinguishable in one or more independent sectors.
Examples include:
Comparable normalized resistivity curves but different Hall responses.
Comparable transport behavior but different local structural geometry.
Comparable high-temperature coherence behavior followed by different low-temperature electronic evolution.
This result would indicate that T* captures a shared component of the state but does not exhaust its physical organization.
5.3 Systematic branch-nonequivalence proposition
Matched-T* conditions display reproducible branch separation across at least two independent primary observable classes, with replication across multiple matched pairs and more than one crystal.
This result would demonstrate that equal resistively defined T* is insufficient to identify an equivalent measured coherence-regime organization under the tested conditions.
5.4 Pair-specific nonequivalence proposition
One matched pair displays separation while other matched pairs satisfy the predefined equivalence criteria.
This result would indicate a localized crossover, anomaly, or pair-specific effect rather than a general cross-branch principle.
5.5 Indeterminacy proposition
The experiment fails to distinguish equivalence from nonequivalence because of inadequate pressure resolution, uncertainty in T*, pressure gradients, sample degradation, insufficient statistical power, incompatible instrumentation, or failure to preserve a common coherence-regime measurement window.
This result would not support either equivalence or nonequivalence.
6. Operational Definition of T*
For continuity with the published pressure study, T* shall initially be defined using the resistivity feature reported by Shi and colleagues:
T* is the temperature of the resistivity maximum associated with the reported coherence feature, identified by the corresponding zero crossing of dρ/dT.
The executing laboratory must preserve separately:
The estimated value of T*.
The uncertainty in T*.
The width of the resistivity feature.
The amplitude of the feature.
The local curvature around the maximum.
The smoothing, interpolation, or fitting parameters used to locate the maximum.
The same algorithm, smoothing procedure, and fitting window must be applied to both pressure branches. Those procedures must be frozen before branch identities are revealed to the primary analyst whenever partial analytical blinding is feasible.
The width, amplitude, asymmetry, and curvature of the resistivity feature shall not be included in the matching criterion. They are potential outcomes of the comparison. Matching on those properties would remove part of the effect that the protocol is designed to test.
A pressure pair is provisionally eligible when both of the following conditions are satisfied:
|T*_L − T*_R| / [(T*_L + T*_R)/2] ≤ 0.05
and
The uncertainty interval for the T* difference remains inside the prospectively defined matching boundary.
The 5 percent criterion is an initial ceiling, not a claim that a 5 percent difference is physically negligible in every implementation. The executing group may adopt a narrower criterion if its pressure and temperature resolution permits. Any revision must occur before the other branch-comparison outcomes are analyzed.
7. Selection of Matched Pressure Pairs
The final pressure pairs must be determined from numerical measurements rather than visual inspection of a published graph.
The selection procedure shall be:
1. Obtain the original numerical T*(P) data or perform a new high-resolution pressure sweep.
2. Separate the descending and ascending monotonic branches surrounding the minimum.
3. Fit or interpolate each branch independently using a method selected before examining the nonmatching observables.
4. Identify a reliably overlapping range of T* values.
5. Select at least three target coherence temperatures within that overlapping range.
6. Solve separately for the corresponding P_L and P_R values.
7. Measure at or near those predicted pressures.
8. calculate T* from the newly measured resistivity curves.
9. admit the pair only if the observed T* values satisfy the matching criterion.
10. Preserve all attempted pairs in the reported record, including failed matches.
Possible target values may lie near 25 K, 30 K, and 35 K, depending on the raw data and experimentally accessible pressure resolution. These values are illustrative and are not locked selections.
At least three matched pairs are preferred because a single comparison could be distorted by a local anomaly, interpolation error, pressure gradient, or sample-specific defect.
8. Reduced-Temperature Comparison
For each branch, define reduced temperature as:
t_L = T/T*_L
t_R = T/T*_R
The two conditions shall be compared at common values of reduced temperature rather than assuming that equal absolute temperatures represent equal positions within their respective coherence crossovers.
Every primary observable must also be reported against absolute temperature. Reduced-temperature scaling may reveal collapse, but it can also conceal physically meaningful absolute-scale differences.
The reporting requirement is therefore:
Raw observable versus absolute temperature.
Raw observable versus reduced temperature.
Prospectively normalized observable versus reduced temperature.
No normalization scheme may be selected solely because it creates stronger collapse or stronger branch separation.
9. Experimental Design
9.1 Samples
The primary protocol should use the same CeSiI crystal across both pressure branches whenever technically possible.
A same-crystal design reduces confounding from composition, disorder, contact geometry, residual resistivity ratio, and crystal quality. It does not eliminate pressure-history effects, contact changes, strain accumulation, or degradation.
The minimum replication requirement should include at least two independently prepared and loaded crystals.
For every crystal, the investigators should document:
Composition and stoichiometry.
Crystal-growth batch.
Dimensions and orientation.
Residual resistivity ratio.
Contact configuration.
Storage and handling conditions.
Evidence of oxidation or moisture exposure.
Pressure-cell loading history.
Changes in contact resistance.
Any visible or inferred degradation.
A replaced sample must be assigned a new sample identity. Data from different crystals must not be represented as an uninterrupted sequence from one specimen.
9.2 Pressure control
Pressure shall be calibrated using a validated standard appropriate to the apparatus.
The protocol must record:
Pressure-cell type.
Pressure medium.
Calibration method.
Pressure uncertainty.
Estimated pressure gradient.
Pressure drift during cooling.
Compression or decompression direction.
Equilibration time at each pressure.
Evidence of nonhydrostatic stress.
Sample orientation.
Electrical-contact geometry.
The published transport study and the structural study used different pressure apparatuses and measurement temperatures. Their pressure environments must not be presumed mechanically equivalent.
Apparatus-specific stress may affect the lattice, resistivity, transition widths, and superconducting response. Cross-apparatus comparisons must therefore be interpreted cautiously.
9.3 Measurement order
The measurement sequence should be prospectively specified.
A preferred design is:
Initial low-pressure characterization.
Ascending-pressure measurements through all candidate P_L and P_R points.
High-pressure endpoint characterization.
Descending-pressure replication where technically feasible.
Post-run sample and contact assessment.
Randomization of pressure itself is generally impractical. The analytical order of matched pairs may nevertheless be randomized after data acquisition.
9.4 Compression and decompression
The core experiment compares different pressures on opposite branches of a nonmonotonic T*(P) relation. That is a branch-conditioned equivalence test.
It is not, by itself, proof of thermodynamic path dependence.
A separate history-dependent extension compares the same nominal pressure and temperature after compression and decompression:
P_L↑ versus P_L↓
P_R↑ versus P_R↓
Reproducible differences at the same pressure, temperature, and measurement conditions would provide evidence of pressure-history dependence, hysteresis, residual strain, metastability, or another path-sensitive process.
Such differences must be distinguished from the core cross-branch comparison.
9.5 Analytical blinding
The experimental operator cannot be blinded to the applied pressure.
Partial analytical blinding remains possible.
After pressure points and matching eligibility have been established, a primary analyst who did not select the pressures may receive datasets labeled only as condition A and condition B.
Before branch identities are disclosed, the analyst should freeze:
Preprocessing rules.
Background subtraction.
Smoothing parameters.
Fitting windows.
Interpolation procedures.
Outlier handling.
Exclusions.
Equivalence margins.
Curve-distance calculations.
The blind should be broken only after the primary analysis record has been preserved.
10. Primary Coherence-Regime Measurements
10.1 Electrical resistivity
Electrical resistivity is necessary because it defines the matching marker T*.
The comparison should preserve:
Absolute resistivity.
Location of the maximum.
Peak width.
Peak height.
Peak asymmetry.
Local curvature.
dρ/dT.
d²ρ/dT².
Reduced-temperature behavior.
Prospectively normalized curve shape.
The matching process constrains only the location of the resistive maximum. Other characteristics of the resistivity curve remain available for comparison.
However, resistivity-shape differences alone cannot support a strong multisector state claim because the pair was selected using the same measurement family. A strong conclusion of branch-conditioned nonequivalence requires confirmation in at least one observable not used to define T*.
10.2 Hall response
The Hall response is a high-priority independent observable because it may be sensitive to carrier density, mobility, multiband transport, anomalous Hall contributions, Kondo coherence, and Fermi-surface reconstruction.
The protocol should compare:
Hall resistivity as a function of magnetic field.
Low-field and high-field slopes where physically justified.
Temperature evolution of the Hall coefficient.
Nonlinearity.
Sign changes.
Hysteresis.
Field-orientation dependence.
Any separation must be interpreted with attention to anomalous Hall contributions and multiband effects. A Hall difference does not uniquely identify its microscopic cause, but it can establish electronic nonequivalence independent of the resistivity maximum.
10.3 Magnetoresistance
Magnetoresistance should be measured using identical field orientation, field range, sweep direction, and thermal protocol for both members of a matched pair.
Relevant comparisons include:
Magnitude.
Sign.
Low-field curvature.
High-field behavior.
Anisotropy.
Scaling relationships.
Crossover fields.
Hysteresis.
Agreement in zero-field resistivity combined with branch-dependent magnetoresistance would support partial equivalence rather than broad equivalence.
10.4 Magnetic response
Where technically feasible, magnetic susceptibility, magnetization, nuclear magnetic resonance, muon spin measurements, or another pressure-compatible magnetic probe may be included.
The objective is not merely to rediscover the low-temperature antiferromagnetic transition. It is to determine whether magnetic correlations or fluctuation spectra already differ within the shared coherence-regime interval.
11. Advanced Independent Measurements
11.1 Hybridization-sensitive spectroscopy
CeSiI has been reported to exhibit highly anisotropic, nodal hybridization. [4]
A matched-T* comparison could therefore examine whether the two branches possess equivalent:
Hybridization energy scales.
Spectral-weight transfer.
Momentum dependence.
Orbital dependence.
Gap or pseudogap behavior.
Coherence-onset profiles.
Direct pressure-compatible angle-resolved photoemission or scanning-tunneling measurements at the required pressures may be impractical. Those methods are not mandatory for the minimum protocol.
Alternative pressure-compatible optical, X-ray, or spectroscopic approaches should be selected by specialists according to actual capability.
11.2 Fermi-surface-sensitive measurements
Quantum oscillations or another Fermi-surface-sensitive method would provide a powerful test of whether matched T* corresponds to comparable electronic topology.
Potential outcomes include:
Comparable frequencies and effective masses.
Different frequencies but similar transport scales.
Abrupt reconstruction on one pressure branch.
Continuous evolution on both branches.
No detectable oscillations because of scattering, pressure limitations, or insufficient field.
Failure to observe quantum oscillations is not automatically evidence of Fermi-surface equivalence.
11.3 Structural measurements
The room-temperature structural study reported anisotropic lattice responses and local coordination changes near the pressure region associated with the T* minimum and suppression of antiferromagnetism. [2]
A structural matched-scale test should compare:
a-axis and c-axis lattice parameters.
Unit-cell volume.
Ce–Ce distances.
Ce–Si distances.
Bond-angle distributions.
Silicon-layer flattening or buckling.
Local coordination symmetry.
Indicators of charge redistribution.
The preferred experiment would measure structure at low temperature under the same pressure conditions used for the electronic comparison.
Room-temperature structural measurements may provide useful contextual information but cannot automatically be treated as the structure of the low-temperature coherence regime.
A smooth unit-cell volume does not establish smooth internal organization. Conversely, an internal structural difference does not by itself identify the electronic mechanism responsible for a transport difference.
12. Secondary Low-Temperature Continuation
After completion of the primary coherence-regime comparison, each matched condition may be followed toward its low-temperature ground state.
Secondary endpoints may include:
Antiferromagnetic transition temperature.
Superconducting onset temperature.
Zero-resistance temperature.
Upper critical field.
Residual resistivity.
Low-temperature resistivity exponent.
Quadratic resistivity coefficient.
Specific heat.
Magnetic susceptibility.
Quantum oscillations.
Other normal-state measurements performed after superconductivity is suppressed by field.
This continuation asks:
Can two conditions with equal or nearly equal coherence markers evolve toward different low-temperature phases or critical behaviors?
That question is scientifically valuable, but its answer must remain secondary because the published phase diagram already indicates differences near the ground state.
13. Statistical and Equivalence Framework
13.1 Equality is not established by a nonsignificant difference
Failure to reject a conventional null hypothesis of no difference does not demonstrate equivalence.
Equivalence requires a prospectively defined margin representing the largest difference that would still be considered scientifically negligible for a specific observable.
For each primary observable class j, define an equivalence margin ε_j before branch identities are analyzed.
13.2 Curve comparison
For an observable O_j sampled at N reduced-temperature points, a weighted curve-separation measure may be defined as:
D_j = sqrt[(1/N) Σ w_k(O_j,L(t_k) − O_j,R(t_k))²]
The weights w_k and temperature grid t_k must be selected prospectively.
Broad equivalence for observable j requires the upper bound of the prespecified confidence interval for D_j to remain below ε_j.
Evidence of separation requires more than one isolated temperature point unless that point was prospectively designated as a primary endpoint.
13.3 Replication structure
The preferred analysis should account for:
Repeated temperatures within each pressure condition.
Matched pressure pairs.
Multiple crystals.
Measurement runs.
Pressure uncertainty.
Within-sample correlation.
A hierarchical or mixed-effects framework may be appropriate, but the final model must be chosen by the executing statistical and experimental team before the confirmatory comparison.
13.4 Strong branch-nonequivalence criterion
A strong branch-nonequivalence classification requires:
At least two independent primary observable classes.
The same directional or structurally consistent separation in at least two matched-T* pairs.
Replication in at least two crystals where technically feasible.
Effect estimates exceeding prospectively defined minimum meaningful differences.
Results that remain after prespecified sensitivity analyses.
A single isolated difference does not satisfy this standard.
13.5 Broad-equivalence criterion
Broad equivalence requires:
Successful T* matching.
Equivalence within the primary resistive coherence profile.
Equivalence in at least one independent electronic observable.
No prespecified primary observable showing a reproducible difference exceeding its nonequivalence threshold.
Adequate power to detect the minimum differences defined as scientifically meaningful.
Broad equivalence applies only to the tested observables. It does not establish complete state identity.
13.6 Power and sample planning
The final power analysis cannot be completed without empirical estimates of instrument reproducibility, pressure uncertainty, sample variability, and scientifically meaningful equivalence margins.
The executing group should use calibration data, repeated measurements, or an explicitly designated pilot dataset to estimate those values.
Pilot observations used to establish variance or feasibility must not be silently merged into the confirmatory dataset unless that decision was made prospectively.
14. Feasibility Gates
Before full execution, the protocol must pass the following gates.
Gate 1: T* can be measured reproducibly on both branches.
Gate 2: At least three matched or near-matched pairs can be established within the attainable pressure resolution.
Gate 3: A valid common coherence-regime temperature interval remains after ordered phases and safety margins are excluded.
Gate 4: The same crystal or appropriately matched crystals can survive the required pressure sequence.
Gate 5: At least one independent observable beyond the defining resistivity curve can be measured with sufficient sensitivity.
Gate 6: Pressure gradients and nonhydrostatic stress can be quantified adequately.
Gate 7: Equivalence margins can be justified scientifically rather than selected after observing the branch differences.
Gate 8: The intended sample number provides adequate precision for the primary classification.
Failure of a gate should be reported. It should not be reclassified as evidence for or against the scientific proposition.
15. Confounds and Failure Conditions
15.1 Pressure remains different
P_L and P_R are different by design. Any result must be described as cross-branch equivalence or nonequivalence among selected observables, not proof that two complete states reached at different pressures are identical or different solely because of their routes.
15.2 Ambiguous microscopic meaning of T*
The resistive maximum may reflect coupled contributions from Kondo coherence, crystal-field excitations, magnetic scattering, hybridization, or another pressure-dependent process.
This ambiguity is part of the motivation for the experiment.
The paper therefore refers to T* as a resistively defined coherence marker rather than assuming that it is a unique microscopic Kondo temperature.
15.3 Circularity
Because resistivity defines T*, resistivity alone cannot provide fully independent confirmation that the matched states differ.
Differences in resistivity shape remain informative, but stronger claims require Hall, magnetoresistance, magnetic, spectroscopic, structural, or another independent observable.
15.4 Interpolation error
A pressure pair predicted from fitted branches must be verified experimentally. Interpolated equality without measured confirmation does not qualify as a matched pair.
15.5 Sample variation
Different crystals may vary in disorder, residual resistivity, transition sharpness, strain, and contact quality.
Strong conclusions require both within-crystal comparisons and cross-crystal replication where possible.
15.6 Pressure gradients and stress
Nonhydrostatic stress or pressure gradients may create apparent branch separation.
Pressure environment, transition width, and apparatus behavior must be included in the uncertainty assessment.
15.7 Contact evolution
Contact resistance or geometry may change during pressure cycling.
Four-terminal measurement reduces but does not eliminate all contact- and geometry-related concerns.
15.8 Thermal history
Different cooling rates, equilibration times, or temperature-sweep directions may alter the measured response.
These variables must be standardized or recorded.
15.9 Ordered-phase contamination
The primary analysis must exclude temperature regions influenced by antiferromagnetic or superconducting order unless that influence is explicitly modeled and prospectively included.
15.10 Normalization artifacts
Raw and normalized data must both be reported.
A normalization procedure selected after viewing the outcome may create an artificial collapse or artificial separation.
15.11 Multiple comparisons
A large collection of observables increases the probability of selectively emphasizing favorable results.
Primary, secondary, and exploratory endpoints must be distinguished before the final analysis.
15.12 Structural temperature mismatch
Room-temperature lattice behavior cannot automatically be assumed to persist unchanged into the low-temperature electronic regime.
Connections between structural and electronic observations must remain provisional unless measured under comparable conditions.
15.13 Insufficient measurement sensitivity
Apparent equivalence may result from instruments that cannot resolve the physically meaningful difference.
Equivalence margins must therefore be considered alongside measurement precision.
16. Outcome Classification
Outcome A: Broad measured coherence-regime equivalence
The matched conditions satisfy the predefined equivalence criteria for the resistive coherence profile and at least one independent primary electronic observable.
Interpretation:
T* functions as a comparatively strong organizing coordinate for the measured coherence regime across the tested pressure branches.
This outcome constrains strong claims that branch position necessarily produces detectable nonequivalence in this system.
Outcome B: Partial equivalence
The matched conditions satisfy equivalence criteria in some sectors but exceed prospectively defined difference thresholds in another independent sector.
Interpretation:
T* identifies a shared component of the coherence regime but does not completely specify its measured organization.
Outcome C: Systematic branch nonequivalence
At least two independent primary observable classes display reproducible branch separation across multiple matched pairs and samples.
Interpretation:
Equal resistively defined T* does not identify an equivalent measured coherence-regime organization under the tested conditions.
Outcome D: Pair-specific nonequivalence
One matched pair differs while the remaining pairs meet the equivalence criteria or remain unresolved.
Interpretation:
The evidence supports a localized anomaly or crossover but not a general branch-wide conclusion.
Outcome E: History dependence
The same nominal pressure and temperature produce reproducibly different results after compression and decompression.
Interpretation:
The system exhibits pressure-history dependence, hysteresis, residual strain, metastability, or another path-sensitive response.
This outcome belongs to the compression–decompression extension and is distinct from ordinary cross-branch nonequivalence.
Outcome F: Indeterminate
Failed matching, excessive uncertainty, inadequate power, sample degradation, pressure gradients, or incompatible measurements prevent classification.
Interpretation:
No conclusion about cross-branch equivalence is justified.
17. Experimental Execution Levels
Level I: Minimum matched-scale transport test
Required elements:
High-resolution resistivity.
Verified matched-T* pairs.
At least three matched pairs.
A common primary coherence-regime temperature window.
Raw and reduced-temperature curve comparison.
At least two crystals where feasible.
This level can determine whether equal resistive maximum temperatures correspond to equivalent resistive coherence profiles.
It cannot establish multisector state equivalence.
Level II: Independent electronic confirmation
Level I plus at least one of the following:
Hall response.
Magnetoresistance.
Pressure-compatible magnetic response.
Another independent transport or electronic observable.
This level can determine whether any observed cross-branch relationship extends beyond the resistivity measurement used to define T*.
Level III: Multisector comparison
Levels I and II plus one or more of the following:
Hybridization-sensitive spectroscopy.
Fermi-surface-sensitive measurement.
Low-temperature structural measurement.
Thermodynamic measurement.
Detailed magnetic measurement.
This level can test whether the cross-branch relationship extends across independent electronic, magnetic, structural, and thermodynamic sectors.
18. Precommitment and Data Reporting
Before the confirmatory branch comparison, the executing investigators should archive:
The operational definition of T*.
The method used to locate T*.
The smoothing and fitting parameters.
The branch-fitting method.
The matching tolerance.
The candidate pressure-pair selection procedure.
The final admitted pressure pairs.
The primary temperature interval.
The safety margins above ordered transitions.
The primary and secondary observables.
The normalization rules.
The equivalence margins.
The minimum meaningful difference thresholds.
The exclusion criteria.
The handling of failed pressure points.
The treatment of missing data.
The statistical model.
The multiplicity procedure.
The outcome-classification rules.
The protocol should require reporting of:
All attempted pressure pairs.
Pairs that fail the matching criterion.
Null results.
Contradictory results.
Technical failures.
Sample replacements.
Pressure-medium failures.
Contact failures.
Deviations from the planned temperature window.
Post-data changes to the analysis.
Any change made after examining branch outcomes must be identified as exploratory.
19. Relationship to Pathways, Boundaries, and Phases
The Pathways, Boundaries, and Phases architecture proposes that a scalar endpoint or state marker may fail to contain all information about the organization through which a physical system expresses and maintains that state.
CeSiI provides a suitable prospective testing environment because:
T*(P) is nonmonotonic.
The same T* may occur at two different pressures.
The two branches surround a pressure region associated with suppression of antiferromagnetism.
Superconductivity appears near the same region.
Normal-state transport changes near the critical region.
Local structural variables respond nonmonotonically while the global unit-cell volume remains smooth.
The architectural proposition is not that matched-T* states must differ.
The narrower proposition is:
Equality of one conventional scalar marker should not be treated as sufficient evidence of measured state equivalence without independent comparison.
The experiment is allowed to constrain the architecture in either direction.
Broad equivalence would limit strong branch-conditioned interpretations for the tested CeSiI observables.
Partial equivalence would indicate that the marker captures some but not all of the relevant organization.
Systematic multisector separation would support the proposition that physically consequential organization can remain unrepresented by a shared scalar marker.
No single result would prove or disprove the complete Pathways, Boundaries, and Phases architecture.
20. Request for Independent Experimental Testing
The author invites adversarial feasibility review and, where technically practical, independent execution of the proposed matched-T* comparison.
The request is directed particularly to:
The original CeSiI investigators.
Laboratories capable of synthesizing and preserving CeSiI crystals.
High-pressure and low-temperature transport groups.
Heavy-fermion and quantum-criticality researchers.
Pressure-compatible Hall and magnetotransport groups.
Hybridization and Fermi-surface spectroscopy specialists.
High-pressure structural researchers.
Statistical specialists experienced in equivalence testing and repeated-measures physical experiments.
The request is not for endorsement of Pathways, Boundaries, and Phases or any wider unification claim.
The request is:
Please determine whether this matched-T* cross-branch comparison is technically executable and scientifically discriminating. Identify any fatal confound, impossible measurement, incorrect physical assumption, circular inference, semantic ambiguity, or outcome that would fail to distinguish the stated propositions.
The principal feasibility questions are:
1. Can T* be estimated with enough precision to establish multiple cross-branch matched pairs?
2. Is the proposed 5 percent maximum matching difference defensible, or should a narrower criterion be required?
3. Can the same crystal survive measurements across both pressure branches?
4. What pressure medium and apparatus best minimize nonhydrostatic stress over the required range?
5. What temperature window best isolates the coherence regime from antiferromagnetic and superconducting order?
6. Which independent observable offers the strongest technically realistic discrimination?
7. Can Hall or magnetoresistance measurements be performed with sufficient resolution at the matched pressures?
8. Can low-temperature structural measurements be obtained under comparable pressure conditions?
9. How should pressure uncertainty be propagated into the matching and equivalence analyses?
10. What equivalence margins are scientifically meaningful for each primary observable?
11. What sample number and replication structure are required?
12. How should compression–decompression effects be separated from ordinary pressure-branch effects?
13. Would a minimum resistivity-only experiment remain scientifically useful?
14. What modifications would be necessary before preregistration or submission as a collaborative Stage 1 Registered Report?
15. Does any known property of CeSiI make the central matched-scale comparison physically meaningless or logically non-discriminating?
A feasibility review concluding that the protocol is invalid would itself be valuable. Fatal weaknesses should be identified before substantial laboratory resources are committed.
21. Interpretive Limits
A result of systematic branch nonequivalence would not establish that:
Pressure history alone caused the separation.
The complete Pathways, Boundaries, and Phases architecture is validated.
Conventional heavy-fermion or Kondo theory is invalid.
A specific microscopic mechanism has been identified.
The superconducting pairing symmetry is known.
The critical region necessarily represents one particular form of quantum criticality.
Every equal scalar marker in every physical system conceals different organizations.
A result of broad equivalence would not establish that:
The complete thermodynamic states are identical.
Pressure is physically irrelevant.
The low-temperature phases are identical.
The local lattice geometry is identical.
All unmeasured degrees of freedom are equivalent.
T* is a universal state variable.
The conclusions of the experiment must remain restricted to:
The tested CeSiI samples.
The measured pressure range.
The operational definition of T*.
The prespecified temperature interval.
The selected observables.
The measurement resolution.
The adopted equivalence margins.
The achieved replication.
22. Conclusion
The V-shaped pressure dependence of the resistively defined coherence temperature in CeSiI creates a natural matched-scale experiment.
Conditions on opposite sides of the pressure-tuned critical region may possess equal or nearly equal values of T* while occupying different pressure branches. This permits a direct test of whether the shared scalar marker identifies an equivalent measured coherence regime.
The primary comparison must remain above the antiferromagnetic and superconducting transitions so that already-known differences in ordered ground states do not determine the result trivially.
The protocol requires multiple matched pairs, explicit uncertainty in T*, prospectively defined equivalence margins, raw and reduced-temperature reporting, independent observables, replication, and preservation of indeterminate outcomes.
Broad equivalence would support T* as a comparatively sufficient organizing coordinate for the tested observables.
Partial equivalence would indicate that T* captures a shared component while leaving other physical sectors unresolved.
Systematic separation across independent measurements would demonstrate that equal resistively defined coherence temperature does not identify an equivalent measured coherence-regime organization under the tested conditions.
Indeterminacy would document the technical limits that must be overcome before the proposition can be resolved.
The proposed experiment does not require the invention of an entirely new material platform. It extends an existing pressure phase diagram using a material, pressure range, cryogenic environment, and transport methodology already demonstrated by the relevant experimental community.
The minimum version requires verified matched-T* resistivity measurements. Stronger versions add Hall response, magnetoresistance, magnetic measurements, hybridization-sensitive spectroscopy, Fermi-surface probes, and low-temperature structural comparison.
The proposal is therefore offered as a prospective scientific protocol and as a formal request for independent adversarial feasibility review and experimental testing.
Its decisive question is:
When coherence temperature is equal, is the measured coherence regime also equivalent, or has the scalar marker concealed two distinguishable physical organizations?
References
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