Temporal Convergence Framework: Distinguishing Structure from Coincidence in High-Precision, Low-Dimensionality Parameter Spaces

1. Introduction

1.1. Universal Challenge: Precision Outpacing Dimensionality

Computational methods across scientific domains share a common trajectory: iterative refinement improves measurement precision faster than dimensional growth expands parameter spaces. When uncertainties decrease to 2–3% relative precision while parameter count remains at three to six values, combinatorial search through simple formulas almost guarantees discovery of statistically significant relationships—whether physically meaningful or coincidental.

The challenge manifests across domains. Particle physics: three light quark masses at 2% precision enable countless ratio tests. Cosmology: six CMB parameters from Planck enable systematic formula mining. Condensed matter: critical exponent relationships at phase transitions. Fundamental constant metrology: searching for time-variation or inter-constant relationships at ${10}^{-9}$ precision. Machine learning: hyperparameter optimization exploring vast hypothesis spaces where spurious correlations achieve significance. In each case, when search space exceeds parameter dimensionality by orders of magnitude, discrimination might be enhanced through additional evaluation criteria.

Computational fluid dynamics provides a concrete industrial parallel. Navier-Stokes equations cannot be solved analytically in turbulent regimes, requiring iterative approximation. CFD validation examines convergence behavior across refinement cycles: agreement at one resolution provides initial evidence, but temporal sequences of improving approximations provide additional discriminatory information. Similarly, lattice QCD employs multiple independent collaborations (BMW, MILC, ETM, HPQCD) using different systematic approaches—analogous to different mesh strategies—where even less precise results contribute valuable information by constraining systematic uncertainties. Our framework applies similar principles: statistical agreement at one measurement vintage provides initial evidence, but convergence behavior as experimental precision improves across independent data releases provides additional discriminatory capability. The temporal dimension helps distinguish genuine structure from numerical artifacts in both domains.

We demonstrate a solution using particle physics quark masses specifically because N=3 represents the methodological worst case. With only three parameters, nearly any algebraically simple formula achieves statistical agreement by chance. If our framework successfully filters coincidences in this regime while preserving historically validated patterns (Koide formula, Gell-Mann-Okubo relations), it provides discrimination in any higher-dimensional context.

The core methodology: temporal convergence through timestamped predictions. Publication (journal articles, preprints, or public repositories) establishes the temporal baseline—the point before subsequent data releases become available. This leverages existing scientific practice: the community already respects that Koide (1982) preceded modern precision lepton measurements, establishing clear temporal ordering. The framework makes this temporal dimension an explicit evaluation criterion.

1.2. Historical Precedent and Methodological Gap

Empirical mass relations have historical precedent in revealing physical structure before theoretical understanding emerges. The Gell-Mann-Okubo formula [4,5] related hadron masses through what was later understood as SU(3) flavor symmetry breaking, identified empirically before the quark model existed. The Koide formula [3] describes charged lepton masses with remarkable precision despite lacking theoretical derivation after four decades. These historical successes demonstrate that empirically robust patterns can guide theoretical development. As precision improves, the field can benefit from objectively defining evaluation criteria through community consensus for distinguishing potentially meaningful patterns from numerical artifacts. Traditional particle physics methodology was optimized for discovery physics: predict new particle, build detector, confirm or exclude. As the landscape shifts toward understanding mathematical structure of already-known parameters, establishing clear standards through community consensus becomes increasingly valuable.

1.3. Cross-Disciplinary Opportunity

Lattice QCD precision improvements create opportunities for productive empirical phenomenology. Current uncertainties enable discriminatory tests while remaining above the noise floor where combinatorial coincidences overwhelm signal. This regime may persist for approximately a decade as systematic improvements continue.

Explicit evaluation criteria can facilitate cross-disciplinary contributions by providing clear operational targets. Independent development of computational results and theoretical models reduces mutual bias: lattice collaborations compute masses independently of phenomenological pattern searches, enabling decoupled timelines for empirical and theoretical work. This separation supports asynchronous, independent validation from both communities. The framework separates empirical validation (criteria 1–6) from theoretical explanation (criterion 7), enabling researchers from computational backgrounds to contribute validated patterns that theorists can subsequently investigate.

This work proposes one framework emphasizing systematic evaluation to complement existing peer review processes. Community discussion and iterative refinement can help establish whether these specific seven criteria and associated thresholds provide useful operational standards as experimental precision enables discriminatory phenomenology.

1.4. Motivating Example: Observing Patterns Through Temporal Evolution

Framework development was motivated by observing how temporal behavior across data releases can reveal information beyond instantaneous statistical agreement. Consider algebraic unification of two consistently reported lattice QCD ratios: $m_s/m_d\approx 20$ and $m_d/m_u\approx 2.16$. These published values approximately satisfy:

$$2{(m_d/m_u)}^3\approx m_s/m_d$$

(1)

With FLAG 2024 central values $m_d/m_u=2.162\pm 0.050$ and $m_s/m_d=20.0\pm 0.5$1, the pattern predicts $2{\left(2.162\right)}^3=20.22$ versus measured 20.0, a deviation of $0.16\sigma$.

We emphasize that this Diagnostic Pattern serves purely as a methodological calibration tool—a relationship we expect to fail—to demonstrate the framework’s filtering capability, not as a physics claim. We intentionally selected this pattern not as physics claim but as methodological demonstration. We will voluntarily self-falsify it using two framework criteria, showing how the framework enables filtering before peer review and creating concrete example for authors of high rigor standards.

We term this the Diagnostic Pattern, emphasizing its role as methodological tool. The pattern achieves statistical consistency (0.16$\sigma$ deviation) and passes Monte Carlo validation with Bonferroni correction, but lacks discriminatory power with only three light quark masses. While such statistical agreement is necessary for pattern consideration, the community correctly recognizes that statistical significance alone does not establish physical validity, motivating multi-criteria evaluation. Despite agreement with lattice world averages within $0.16\sigma$, scale invariance, and cross-collaboration consistency, directional trends and theoretical considerations suggest physical implausibility. The pattern fails criterion 4 through directional divergence (central values trend away from prediction as precision improves). Additionally, extensive survey of flavor symmetry frameworks revealed no structural precedent for the implied Yukawa texture $2y_d^4=y_u^3{y}_s$. The relationship appears incongruent with established symmetry-breaking patterns to the point of lacking physical sensibility. While this constitutes author judgment rather than rigorous proof of impossibility, the combination of directional divergence and apparent physical implausibility motivates self-falsification.

This double failure (directional divergence at criterion 4 combined with physically implausible structure at criterion 7) motivates self-falsification rather than journal submission. Were the pattern converging despite lacking theory, submission to empirical journals would be appropriate. Were theory identified despite current divergence, extended temporal validation might be warranted. Authors bear responsibility for judging pattern severity: framework criteria enable evaluation, venue selection (high-impact journal, empirical journal, preprint archive, community repository, or self-falsification) reflects pattern status across multiple dimensions.

Framework development was motivated by practical experience: establishing a pattern, tracking it across data releases, and observing how temporal behavior revealed information beyond instantaneous statistical agreement. Making these evaluation criteria explicit can help researchers from outside the immediate field understand the standards the community applies when assessing empirical patterns.

2. Methods

2.1. Framework Architecture: Self-Falsification Before Expert Review

The proposed framework addresses complementary objectives: protecting theoretical community resources while enabling legitimate empirical contributions. Current practice offers an opportunity to address both objectives through systematic criteria: explicit standards can enable more consistent evaluation while supporting legitimate empirical contributions. The seven criteria operationalize this through filter pipeline:

Criteria 1–6 (Objective Self-Falsification Gates): If the community establishes consensus standards, authors can reference these criteria when demonstrating pattern satisfaction, providing reviewers with simplified structure for first-pass evaluation. Domain expertise in phenomenology is sufficient; patterns failing objective tests filter pre-submission.

Criterion 7 (Theoretical Viability as Benchmark): Only patterns surviving criteria 1–6 warrant theoretical attention. Criterion 7 stratifies patterns by theoretical grounding for theorists developing flavor models. When theory predicts X and framework-validated pattern shows X at sub-$\sigma$ precision, productive collaboration emerges naturally. Absence of explanation with passing empirical criteria does not constitute automatic rejection; patterns stratify by venue type based on Criterion 7 status.

The framework’s computational and temporal requirements self-select for serious contributions: patterns must survive major experimental releases before publication, transforming enthusiasm into evidence through sustained agreement.

2.2. The Seven Criteria with Sample Operational Thresholds

2.2.1. Criterion 1: Scale Invariance Under Renormalization Group Evolution

Mass ratios must be scale-invariant under QCD running (deviations $<{10}^{-4}$ across 1 GeV to TeV scales; indicative order of magnitude, community to refine as precision improves). This basic prerequisite eliminates scale-dependent relationships.

2.2.2. Criterion 2: Compression of Degrees of Freedom

Patterns must reduce N parameters to N–1 degrees of freedom through unified constraints (e.g., $2{(m_d/m_u)}^3=m_s/m_d$ reduces three masses to two DOF).

2.2.3. Criterion 3: Statistical Agreement at Discriminatory Precision

Patterns must agree with measurements within statistical bounds at discriminatory precision. Community to determine thresholds: $<1\sigma$? $<2\%$ relative uncertainty? The key requirement: precision sufficient to distinguish between competing formulations.

2.2.4. Criterion 4: Temporal Persistence

Patterns must (a) be pre-registered via timestamped repository before new data releases, and (b) maintain directional convergence or stability through experimental cycles. Community to refine as release frequency evolves.

Why Temporal Convergence Provides Robust Protection:

Pre-registration via timestamped public repositories (Zenodo, institutional archives, or preprint servers) creates an immutable record before new measurements become available. Authors cannot:

• Use selective data usage favoring particular vintages
• Adjust formulas post-hoc to match updated values
• Mine through multiple hypothesis variations retroactively
• Claim prescience after observing convergence patterns

This transforms pattern evaluation from statistical (where combinatorial search can make it difficult to discern physically grounded relationships from numerical coincidence) to temporal (providing robust protection against data mining). The only way to pass Criterion 4 is genuine predictive success across independent experimental cycles—precisely the evidence distinguishing structure from coincidence.

2.2.5. Criterion 5: Mathematical Simplicity

Patterns should minimize complexity. Community to decide: Kolmogorov complexity threshold? Allow transcendentals? Maximum operations? For now: basic arithmetic, small integers ($\le 5$), standard constants ($\pi$, e).

2.2.6. Criterion 6: Independent Validation Across Multiple Determinations

Patterns must show consistency across independent experimental/computational approaches.

Threshold: Agreement across $\ge 3$ independent collaborations/methods with different systematic uncertainties.

2.2.7. Criterion 7: Theoretical Viability as Benchmark Stratification

Empirical relationships must not be demonstrably incompatible with existing theoretical frameworks. Critically: absence of explanation does not constitute automatic failure. Criterion 7 has three possible outcomes:

PASS (Compatible): Flavor theorist identifies mechanism within existing frameworks producing the pattern.

PASS (Unknown): No known incompatibility with gauge invariance, anomaly cancellation, or existing symmetry structures. Pattern awaits theoretical investigation but is not ruled out. This constitutes legitimate publication; empirically robust observations merit documentation even without explanation.

FAIL (Incompatible at present): Theorist demonstrates pattern violates fundamental constraints through explicit proof.

Patterns in Unknown status merit documentation as empirical benchmarks. The field benefits from a searchable repository of published patterns where theorists can identify observations matching their predictions.

Division of labor: Empiricists validate patterns through criteria 1–6; theorists may provide explanations or identify incompatibilities. Neither group bears obligation to the other. The framework separates empirical validation from theoretical explanation: different expertise, different contributions.

Self-falsification option: Authors may choose to self-falsify at criterion 7 if they survey existing frameworks, find no support, and possess no new theory to propose. This demonstrates intellectual rigor and filters patterns pre-submission. However, such self-falsification is voluntary; a pattern lacking known mechanism but compatible with fundamental constraints remains valid submission.

Application to the Diagnostic Pattern: We surveyed existing flavor symmetry mechanisms and identified no framework naturally producing the Yukawa texture $2y_d^4=y_u^3{y}_s$. We possess no new theoretical framework to propose. We therefore choose to self-falsify the Diagnostic Pattern at criterion 7. We present this pattern not as viable submission but as demonstration of how authors should filter implausible patterns before peer review.

3. Results: Framework Validation Through Test Cases

We demonstrate framework operation through three test cases: established patterns (Koide formula, Gell-Mann-Okubo relation) that should pass, and the Diagnostic Pattern that correctly self-falsifies.

3.1. Koide Formula: Validated Pattern with Unknown Mechanism

The Koide formula [3] relates charged lepton masses: $(m_e+m_{\mu }+m_{\tau })/{(\sqrt{m_e}+\sqrt{m_{\mu }}+\sqrt{m_{\tau }})}^2=2/3$. Framework evaluation:

Table 1. Koide Formula Evaluation

Criterion	Assessment
1. Scale Inv.	PASS - ratio form is scale-invariant
2. Compression	PASS - reduces 3 lepton masses to 2 DOF
3. Statistical	PASS - agrees within $<0.1\sigma$ consistently
4. Temporal	PASS - persistent/improving over decades
5. Simplicity	PASS - single equation, simple constants
6. Independent	PASS - multiple independent measurements
7. Theoretical	PASS (Unknown) - no mechanism, not ruled out
Status: Legitimate publication

Table 1. Koide Formula Evaluation

Criterion	Assessment
1. Scale Inv.	PASS - ratio form is scale-invariant
2. Compression	PASS - reduces 3 lepton masses to 2 DOF
3. Statistical	PASS - agrees within $<0.1\sigma$ consistently
4. Temporal	PASS - persistent/improving over decades
5. Simplicity	PASS - single equation, simple constants
6. Independent	PASS - multiple independent measurements
7. Theoretical	PASS (Unknown) - no mechanism, not ruled out
Status: Legitimate publication

Koide’s formula demonstrates that empirically robust patterns merit documentation even when theoretical mechanism remains unknown.

3.2. Gell-Mann-Okubo Relation: Historical Precedent

The GMO relation [4,5] predicted hadron mass relationships before the quark model existed. Framework evaluation (retrospective):

Table 2. GMO Relation Evaluation

Criterion	Assessment
1. Scale Inv.	PASS - hadronic scale relationship
2. Compression	PASS - relates multiple hadron masses
3. Statistical	PASS - agreed with measurements
4. Temporal	PASS - validated by subsequent data
5. Simplicity	PASS - simple SU(3) symmetry structure
6. Independent	PASS - multiple hadron measurements
7. Theoretical	PASS (Unknown at time) - emerged later
Status: Would pass BEFORE quark model

Table 2. GMO Relation Evaluation

Criterion	Assessment
1. Scale Inv.	PASS - hadronic scale relationship
2. Compression	PASS - relates multiple hadron masses
3. Statistical	PASS - agreed with measurements
4. Temporal	PASS - validated by subsequent data
5. Simplicity	PASS - simple SU(3) symmetry structure
6. Independent	PASS - multiple hadron measurements
7. Theoretical	PASS (Unknown at time) - emerged later
Status: Would pass BEFORE quark model

GMO relation demonstrates framework would enable historically important empirical observations even before theoretical understanding emerges. The pattern passed empirical criteria (1–6) and was compatible with fundamental constraints (criterion 7: Unknown), allowing publication that guided subsequent theory development.

3.3. Diagnostic Pattern: Demonstration of Self-Falsification

The Diagnostic Pattern $2{(m_d/m_u)}^3\approx m_s/m_d$ demonstrates framework filtering capability:

Table 3. Diagnostic Pattern Evaluation

Criterion	Assessment
1. Scale Inv.	PASS - preserved under QCD RG 1 GeV–1 TeV
2. Compression	PASS - reduces 3 masses to 2 DOF
3. Statistical	PASS - FLAG 2024: within $0.16\sigma$
4. Temporal	FAIL - directional divergence
5. Simplicity	PASS - single equation, integer coefficient
6. Independent	PASS - consistent across ETM, BMW, MILC
7. Theoretical	FAIL - no mechanism; no theory
Status: Doubly self-falsified at 4 and 7

Table 3. Diagnostic Pattern Evaluation

Criterion	Assessment
1. Scale Inv.	PASS - preserved under QCD RG 1 GeV–1 TeV
2. Compression	PASS - reduces 3 masses to 2 DOF
3. Statistical	PASS - FLAG 2024: within $0.16\sigma$
4. Temporal	FAIL - directional divergence
5. Simplicity	PASS - single equation, integer coefficient
6. Independent	PASS - consistent across ETM, BMW, MILC
7. Theoretical	FAIL - no mechanism; no theory
Status: Doubly self-falsified at 4 and 7

Detailed technical analysis (RG evolution, cross-collaboration comparisons, statistical methodology, temporal tracking) provided in Appendix A.

Key findings:

Criterion 4 (Directional Divergence): Central values moved from $m_d/m_u=2.16\pm 0.08$ to $2.162\pm 0.050$ (away from predicted $2.154$) between consecutive reviews. While both measurements remain within $1\sigma$, the 37% uncertainty reduction caused statistical significance to double from $0.075\sigma$ to $0.16\sigma$—measurements are converging toward 2.162, not the predicted 2.154. The critical observation is not just the +0.002 directional movement, but that shrinking uncertainties are converging around a value systematically above the prediction, doubling the statistical significance of the deviation. See Appendix A for detailed interpretation of why shrinking error bars revealing directional offset constitute stronger falsification than traditional statistical disagreement.

Criterion 7 (Absence of Support): Survey of existing flavor frameworks identified no mechanism producing Yukawa texture $2y_d^4=y_u^3{y}_s$. Absent new theoretical framework to propose, pattern self-falsifies at Criterion 7. Theoretical rescue remains possible if flavor theorist identifies compatible mechanism.

The pattern remains open to two rescue pathways: temporal convergence through future measurements (as Koide achieved over decades), or theoretical rescue through identifying compatible mechanisms. However, theoretical rescue would require mechanisms beyond current flavor symmetry formulations, as no existing framework naturally produces the implied Yukawa texture $2y_d^4=y_u^3{y}_s$. We conclude the pattern is likely a numerical coincidence despite current statistical consistency.

Statistical consistency was achieved with relative ease given the low parameter count. This observation led us to explicitly exclude Monte Carlo p-values from our framework criteria entirely. When statistical significance becomes more easily obtained through combinatorial search, it provides less discriminatory power. With only three light quarks and $\mathcal{O}({10}^2$–${10}^3)$ possible simple formulas to test, finding statistically significant relationships becomes more easily obtained. This constraint on statistical power makes multi-criteria evaluation essential: any single test is weak, but multiple independent criteria provide discriminatory capability.

The Diagnostic Pattern’s double self-falsification validates framework discriminatory capability: patterns pass traditional metrics yet correctly self-falsify through directional persistence and theoretical viability criteria.

4. Discussion

4.1. Framework Operation and Methodological Questions

The Diagnostic Pattern identified two failure points: directional divergence (criterion 4) and absence of theoretical support (criterion 7). Rather than submitting this pattern and asking reviewers to make these determinations, we self-falsify, exemplifying the framework’s protective function: authors bear responsibility for evaluation before requesting expert attention. Such self-falsification represents valuable scientific contribution, documenting what does not work with the same rigor as what does.

The Diagnostic Pattern’s self-falsification raises questions for community debate: (1) Should theoretical viability be first rather than last? (2) Does scale invariance provide sufficient discrimination? (3) How strict should directional convergence be? (4) What constitutes adequate temporal validation?

The Diagnostic Pattern’s dual self-falsification illustrates two distinct rescue pathways with different barriers. Temporal rescue requires only future measurement convergence—subsequent FLAG reviews showing $m_d/m_u$ approaching ${10}^{1/3}$ would rehabilitate the pattern, as Koide’s formula achieved through decades of validation. Theoretical rescue presents a higher barrier: extensive survey of flavor symmetry frameworks found no precedent for the Yukawa texture $2y_d^4=y_u^3{y}_s$, suggesting rescue would require revolutionary new mass generation models beyond current formulations rather than incremental refinement of existing approaches.

The framework treats these rescue pathways asymmetrically by design. Temporal evolution is monitored through Criterion 4’s timestamped prediction requirement, enabling patterns to demonstrate convergence across multiple experimental cycles. Theoretical viability stratifies patterns through Criterion 7 into a searchable benchmark database, allowing theorists to identify empirical candidates matching model predictions. Patterns failing Criterion 4 but passing Criterion 7 (Unknown status) merit continued temporal monitoring and documentation. Patterns failing both criteria, as demonstrated here, warrant self-falsification despite current statistical consistency—the framework successfully filters likely numerical coincidences before peer review.

4.2. Historical Context: Complementary Methodologies

Contemporary particle physics benefits from multiple complementary approaches. Theoretical frameworks and experimental precision have both advanced substantially, creating opportunities for empirical phenomenology to bridge these domains. Explicit evaluation criteria can help identify robust patterns that warrant theoretical investigation while filtering numerical artifacts efficiently.

4.3. Framework Benefits for Collaborative Research

The framework can facilitate collaboration between computational and theoretical researchers. Framework-validated patterns published in appropriate venues become accessible benchmarks. When theorists develop flavor models, they can check whether their predictions match documented empirical observations, creating natural opportunities for productive collaboration. This enables temporal decoupling: patterns documented today may find theoretical explanation when relevant framework development reaches that parameter space, without requiring simultaneous empirical validation and theoretical explanation from the same authors.

4.4. The Meta-Pattern Horizon

The framework’s long-term value may emerge through meta-pattern synthesis: relationships between multiple validated patterns revealing latent structure inaccessible to individual observations. Without systematic documentation—including patterns passing empirical criteria (1-6) but lacking mechanisms—such higher-order correlations remain undiscoverable. Historical precedent demonstrates viability: the Balmer series appeared numerological until spectroscopic multiplicity collectively revealed atomic structure. Our framework transforms isolated observations into a curated corpus where emergent mathematical structure in fundamental parameters becomes systematically tractable.

4.5. Limitations and Community Refinement

This framework represents a proposed starting point requiring community evaluation and refinement. The goal is initiating systematic discussion: what standards should govern empirical pattern evaluation as precision enables discriminatory tests? Whether these specific seven criteria and thresholds prove optimal matters less than initiating discussion about explicit, debatable standards that can complement community evaluation practices.

5. Conclusion

Recent lattice QCD precision achievements create opportunities for systematic empirical phenomenology in particle physics. As experimental precision enables discriminatory tests of mathematical structure in fundamental parameters, the field would benefit from explicit evaluation methodology to complement existing peer review processes.

We propose seven explicit criteria forming a filter pipeline that enables self-falsification through objective criteria while creating validated benchmarks for theory testing. The Diagnostic Pattern demonstrates intended operation: passing multiple objective criteria yet self-falsifying at directional persistence and theoretical viability. The framework successfully filters potentially spurious patterns while allowing robust observations (exemplified by Koide and GMO) to proceed, creating empirical benchmarks for theoretical development.

Future work should focus on: (1) community refinement of criterion thresholds and weighting, (2) application to other fermion mass hierarchies and mixing parameters, (3) theoretical investigation of patterns currently lacking known mechanisms, (4) testing framework utility through practical application, (5) iterative improvement based on community feedback.

This framework represents a starting point for community discussion on systematic evaluation standards as experimental precision enables increasingly discriminatory tests of mathematical structure in fundamental parameters.

Appendix A.1. Scale Invariance: QCD Renormalization Group Evolution

Implementation of MILC collaboration RG running algorithms with two-loop anomalous dimensions, flavor threshold matching, and 200-step numerical integration. Cross-validated against published tabulated values at 25 reference scales.

Table A1. Scale invariance verification using FLAG 2024 central values

Scale (GeV)	${\mathit{m}}_{\mathit{d}}/{\mathit{m}}_{\mathit{u}}$	${\mathit{m}}_{\mathit{s}}/{\mathit{m}}_{\mathit{d}}$	$2{({\mathit{m}}_{\mathit{d}}/{\mathit{m}}_{\mathit{u}})}^3$	Deviation
1.0	2.162	20.0	20.22	0.22
2.0	2.162	20.0	20.22	0.22
4.2	2.162	20.0	20.22	0.22
10.0	2.162	20.0	20.22	0.22
91.2	2.162	20.0	20.22	0.22
173.0	2.162	20.0	20.22	0.22
1000.0	2.162	20.0	20.22	0.22
Note: Values remain constant across all scales, confirming scale invariance
of mass ratios under QCD running. Deviation of 0.22 corresponds to
0.16$\sigma$ when propagating uncertainties from $m_d/m_u=2.162\pm 0.050$.

Table A1. Scale invariance verification using FLAG 2024 central values

Scale (GeV)	${\mathit{m}}_{\mathit{d}}/{\mathit{m}}_{\mathit{u}}$	${\mathit{m}}_{\mathit{s}}/{\mathit{m}}_{\mathit{d}}$	$2{({\mathit{m}}_{\mathit{d}}/{\mathit{m}}_{\mathit{u}})}^3$	Deviation
1.0	2.162	20.0	20.22	0.22
2.0	2.162	20.0	20.22	0.22
4.2	2.162	20.0	20.22	0.22
10.0	2.162	20.0	20.22	0.22
91.2	2.162	20.0	20.22	0.22
173.0	2.162	20.0	20.22	0.22
1000.0	2.162	20.0	20.22	0.22
Note: Values remain constant across all scales, confirming scale invariance
of mass ratios under QCD running. Deviation of 0.22 corresponds to
0.16$\sigma$ when propagating uncertainties from $m_d/m_u=2.162\pm 0.050$.

Appendix A.2. Cross-Collaboration Validation

Table A2. Cross-collaboration consistency analysis

Collaboration	${\mathit{m}}_{\mathit{d}}/{\mathit{m}}_{\mathit{u}}$	Uncertainty	Action	Deviation
ETM	2.15 ± 0.08	3.7%	Twisted mass	0.05$\sigma$
BMW	2.17 ± 0.07	3.2%	Stout smearing	0.23$\sigma$
MILC	2.15 ± 0.08	3.7%	Staggered	0.05$\sigma$
HPQCD	2.14 ± 0.06	2.8%	HISQ	0.24$\sigma$
World Average	2.162 ± 0.050	2.3%	Combined	0.16$\sigma$

Table A2. Cross-collaboration consistency analysis

Collaboration	${\mathit{m}}_{\mathit{d}}/{\mathit{m}}_{\mathit{u}}$	Uncertainty	Action	Deviation
ETM	2.15 ± 0.08	3.7%	Twisted mass	0.05$\sigma$
BMW	2.17 ± 0.07	3.2%	Stout smearing	0.23$\sigma$
MILC	2.15 ± 0.08	3.7%	Staggered	0.05$\sigma$
HPQCD	2.14 ± 0.06	2.8%	HISQ	0.24$\sigma$
World Average	2.162 ± 0.050	2.3%	Combined	0.16$\sigma$

Appendix A.3. Statistical Analysis

For the pattern $2{(m_d/m_u)}^3\approx m_s/m_d$:

• Using FLAG 2024: $m_d/m_u=2.162\pm 0.050$, $m_s/m_d=20.0\pm 0.5$
• Left side: $2{\left(2.162\right)}^3=20.22$
• Right side: $20.0$
• Propagated uncertainty: $\sigma =\sqrt{{(6·2.{162}^2·0.050)}^2+0.5^2}\approx 1.42$
• Normalized deviation: $(20.22-20.0)/1.42=0.16\sigma$

We treat $m_d/m_u$ and $m_s/m_d$ as uncorrelated for simplicity; correlated uncertainty treatment would not alter the methodological conclusion.

Appendix A.4. Temporal Persistence Analysis

Table A3. Temporal evolution of $m_d/m_u$ determinations

Review	${\mathit{m}}_{\mathit{d}}/{\mathit{m}}_{\mathit{u}}$	Rel. Uncert.	Distance from ${10}^{1/3}$
FLAG 2019	2.16 ± 0.08	3.7%	+0.006 (0.07$\sigma$)
FLAG 2024	2.162 ± 0.050	2.3%	+0.008 (0.16$\sigma$)
Direction: Away from prediction
Uncertainty improvement: 37.5%
Criterion 4 verdict: FAILS (directional divergence)

Table A3. Temporal evolution of $m_d/m_u$ determinations

Review	${\mathit{m}}_{\mathit{d}}/{\mathit{m}}_{\mathit{u}}$	Rel. Uncert.	Distance from ${10}^{1/3}$
FLAG 2019	2.16 ± 0.08	3.7%	+0.006 (0.07$\sigma$)
FLAG 2024	2.162 ± 0.050	2.3%	+0.008 (0.16$\sigma$)
Direction: Away from prediction
Uncertainty improvement: 37.5%
Criterion 4 verdict: FAILS (directional divergence)

Central value moved away from predicted 2.154 while uncertainty tightened. Under directional convergence requirement, this constitutes self-falsification despite remaining within statistical bounds.

Appendix A.5. Interpreting Temporal Falsification

Traditional statistical evaluation would note that both FLAG 2019 and FLAG 2024 measurements remain well within $1\sigma$ of the predicted value $m_d/m_u={10}^{1/3}\approx 2.154$, seemingly indicating robust temporal stability. However, this perspective misses critical information revealed by simultaneous error bar reduction and directional movement.

Why Error Bar Dynamics Matter: Between FLAG 2019 and FLAG 2024, relative uncertainty decreased by 37.5% (from 3.7% to 2.3%), representing substantial systematic and statistical improvements by the lattice QCD community. This precision gain creates an effective "zoom in" on the true value. When error bars shrink while measurements remain stable around a predicted value, this provides strong validation (as observed with Koide’s formula over decades). Conversely, when error bars shrink while central values converge around a different location than predicted, this reveals a systematic offset that broader uncertainties previously obscured.

Statistical Significance Evolution: The Diagnostic Pattern demonstrates the problematic scenario. Despite both measurements remaining within $1\sigma$, statistical significance actually increased:

• FLAG 2019: Measured 2.16, predicted 2.154, uncertainty 0.08 → $(2.16-2.154)/0.08=0.075\sigma$
• FLAG 2024: Measured 2.162, predicted 2.154, uncertainty 0.050 → $(2.162-2.154)/0.050=0.16\sigma$

The deviation in units of uncertainty doubled from 0.075$\sigma$ to 0.16$\sigma$ despite only 0.002 absolute movement. This occurs because the denominator (uncertainty) decreased faster than any convergence of the central value toward prediction. The measurements are converging toward approximately 2.162, not the predicted 2.154.

Contrast with Physical Patterns: When genuine physical relationships exist, precision improvements reveal convergence toward predicted values, with statistical significance improving (deviations decreasing in units of $\sigma$). The Diagnostic Pattern exhibits the opposite behavior: precision improvements reveal convergence around the wrong value, causing statistical significance to worsen (deviations increasing in units of $\sigma$).

Physical Interpretation: This temporal behavior suggests the pattern’s current statistical consistency results from limited discriminatory power at 2-3% precision rather than underlying physical mechanism. The relationship $2{(m_d/m_u)}^3=m_s/m_d$ appears to be a numerical near-coincidence: close enough to pass tests at current precision, but systematic improvements reveal the offset. Projecting forward, continued uncertainty reduction to sub-1% precision would likely expose statistically significant disagreement, revealing the coincidental nature before theoretical investment occurs.

Framework Operation: Criterion 4 intentionally flags this scenario for self-falsification. The requirement for directional convergence or stability prevents patterns from reaching advanced review stages when precision improvements systematically reveal offsets. This filtering occurs despite formal statistical consistency, demonstrating how multi-criteria evaluation provides discrimination beyond hypothesis testing alone. The Diagnostic Pattern’s temporal behavior, combined with absence of theoretical support (Criterion 7), motivated voluntary self-falsification rather than submission.