Structural Selection Principle

Companion code and papers for an ongoing research series on structural selection, coherence, and chaos in nonlinear field theories.

This work is part of an ongoing research programme on structural selection principles for dynamically consistent field configurations.

Author: Christof Krieg — Independent Researcher Academia.edu · GitHub: Chris4081

Start Here — Core Formalism Primer

If you are new to the series, start with Paper 00:

Core Formalism of Structural Selection
A Reader's Primer for the MAAT / Structural Selection Paper Series

This primer collects the common notation, master functional, support geometry, covariance-response closure, emergent robustness closure, CCI diagnostics, and cosmological projection layer used throughout the later papers. It is intended as the conceptual and mathematical entry point before reading the numbered research papers.

File	Role
`documentation/00_Core_Formalism_of_Structural_Selection.pdf`	compiled reader primer
`documentation/00_Core_Formalism_of_Structural_Selection.tex`	LaTeX source

Papers in this Series

Paper 01 — General Framework

A Structural Selection Principle for Dynamically Consistent Field Configurations Maximum-Entropy Weighting on Physical Solution Spaces

Core idea: δS[Φ]=0 defines possible configurations; a structural energy functional E[Φ] selects the physically preferred ones via a MaxEnt Boltzmann weight.

Paper 02/03 — 1D and 2D Topological Selection

Structural Selection in Nonlinear Field Theories: Kink Selection in 1D and Activity-Dependent Domain-Wall Selection in 2D

Core results:

System	Result
1D	Kinks dominate 81/81 parameter settings, weight 99.96%
2D	Selection transition at s★_c ∈ (0.20, 0.42)

Paper 04 — CCI Framework

A Structural Free Energy on the Solution Manifold Coherence Diagnostics, Critical Coherence Index, and Regime Classification

Core idea: CCI = destabilising / stabilising — a Reynolds-type number for field systems.

CCI < τ₁       →  Ordered   (solitons, kinks)
τ₁ ≤ CCI < τ₂  →  Critical  (intermittency)
CCI ≥ τ₂       →  Chaotic   (instability-dominated)

Paper 05 — Physical Grounding of the CCI

Physical Grounding of the Critical Coherence Index: Entropy Production and Structural Information

Core result: CCI correlates with Ṡ/I ratio:

Quantity	Spearman r
Ṡ/I ratio	0.760
Ṡ alone	0.530
I alone	0.246

Paper 06 — Continuous Ensemble Study

Structural Free Energy and the CCI in a Continuous Ensemble of φ⁴ Field Configurations An Extended Numerical Study with Mixed Initial Conditions

Core result: CCI is primarily a proxy for structural free energy (160-trajectory ensemble):

Quantity	Spearman r
F_struct	0.878 ← dominant
Ṡ/I ratio	0.288
I_nn alone	−0.341

Note: The entropy–information ratio should be understood as an empirical scaling proxy, while the structural free energy provides a more global effective description of solution selection.

Paper 07 — Entropy–Information Scaling (1D)

Entropy–Information Scaling of the Critical Coherence Index in Nonlinear Field Dynamics

Core result: CCI consistent with power-law scaling in 1D:

CCI ∝ Ṡ_cg⁺ / I_nn^α,   α ≈ 2.5
Spearman r_s = 0.734,  p ≈ 1.6×10⁻²¹

Phase diagrams over (σ_π, σ_noise) reveal a clear ordered → chaotic transition. The exponent α is empirical; its theoretical origin is an open problem.

Paper 08 — Landau-Type Theory

Landau-Type Theory of Structural Selection in Nonlinear Field Systems Entropy–Information Scaling and the Critical Coherence Index

Core idea: CCI and structural free energy fit a Landau framework:

F_struct[O] = a(μ)·O² + b·O⁴ + γ(∇O)²

Order parameter:   O ~ I_nn  (structural information)
Control parameter: μ ~ CCI

μ < μ_c  →  Ordered   (high coherence)
μ = μ_c  →  Critical  (phase boundary)
μ > μ_c  →  Chaotic   (low coherence)

The Landau coefficients are not derived from first principles — this is a theoretical proposal and interpretive framework.

Paper 09 — Entropy–Information Scaling (2D)

Entropy–Information Scaling of the CCI in Two-Dimensional φ⁴ Field Dynamics Dimensional Dependence and Robustness of Structural Instability

Core result: Scaling persists in 2D with shifted exponent:

Quantity	1D (Paper 07)	2D (Paper 09)
Spearman r_s	0.734	0.870
Best α	≈ 2.5	≈ 3.0 (within tested range)
Dominant factor	Ṡ/I balance	I_nn loss

Structural information (I_nn) becomes the dominant factor in 2D. Log-linear regression: R² ≈ 0.74.

Paper 10 — Dimensional Dependence of Scaling

Dimensional Dependence of Entropy–Information Scaling in Nonlinear Field Systems Evidence for a Dimension-Dependent Universality Class

Core result: Systematic dimensional trend of the scaling exponent:

Dimension	α_eff	Spearman r_s	Dominant factor
1D	≈ 2.5	0.734	Ṡ/I balance
2D	≳ 3.0 (within tested range)	0.870	I_nn loss

The effective exponent increases with dimension; the 2D value is a lower bound. The Spearman correlation strengthens from 1D to 2D, indicating increased robustness.

The dimensional dependence of α is a property of the projection from the structural free energy onto the entropy–information plane, rather than a fundamental property of the selection principle itself.

Paper 11 — Entropy–Information Scaling (3D)

Persistence of Entropy–Information Scaling in Three Dimensions Breakdown of a Simple Exponent Trend in Nonlinear Field Systems

Core result: Scaling persists in 3D but the exponent structure changes:

Quantity	1D	2D	3D
Spearman r_s	0.734	0.870	0.850
α_eff	≈ 2.5	≳ 3.0	plateau [2.8, 3.5]
R²	moderate	0.74	0.37
Dominant	Ṡ/I balance	I_nn loss	balanced

The notion of a single effective exponent breaks down in 3D: instead of a sharp optimum, a broad plateau of near-maximal correlation appears over α ∈ [2.8, 3.5]. The reduced R² indicates a degeneracy of effective scaling descriptions.

The entropy–information scaling remains robust across dimensions, but transitions from a well-defined power-law exponent to a broad effective scaling regime in higher dimensions.

Paper 12 — Cross-Dimensional Collapse Analysis

Collapse Analysis of Entropy–Information Scaling Across Spatial Dimensions Shared Structure and Dimensional Degeneracy in Nonlinear Field Systems

Core idea: Direct visual comparison of the scaling relation across 1D, 2D, and 3D via collapse plots, testing whether a unified cross-dimensional description exists.

Core result: Partial collapse observed — shared monotonic structure, no universal curve.

View	Observation
Log-x	shared monotonic trend across 1D, 2D, 3D
Log-log	similar trend with increasing 3D dispersion
Normalized	partial overlap; 3D scatter broader

Scaling survives, but the exponent does not. The partial collapse reflects a shared structural principle expressed through dimension-dependent effective parameters.

Script: Collapse-Plot.py

Requires the following CSV files (how to generate them):

CSV file	Generated by
`cci_entropy_information_test.csv`	`python cci_alpha_scaling.py`
`2d_cci_entropy_information_test.csv`	`python cci_entropy_scaling_2d.py`
`3d_cci_entropy_information_test.csv`	`python cci_entropy_scaling_3d.py`

Paper 13 — Multi-Parameter Scaling Analysis

Beyond Single-Exponent Scaling: Multi-Parameter Entropy–Information Relations in Nonlinear Field Systems

Core idea: Extends the single-exponent ansatz CCI ∝ Ṡ^α / I_nn^α to a two-parameter model CCI ∝ Ṡ^a · I_nn^{-b}, testing whether the 3D exponent plateau reflects a deeper multi-channel structure.

Core result: The entropy and information weights vary systematically with dimension:

Dimension	a	b	R²
1D	−1.60	0.33	0.76
2D	−0.08	0.50	0.74
3D	+0.30	0.30	0.37

The entropy contribution changes sign from 1D to 3D — a qualitative shift in the role of dynamical activity across dimensions.

The exponent description fails not because the scaling breaks, but because the system transitions to a higher-dimensional multi-parameter scaling structure.

Script: multiparameter_fit.py

Requires the following CSV files (see Paper 12 for how to generate them):

CSV file	Generated by
`cci_entropy_information_test.csv`	`python cci_alpha_scaling.py`
`2d_cci_entropy_information_test.csv`	`python cci_entropy_scaling_2d.py`
`3d_cci_entropy_information_test.csv`	`python cci_entropy_scaling_3d.py`

Paper 14 — Scaling Manifold Framework

Scaling Laws as Projections: Evidence for a Dimension-Dependent Scaling Manifold in Nonlinear Field Systems

Core idea: The observed power-law scaling is not fundamental, but represents a local projection of a higher-dimensional geometric object — the scaling manifold:

M_scale = { (Ṡ, I_nn, d, CCI) | CCI = F(Ṡ, I_nn, d) }

Effective exponents are local tangent directions of this manifold; exponent degeneracy in 3D reflects its higher-dimensional geometry.

Empirical evidence:

Evidence	Result
PCA of log-space observables	PC1 dominates in 1D/2D; PC2/PC3 gain in 3D
Joint PCA (1D+2D+3D)	shared structure with dimension-dependent spread
ξ_aniso quartile-split (3D)	plateau width Δα varies from 0.20 to 2.20 across quartiles
Continuous correlation	r_s(ξ_aniso, R_α) = −0.721, p = 0.0001

Key result:

Scaling laws are coordinate-dependent projections of a scaling manifold. Universality may extend from scaling exponents to geometric structures.

Scripts: manifold_geometry_plot.py, xi_aniso_full_test.py

Requires the following CSV files (see Paper 12 for how to generate them):

CSV file	Generated by
`cci_entropy_information_test.csv`	`python cci_alpha_scaling.py`
`2d_cci_entropy_information_test.csv`	`python cci_entropy_scaling_2d.py`
`3d_cci_entropy_information_v2.csv`	`python cci_entropy_scaling_3d_v2.py`

Paper 16 — Plateau Degeneracy Measure

Plateau Degeneracy in Entropy–Information Scaling: A Quantitative Measure for the Breakdown of Single-Exponent Descriptions in Nonlinear Field Systems

Core idea: Formalises the breakdown of single-exponent scaling observed in 3D by introducing a rigorous plateau degeneracy framework.

Key definitions:

Measure	Formula	Meaning
Plateau set	P_ε = {α \| r_s(α) ≥ r_s^max − ε_plat}	near-optimal exponents
Width	D_width = α_max^(ε) − α_min^(ε)	plateau size
Normalised	D_norm = D_width / (α_max − α_min)	in [0, 1]
Flatness	D_flat = Var(r_s) within P_ε	0 = perfectly flat
Combined	D = D_width / (1 + D_flat)	joint measure

Exact results (ε_plat = 0.01 · r_s^max, scan range α ∈ [0.5, 3.5], step h = 0.1):

Dimension	α*	r_s^max	Plateau	D_norm	n
1D	3.5	0.748	[2.9, 3.5]	0.200	7
2D	3.5	0.881	[3.1, 3.5]	0.133	5
3D	2.8	0.849	[2.6, 3.5]	0.300	10

Key finding:

D_norm(2D) < D_norm(1D) < D_norm(3D) — non-monotonic dimensional dependence. The breakdown of single-exponent scaling is not gradual but occurs via a dimensional transition between 2D and 3D.

Script: plateau_degeneracy_exact.py

Requires the following CSV files:

CSV file	Used for
`cci_entropy_information_test.csv`	1D r_s(α) scan (Paper 07)
`2d_cci_entropy_information_test.csv`	2D r_s(α) scan (Paper 09)
`3d_cci_alpha_scan.csv`	3D r_s(α) scan (Paper 11)

Paper 17 — Predictive Power of the CCI

Predictive Power of the Critical Coherence Index: From Structural Diagnostics to Regime Classification

Core idea: Tests whether the CCI can predict the dynamical regime of a system using a threshold-based classifier on a 120-sample 1D benchmark dataset.

Core results:

Classifier	Accuracy	τ₁	τ₂
CCI-only	1.000 (120/120)	0.2785	0.3527
F_struct-only	0.8417 (101/120)	0.4204	0.4984

5-fold cross-validation (CCI-only): Ā = 0.992 ± 0.019 (folds: [1.000, 1.000, 1.000, 0.958, 1.000])

The CCI alone is sufficient to fully separate ordered, critical, and chaotic regimes. The near-perfect cross-validated accuracy indicates robustness across data splits.

Script: paper17_analysis.py

Requires: cci_entropy_information_test.csv (1D benchmark dataset)

Paper 18 — Dynamical Origin of Structural Selection

A Minimal Dynamical Model for Structural Selection: A Field Equation for the Critical Coherence Index

Core idea: Promotes the Critical Coherence Index (CCI) from a diagnostic observable to a dynamical field C(x,t), governed by a minimal nonlinear relaxation equation with gradient-flow structure.

Model equation:

Term	Expression	Meaning
Dynamics	∂ₜC = D ∇²C + aC − bC³	coherence-field evolution
Free energy	F[C] = ∫ [ (D/2)\|∇C\|² − (a/2)C² + (b/4)C⁴ ] dx	structural potential
Gradient flow	∂ₜC = −δF/δC	energy minimisation

Stationary structure:

Regime	Condition	Solution
Disordered	a < 0	C = 0 (stable)
Critical point	a = 0	pitchfork bifurcation
Ordered	a > 0	C = ±√(a/b)

Numerical results (a = b = 1, D = 0.8):

Quantity	Value	Interpretation
Fixed points	±1.0	stable coherence states
⟨C⟩ (1D)	-0.080	domain balance
std(C) (1D)	0.939	saturation near ±1
Energy drop	0.057 → -27.076	monotonic decay

Key findings:

The coherence-field equation reproduces spontaneous symmetry breaking, domain formation, and energy minimisation within a minimal framework. Structural selection emerges dynamically as gradient descent in a quartic free-energy landscape.

Script: coherence_simulation.py

Features:

Component	Description
0D dynamics	relaxation to fixed points
1D field	domain formation with periodic BC
Energy tracking	verifies gradient-flow behaviour

Paper 19 — Active Structural Control

Active Structural Control in Nonlinear Field Systems: Parameter-Dependent Steering of Coherence

Core idea: Extends the CCI framework from a passive diagnostic into an active control paradigm. A feedback control term is introduced into the coherence-field dynamics, formalising the complete loop φ → C → U → φ. Structural coherence becomes a controllable quantity rather than a purely emergent observable.

Control scheme:

Step	Expression	Role
Field → Coherence	C = G_σ[φ²] − (G_σ[φ])²	coarse-grained variance
Coherence → Control	U = −(C − C*)	feedback signal
Control → Field	φ̈ = ∇²φ − (φ³−φ) − γφ̇ + λU	controlled dynamics

Best-performing configuration (γ = 0.02, λ = 0.24):

Observable	Baseline	Controlled	Change
CCI	0.0579	0.0540	−6.7%
F_struct	3.3433	3.1322	−6.3%
I_nn	0.1080	0.2746	+154%

Two operational regimes:

Regime	Damping	Effect
Structure refinement	γ ≳ 0.05	F_struct ↓, I_nn ↑, CCI stable
Regime steering	γ ≲ 0.03	CCI ↓, genuine regime transition

Structural selection can be formulated as a control problem. The feedback loop φ → C → U → φ enables target-driven steering of the system in coherence space, with qualitatively distinct effectiveness regimes depending on damping strength.

Script: active_control_phi4.py

Requires: 2D φ⁴ field simulation with RK4 integration (see repository)

Paper 20 — Universal Stability Test

Universal Stability of Structural Observables Across Dimensions: A Comparative Test of CCI, Structural Free Energy, and Entropy–Information Scaling

Core idea: Tests which structural observables remain dimensionally robust across 1D, 2D, and 3D φ⁴ systems. Introduces relative drift D_rel = max(μ_d)/min(μ_d) as a scale-invariant measure of dimensional stability.

Core results:

Observable	1D Mean	2D Mean	3D Mean	Max/Min Drift
CCI	0.303	0.457	0.601	1.98
F_struct	0.430	0.641	0.824	1.92
MI	0.574	0.449	0.393	1.46
Ṡ⁺	0.263	0.080	0.111	3.30
Ratio	6.81	71.64	13563	1992

Key finding:

CCI and F_struct remain comparatively stable across dimensions (drift < 2), while the entropy–information ratio loses consistency as a dimensionally robust observable (drift ≈ 2000). The instability is not merely quantitative but structural: it reflects a projection-dependent construction that cannot define a consistent observable across dimensions.

Hierarchical interpretation:

Observable	Role
CCI	comparatively robust structural coordinate
F_struct	stable global effective functional
Ratio	structurally unstable projection-dependent proxy

Structural observables (CCI, F_struct) define intrinsic coordinates on the solution manifold. Scaling ratios correspond to dimension-dependent projections.

Script: uni_stability_test.py

Requires the following CSV files (how to generate them):

CSV file	Generated by
`cci_entropy_information_test.csv`	`python cci_alpha_scaling.py`
`2d_cci_entropy_information_test.csv`	`python cci_entropy_scaling_2d.py`
`3d_cci_entropy_information_test.csv`	`python cci_entropy_scaling_3d.py`

Paper 22 — Numerical Validation of Structural Selection

Numerical Validation of Structural Selection: Evidence Beyond Energy-Based Ranking in a Minimal φ⁴ Model

Core idea: First static lattice validation of the structural-selection functional in a minimal 1+1 dimensional φ⁴ model. Tests whether F_struct does more than reproduce ordinary energy ranking by benchmarking vacua, the unstable homogeneous saddle, the discrete kink, and a sector-matched distortion ensemble.

Core results:

Benchmark	Result	Interpretation
Vacuum discrimination	Δ_vac = 2.583852	true vacua cleanly separated from unstable saddle
Kink discrimination	p_K = 1.000	discrete kink outranks all tested distortions
Ranking test (20%)	0.010363 < 0.012128	structural ranking outperforms energy ranking at the 20% percentile

Key finding:

Structural selection becomes numerically testable in a minimal field-theory setting. The structural layer is not merely a relabeling of on-shellness or static energy: it distinguishes stable from unstable exact solutions and adds measurable ranking power within a nontrivial topological sector.

Scripts:

Script	Role
`structural_selection_phi4_protocol.py`	runs the static lattice benchmark and generates `structural_selection_phi4_results.json`
`structural_selection_phi4_plots.py`	reads `structural_selection_phi4_results.json` and generates the validation figures

To reproduce the results:

python3 structural_selection_phi4_protocol.py --json-output structural_selection_phi4_results.json --pretty-json
python3 structural_selection_phi4_plots.py --input structural_selection_phi4_results.json --output-dir structural_selection_phi4_plots

This produces:

Output	Generated by
`structural_selection_phi4_results.json`	`structural_selection_phi4_protocol.py`
`structural_selection_phi4_plots/`	`structural_selection_phi4_plots.py`

Documentation PDF: documentation/22_Numerical_Validation_of_Structural_Selection__Evidence_Beyond_Energy_Based_Ranking_in_a_Minimal_phi4_Model.pdf

Paper 26 — Structural Selection of Effective Constants

Structural Selection of Effective Constants: From MAAT Basins to MaxEnt-Weighted RG Bridge Tests

Core idea: Extends structural selection from nonlinear field configurations to effective constants. The paper tests whether MAAT-type structural scores define low-defect basins in constant space and whether Standard-Model-like RG flow preserves cross-sector structural constraints. The v12/v13 extension calibrates the sector weights lambda_a through a maximum-entropy procedure.

Scientific status: This is a phenomenological benchmark, not a first-principles derivation of the constants, not a precision Standard Model fit, and not a solution of the cosmological-constant problem.

Core results:

Benchmark	Result	Interpretation
Natural-constants basin	observed proxy and MAAT optimum lie in same low-defect basin	basin-level compatibility, not exact prediction
SM RG bridge	`F_bridge = 0.210446 < F_obs = 0.231789`	selected bridge point lies in same structural region as observed SM proxy
v11 holdout: α_em	predicted/observed = 1.581	direct electromagnetic term can be removed while cross-sector constraints remain informative
v11 holdout: sin²θ_W	predicted/observed = 1.543	weak-sector proxy remains order-of-magnitude constrained
v11 holdout: λ_H	predicted/observed = 0.170	Higgs quartic sector is the main limitation of the current toy bridge
v12 MaxEnt weights	`R > V ≈ S > B > H`	equal sector weighting is replaced by calibrated effective weights
v13 MaxEnt bridge	stability = 0.9936	MaxEnt-weighted run preserves the low-defect constants basin

Key finding:

Structural selection identifies a compatible basin of effective constants rather than a unique point. The v11 holdout test provides a first falsifiable cross-sector prediction protocol, while v12/v13 reduce the earlier equal-weight assumption by MaxEnt-calibrating effective sector weights.

Reference data sources:

Source	Used for
CODATA/NIST fundamental constants	low-energy constants and dimensionless comparison values
Particle Data Group 2024	Standard-Model proxy values near the electroweak scale

Data attribution and license note: PDG 2024 content is licensed under CC BY 4.0 except where otherwise noted. NIST web information is generally public information unless marked otherwise, but NIST Standard Reference Data may have separate copyright and licensing conditions. The benchmark uses only reference comparison values; users should cite CODATA/NIST and PDG when reusing or discussing the numerical inputs.

Scripts and reproducibility:

Folder	Role
`experiments/natural_constants_selection/`	v1--v13 natural-constants basin, robustness, ablation, gradient-flow, and MaxEnt-weight tests
`experiments/standard_model_bridge/`	one-loop SM-like RG bridge, four-panel summary figure, and v11 holdout test

Documentation PDF: documentation/26_Structural_Selection_of_Effective_Constants.pdf

Paper 27 — Boundary-Aware Calibration of MAAT Structural Weights

Boundary-Aware Calibration of MAAT Structural Weights: A Reproducible Benchmark for Constraint-Dominated Structural Selection

Core idea: Calibrates MAAT structural weights on a fused dataset containing SAT hardness instances, unconstrained MAAT-Core states, and explicit constraint-boundary regimes. The key question is whether boundary information changes the inferred selection hierarchy.

Core results:

Quantity	Result
Fused samples	3400
Sources	SAT hardness atlas, MAAT-Core, MAAT-Core boundary
Lambda hierarchy	`R > V > H > S ~= B`
Dominant share	`R = 0.3917`
Fit loss	`0.0029543310`

Key finding:

Robustness / Respect becomes the dominant structural selector once explicit boundary, margin, and violation information are included.

Scripts and reproducibility:

Folder	Role
`experiments/boundary_aware_lambda_calibration/`	fused defect dataset, closed lambda fit, plots, and result JSON

Documentation PDF: documentation/27_Boundary_Aware_Calibration_of_MAAT_Structural_Weights.pdf

Paper 28 — Cosmological Critical Coherence Index

Cosmological Critical Coherence Index: A Structural-Stress Observable for Cosmic Evolution

Core idea: Defines a dimensionless cosmological CCI diagnostic combining expansion stress, redshift activity, linear growth coherence, and structural imbalance. The goal is to visualise the balance between expansion history and structure-growth history, not to replace standard cosmological inference.

Core results:

Quantity	Result
Reference cosmology	Planck-2018 flat LCDM
Chronometer points	31
Model range	`0 <= z <= 10`
Normalised CCI at `z=1`	`~6.66`
Normalised CCI at `z=2`	`~22.6`
Normalised CCI at `z=10`	`~803.8`
Chronometer residual RMS	`~1.01`

Key finding:

Cosmic evolution can be represented not only as expansion history, but as a structural-stress history comparing expansion/activity against growth coherence.

Operational convention: This first CCI-cosmology projection uses unit weights, fixes kappa = 1, and does not fit regime cutoffs. Full MAAT connectivity (V) and robustness (R) sectors are specified in the paper as operational targets for future multi-probe work, not as measured sectors in the chronometer-only projection.

Scripts and reproducibility:

Folder	Role
`experiments/cosmological_cci/`	Cosmological CCI script, chronometer input table, generated CSVs, and plots

Data attribution and license note: Planck-2018 parameter values and Cosmic Chronometer measurements are external scientific data and should be cited to the original publications/collaborations when reused or discussed. The CSV tables and figures in this repository are derived analysis artifacts generated for reproducibility of the CCI diagnostic. No endorsement by the Planck Collaboration or the chronometer-data authors is implied.

Documentation PDF: documentation/28_Cosmological_Critical_Coherence_Index.pdf

Paper 29 — Cosmological CCI with Growth Connectivity and Robustness

Cosmological Critical Coherence Index with Growth Connectivity and Robustness Margins: A Multi-Sector Structural-Stress Benchmark Using H(z) and f sigma_8(z)

Core idea: Extends the chronometer-only cosmological CCI into a multi-sector diagnostic by making the previously open connectivity (V) and robustness (R) sectors operational. Cosmic Chronometer H(z) residuals define expansion consistency, BOSS DR12 f sigma_8 residuals define growth connectivity, and joint expansion-growth consistency defines robustness.

Core results:

Quantity	Result
Chronometer points	31
BOSS DR12 f sigma_8 points	3
H(z) pull RMS	`0.693`
f sigma_8 pull RMS	`0.697`
Transition proxy	`z_c ~= 1.114`
Normalised v0.3 CCI at `z=1`	`~6.60`
Normalised v0.3 CCI at `z=2`	`~15.42`
Lambda hierarchy	`S > H > R > V`

Key finding:

The cosmological CCI can be extended from an expansion-growth scalar into a multi-sector observable with measured growth connectivity, robustness margins, companion MaxEnt weights, and a reproducible transition marker.

Scientific status: This is a diagnostic benchmark, not precision cosmological inference. It does not fit cosmological parameters, replace LCDM, or claim evidence for modified gravity.

Scripts and reproducibility:

Folder	Role
`experiments/cosmological_cci_v03/`	v0.3 cosmological CCI script, H(z) and f sigma_8 input tables, generated CSV/JSON outputs, and plots

Data attribution and license note: Paper 29 uses Planck-2018 reference parameters, Cosmic Chronometer H(z) values, and BOSS DR12 consensus f sigma_8 measurements from the cited literature. Repository CSV/PNG files are derived reproducibility artifacts only. No endorsement by the Planck Collaboration, SDSS/BOSS Collaboration, or the chronometer-data authors is implied.

Documentation PDF: documentation/29_Cosmological_CCI_Growth_Connectivity_and_Robustness.pdf

Paper 30 — Response-Based Derivation of MAAT Structural Weights

Response-Based Derivation of MAAT Structural Weights: From Covariance Geometry to Selection Pressure

Core idea: Provides the missing response-theoretic interpretation of lambda_a. Instead of treating the weights as arbitrary fitted constants, the addendum derives them as linear-response coefficients of the empirical defect ensemble:

lambda = (Cov_mu0[d] + eta tr(C)/A I)^(-1)(<d>_mu0 - <d>_target)

Here mu0 is the empirical reference measure over the fused defect ensemble, Cov_mu0[d] is the defect covariance matrix, and <d>_target is the selected target sub-ensemble.

Core results:

Target ensemble	Main hierarchy	Interpretation
Low-defect 20%	`R > B > S > H > V`	generic low-defect selection gives dominant but not exclusive robustness
Safe + core-safe	`R >> H > B > V`	explicit safety targets are overwhelmingly robustness-driven
Safe boundary only	`R > S > B > V > H`	boundary-safe states reproduce a balanced but R-dominant hierarchy
Not violated	`R > S`	excluding violations activates mainly robustness and activity

Key finding:

R dominance emerges as a covariance-response effect of safety/boundary target selection. The weights are still effective and ensemble-dependent, but they are no longer merely hand-chosen fit parameters.

Scripts and reproducibility:

Folder	Role
`experiments/lambda_response_closure/`	response-theoretic lambda derivation, result JSON, and comparison plots

Documentation PDF: documentation/30_Response_Theoretic_Closure_of_MAAT_Structural_Weights.pdf

Paper 31 — Dynamic Structural Selection

Dynamic Structural Selection: From Response Fields to Observable Cosmology in the MAAT Framework

Core idea: Consolidates the MAAT versioned development from static structural ranking into a dynamical, local, gravitational, and observable effective-theory pipeline:

d_a -> C_ab -> lambda_a(t) -> lambda_a(x,t) -> T_MAAT -> FLRW -> observables

The paper combines response-field dynamics, local selection-pressure fields, effective gravitational coupling, a scalar worked example, an FLRW stability scan, and an observable-projection layer.

Core results:

Layer	Result	Interpretation
v0.5 response flow	final tracking error `9.91e-7`	lambda follows covariance-response fixed point
v0.6 local fields	final residual ratio `0.1648`	local fields reduce equation residuals with energy unchanged
v0.9 FLRW scan	`666/784` stable, `618/784` background-safe	broad stable region in the toy scalar scan
v0.10 observables	max `	Delta H/H
v0.10 growth proxy	max `	Delta f sigma_8/f sigma_8

Key finding:

Structural selection can be represented as a reproducible effective-theory ladder from defect diagnostics to observable cosmological signatures, while remaining explicitly toy-level and not a completed cosmological model.

Scripts and reproducibility:

Folder	Role
`experiments/maat_dynamic_fields_v05_v09/`	v0.5 response flow, v0.6 local field benchmark, v0.9 FLRW stability scan
`experiments/maat_observable_predictions_v10/`	observable projection used in Paper 31

Documentation PDF: documentation/31_Dynamic_Structural_Selection_From_Response_Fields_to_Observable_Cosmology.pdf

Paper 32 — First H(z) Data Comparison

First H(z) Data Comparison of MAAT Structural-Selection Cosmology: Cosmic Chronometer Diagnostic and Chi-Square Scan

Core idea: Moves the MAAT cosmology stack from internal observable projection to a first direct comparison with observational expansion data. The paper uses Cosmic Chronometer H(z) measurements as a diagnostic goodness-of-fit test for the fixed v0.10 branch and for a two-parameter MAAT scalar scan.

Core results:

Quantity	Result
Chronometer points	`31`
LCDM reference	`chi2 = 14.8759`, `chi2/N = 0.4799`
Fixed v0.10 MAAT	`chi2 = 15.3661`, `chi2/N = 0.4957`
Best scanned MAAT branch	`chi2 = 15.3674`, `Delta chi2 = +0.4915` vs LCDM
Best scan reduced chi-square	`chi2_nu = 0.5299` for `N-k = 29`
Stable scan points	`581/616`
Best branch maximum `Omega_MAAT`	`0.02668`
Best branch maximum `	Delta H/H

Key finding:

In this first H(z) diagnostic, MAAT remains stable, subdominant, and close to LCDM, but it is not favoured over LCDM by the Cosmic Chronometer chi-square. The value of the result is therefore falsifiability and pipeline closure, not a detection claim.

Scripts and reproducibility:

Folder	Role
`experiments/maat_hz_chi2_paper32/`	fixed v0.10 chronometer comparison and two-parameter H(z) chi-square scan

Documentation PDF: documentation/32_First_Hz_Data_Comparison_of_MAAT_Structural_Selection_Cosmology.pdf

Paper 33 — CCI as Projection Observable

The Critical Coherence Index as a Projection Observable: Breadth--Depth Compression and Transition Robustness in a Minimal Cosmological Benchmark

Core idea: Clarifies what the CCI represents in the cosmological setting: not an energy density and not a fitted cosmological parameter, but a projection observable measuring stress between expansion-driven breadth and accessible structural depth.

Core results:

Quantity	Result
v0.3 reference background	Planck-like flat LCDM
v0.3 transition estimate	`z_tr = 0.84985`
v0.3 `C_norm(z_tr)`	`12.0466`
v0.3 final `C_norm(z=3)`	`474.184`
v0.4 sensitivity points	`6930`
v0.4 mean transition redshift	`0.84491`
v0.4 median transition redshift	`0.88889`
v0.4 fraction in `0.5 <= z_tr <= 1.1`	`0.55657`

Key finding:

The CCI can be operationally interpreted as a stabilized projection stress: it grows when expansion breadth outpaces accessible coherent depth. The resulting transition marker is reproducible across a broad proxy-parameter grid, but it is not a measured cosmological transition or a data-fit result.

Scripts and reproducibility:

Folder	Role
`experiments/maat_cci_projection_paper33/`	v0.2 stabilization predecessor, v0.3 LCDM-growth projection benchmark, and v0.4 sensitivity scan

Documentation PDF: documentation/33_CCI_as_Projection_Observable.pdf

Paper 34 — Projection Observable vs Growth and Expansion Data

MAAT Projection Observable vs Growth and Expansion Data: Transition Marker and Parameter Sensitivity in v0.11

Core idea: Tests whether the MAAT projection observable C_proj collapses onto standard growth data. It does not: the large mismatch against f sigma_8 is interpreted as evidence that C_proj is a projection-level diagnostic, not a matter-growth observable.

Core results:

Quantity	Result
v0.11 baseline transition marker	`z_tr = 1.04356`
`C_norm(z_tr)`	`22.64463`
`R_proj(z_tr)`	`1.11151e-03`
`C_norm(z=3)`	`647.49295`
LCDM `f sigma_8` chi-square/dof	`1.04813`
Rescaled `C_proj` vs `f sigma_8` chi-square/dof	`42.76835`
LCDM `H(z)` chi-square/dof	`0.47944`
Gamma-alpha scan points	`625`
Scan median transition	`1.18715`
Scan transition range	`[0.28073, 1.58353]`

Key finding:

Paper 34 demonstrates that the MAAT projection observable does not reduce to known growth observables, establishing it as a distinct layer of cosmological information. The result is diagnostic, not a cosmological fit or evidence claim.

Scripts and reproducibility:

Folder	Role
`experiments/maat_projection_growth_paper34/`	v0.11 projection residual, growth-data mismatch diagnostic, H(z) reference, and gamma-alpha transition scan

Documentation PDF: documentation/34_MAAT_Projection_Observable_vs_Growth_and_Expansion_Data.pdf

Paper 35 — Linear Growth Embedding

Linear Growth Embedding of MAAT Structural Selection: Perturbation Stability, Positivity, and Sub-Percent Growth Signatures

Core idea: Moves beyond the direct C_proj versus f sigma_8 template comparison of Paper 34 by embedding the representative MAAT branch into a minimal linear growth equation. The goal is to test whether MAAT produces small and stable growth-sector effects when treated as a forward model.

Core results:

Quantity	Result
`Omega_MAAT,0`	`0.00331`
`w_MAAT`	`-0.801`
Max `	Delta D/D
Max `	Delta f sigma_8/f sigma_8
NEC margin	`+0.199 rho`
Positivity stress tests	`3/3 positive`
Fixed-point mode stability	all tested `gamma_k > 0`

Key finding:

A minimal MAAT growth embedding produces sub-percent growth-sector deviations while preserving linear stability, positivity, and acceptable energy-condition behaviour for the tested branch. This is a first perturbative toy embedding, not a precision cosmological constraint.

Scripts and reproducibility:

Folder	Role
`experiments/maat_growth_perturbation_paper35/`	corrected LCDM reference, MAAT growth equation, selection-pressure perturbations, positivity stress test, and energy-condition check

Documentation PDF: documentation/35_Linear_Growth_Embedding_of_MAAT_Structural_Selection.pdf

Paper 36 — The Maturation of MAAT

The Maturation of MAAT: From Projection Observables (v0.11) to Structural Selection (v1.0) to Emergent Robustness Closure (v1.2.1)

Core idea: Reconstructs the conceptual evolution of the MAAT framework from projection diagnostics to structural selection and finally to the v1.2.1 robustness closure. The paper resolves the ambiguity of the old independent R sector by replacing it with derived support-level quantities: R_resp for structural respect and R_rob for emergent robustness.

Core results:

Quantity	Result
Primary support sectors	`H, B, S, V`
Old v1.0 input	independent `R`
v1.2.1 respect closure	`R_resp = (H B V)^(1/3)`
v1.2.1 robustness closure	`R_rob = min(R_resp, (H B S V)^(1/4))`
Stability convention	`Stability = R_rob`
Closure coefficient	`lambda_cl`, covariance/response-derived
Primary-field count	reduced from `5` to `4`

Key finding:

MAAT v1.2.1 does not add another sector. It reduces the framework by reconstructing robustness as a derived closure quantity on normalised supports. The result is a more closed and interpretable selection framework with the same baseline observable pipeline.

Scientific status: This is a conceptual closure and synthesis paper. It does not introduce a new numerical benchmark. Its role is to make the notation, support-level interpretation, and R-closure logic internally consistent across Papers 30--35 and the Paper 37 v1.2.1 benchmark.

Documentation PDF: documentation/36_The_Maturation_of_MAAT.pdf

Paper 37 — MAAT v1.2.1 Observable Proxy Predictions

MAAT v1.2.1 Observable Proxy Predictions and Stability Landscape: Structural Fields, Projection Layer, CCI Diagnostics, and Parameter Sensitivity

Core idea: Provides the numerical companion benchmark for Paper 36. The paper applies the v1.2.1 closure convention with four primary support sectors H, B, S, V, derived structural respect R_resp=(H B V)^(1/3), and emergent robustness R_rob=min(R_resp,(H B S V)^(1/4)). It then tests the baseline observable proxy and scans a two-parameter projection/damping landscape.

Core results:

Quantity	Result
Trajectory points	`382`
Redshift range	`0 <= z <= 2.33`
Baseline max `	Delta H/H
Baseline max `Omega_MAAT`	`3.31%`
Baseline max `	Delta f sigma_8 / f sigma_8
Mean structural respect	`<R_resp> = 0.818`
Mean emergent robustness	`<R_rob> = 0.805`
Mean CCI v1.2.1	`0.208`
Stability scan	`1089/1089` internally stable points
Best scan point	`eta = 0.000`, `gamma = 0.050`
SAT companion correlations	`rho(V, runtime)=0.647`, `rho(R_rob, runtime)=0.484`

Key finding:

The v1.2.1 closure does not destabilise the observable proxy pipeline. Emergent robustness remains finite across the scanned parameter space, and the baseline is not an isolated fine-tuned point within the tested two-parameter landscape.

Scientific status: This is a toy/proxy benchmark, not an observational cosmology fit. Growth deviations are simplified proxy quantities, not Boltzmann-code predictions.

Scripts and reproducibility:

Folder	Role
`experiments/maat_v121_observables_stability_paper37/`	Paper 37 observable proxy run, stability landscape scan, SAT companion validation, output CSV/JSON files, and figures

Documentation PDF: documentation/37_MAAT_v121_Observable_Proxy_Predictions_and_Stability_Landscape.pdf

Paper 38 — MAAT v1.2.1 Robustness Closure in Linear Growth

MAAT v1.2.1 Robustness Closure in Linear Growth: Sub-Percent Growth Signatures, Emergent Robustness, and Selection-Field Stability

Core idea: Applies the v1.2.1 closure convention to a minimal linear-growth benchmark. The paper updates the Paper 35 growth pipeline with the Paper 36/Paper 37 closure logic: R_resp=(H B V)^(1/3), R_rob=min(R_resp,(H B S V)^(1/4)), and Stability=R_rob. It also checks selection-field perturbations, positivity of the lambda-relaxation equation, fixed-point decay, and basic energy-condition diagnostics.

Core results:

Quantity	Result
Max `	Delta D/D
Max `	Delta f sigma_8 / f sigma_8
Mean `R_resp`	`0.9543`
Mean `R_rob`	`0.9313`
Mean `CCI_min`	`0.3114`
Mean `CCI_diag`	`0.2348`
Selection-field positivity	passed all tested scenarios
Fixed-point perturbations	stable for all tested modes

Key finding:

The v1.2.1 robustness closure can be inserted into a growth-sensitive benchmark without producing large observable deviations or obvious selection-field instabilities. The result is an internal consistency check, not an observational detection.

Scientific status: This is an effective-theory growth benchmark, not a Boltzmann-code calculation and not a cosmological likelihood.

Scripts and reproducibility:

Folder	Role
`experiments/maat_paper38_v121_robustness_closure/`	Paper 38 linear-growth closure benchmark, selection-field perturbation test, output CSV/JSON files, and figures

Documentation PDF: documentation/38_MAAT_v121_Robustness_Closure_in_Linear_Growth.pdf

Paper 39 — MAAT v1.2.1 Observable Growth Signature Proxy

MAAT v1.2.1 Observable Growth Signature Proxy: Projection-Modulated f sigma_8, Emergent Robustness, and a Boundary-Limited Diagnostic Scan

Core idea: Tests whether the MAAT projection observable can be inserted as a small explicit signature template in the growth observable f sigma_8(z) while preserving the v1.2.1 closure convention: R_resp=(H B V)^(1/3), R_rob=min(R_resp,(H B S V)^(1/4)), and Stability=R_rob.

Core results:

Quantity	Result
Growth comparison points	`13`
Projection transition estimate	`z_tr ~= 0.6508`
Epsilon scan range	`[-0.01, 0.01]`
Best epsilon	`-0.0100`
LCDM chi2	`12.4373`
MAAT proxy chi2	`12.3772`
Delta chi2	`-0.0601`
Max `	Delta f sigma_8 / f sigma_8
Mean `	Delta f sigma_8 / f sigma_8
Mean `R_resp`	`0.7247`
Mean `R_rob`	`0.6673`
Mean `CCI_min`	`0.2662`
Mean `CCI_diag`	`0.2036`

Key finding:

A bounded MAAT projection template can modulate f sigma_8(z) at the sub-percent level while preserving the v1.2.1 robustness closure. The best epsilon value lies at the scan boundary, so the result is a diagnostic compatibility check, not a measured parameter or evidence for modified growth.

Scientific status: This is an observable-signature proxy, not a full Boltzmann-code calculation and not a precision cosmological likelihood.

Scripts and reproducibility:

Folder	Role
`experiments/maat_paper39_observable_growth_signature/`	Paper 39 projection-modulated `f sigma_8` proxy, epsilon chi2 scan, v1.2.1 closure diagnostics, output CSV/JSON files, and figures

Documentation PDF: documentation/39_MAAT_v121_Observable_Growth_Signature_Proxy.pdf

Paper 40 — MAAT v1.2.1 Structural Signature Test in Growth Data

MAAT v1.2.1 Structural Signature Test in Growth Data: Projection CCI, Residual Magnitudes, and Exploratory Null Tests

Core idea: Tests whether MAAT v1.2.1 structural diagnostics align with the residual structure of a compact f sigma_8(z) growth comparison set relative to a Planck-normalised LCDM baseline. Unlike Paper 39, this paper does not ask whether MAAT improves the fit. It asks whether structural diagnostics track where the reference model shows larger residual stress.

Core results:

Quantity	Result
Growth comparison points	`13`
Projection transition estimate	`z_tr ~= 0.6508`
Spearman `R_proj` vs signed residual	`0.6319`, `p = 0.0228`
Spearman `V` vs signed residual	`-0.6319`, `p = 0.0228`
Spearman `CCI_diag` vs `	residual_sigma
Random-field null for `CCI_diag`	`p = 0.0363`
Redshift-shuffle null for `CCI_diag`	`p = 0.0359`
Spearman `B` vs `	residual_sigma

Key finding:

The diagnostic CCI tracks residual magnitude more clearly than residual direction in this compact growth-data benchmark. Because the balance support B is residual-sensitive, the result is a semi-supervised structural consistency test, not a blind prediction or detection claim.

Scientific status: This is an exploratory residual-structure diagnostic, not a Boltzmann-code calculation and not a full cosmological likelihood.

Scripts and reproducibility:

Folder	Role
`experiments/maat_paper40_structural_signature_test/`	Paper 40 CCI/residual structural signature test, permutation null tests, output CSV/JSON files, and figures

Documentation PDF: documentation/40_MAAT_v121_Structural_Signature_Test_in_Growth_Data.pdf

Paper 41 — MAAT v1.2.1 Variable Closure and Measurement Definitions

MAAT v1.2.1 Variable Closure and Measurement Definitions: Formal Definition of Structural Supports, Activity, Response-Derived Closure, and Projection Observable Derivation

Core idea: Provides the formal variable-definition layer for the v1.2.1 framework used in Papers 33--40. It defines the four primary support sectors H, B, S, V from non-negative defects, keeps robustness emergent through R_resp=(H B V)^(1/3) and R_rob=min(R_resp,(H B S V)^(1/4)), and makes clear that lambda_cl is a derived closure coefficient rather than a fifth primary sector weight.

Core definitions:

Quantity	Definition / role
Support scores	`Gamma_a = 1 / (1 + d_a)`
Primary sectors	`H`, `B`, `S`, `V` only
Implicit respect	`R_resp = (H B V)^(1/3)`
Emergent robustness	`R_rob = min(R_resp,(H B S V)^(1/4))`
Stability	`Stability = R_rob`
Closure coefficient	`lambda_cl = sum_a w_a lambda_a`, sensitivity-projected from primary weights
CCI diagnostic	`CCI_diag = S / (H + B + V + R_rob + epsilon)`

Key finding:

MAAT v1.2.1 is operationally closed at the variable-definition level: no independent fifth robustness sector is introduced, R_resp is not double-counted in the CCI denominator, and projection parameters are fixed by the stated response-map convention rather than fitted as independent observables.

Scientific status: This is a formal closure and measurement-definition note. It introduces no new numerical data set, likelihood analysis, or observational fit.

Reproducibility: Paper 41 has no standalone code folder. It documents the variable conventions used by the existing reproducibility folders: experiments/lambda_response_closure/, experiments/maat_cci_projection_paper33/, experiments/maat_paper39_observable_growth_signature/, and experiments/maat_paper40_structural_signature_test/.

Documentation PDF: documentation/41_MAAT_v121_Variable_Closure_and_Measurement_Definitions.pdf

Paper 42 — Blind Projection Test

Blind Projection Test: Response-Derived Projection, No Projection-Shape Tuning, and Residual Structure Diagnostics

Core idea: Removes the projection-shape tuning used in earlier residual diagnostics. The observable parameters are derived from response-closed MAAT v1.2.1 sector weights:

defects -> covariance C -> lambda -> response shares pi_a
        -> gamma_lambda, Bstar_lambda, alpha_lambda
        -> C_proj_lambda(z)

The test uses a compact f sigma_8(z) growth comparison set and asks whether the response-derived projection observable and CCI diagnostics retain a residual-structure alignment without fitting epsilon, gamma, Bstar, or alpha. The small denominator regulator is fixed numerically and is not treated as a physical parameter.

Core results:

Quantity	Result
Growth comparison points	`13`
Response shares	`pi_H=0.0960`, `pi_B=0.1257`, `pi_S=0.3879`, `pi_V=0.3903`
Derived `gamma_lambda`	`1.3305`
Derived `Bstar_lambda`	`4.5091`
Derived `alpha_lambda`	`1.9937`
Transition marker	`z_tr = 0.6986`
Spearman `CCI_diag` vs `	residual_sigma
Spearman `C_proj_lambda` vs `	residual_sigma

Key finding:

The residual-structure direction survives blind projection-shape closure, but the signal is reduced to a positive, non-significant exploratory trend. Paper 42 is therefore a robustness stress test, not a detection claim.

Scripts and reproducibility:

Folder	Role
`experiments/maat_paper42_blind_projection_test/`	Paper 42 blind projection test, response-derived parameters, residual/CCI correlations, permutation null tests, output CSV/JSON files, and figures

Data attribution and license note: The Planck-normalised reference parameters and compact f sigma_8 comparison points are external scientific data and should be cited to the original publications/collaborations. The repository CSV/PNG files are derived reproducibility artifacts only. No endorsement by the Planck Collaboration, survey collaborations, or original data authors is implied.

Documentation PDF: documentation/42_Blind_Projection_Test.pdf

Paper 43 — Linear Perturbations and Structure Growth

Linear Perturbations and Structure Growth: From Projection Signatures to Effective Growth Dynamics in Structural-Selection Cosmology

Core idea: Promotes the MAAT projection layer from a direct growth signature template to an explicit effective linear matter-growth equation. The bounded projection kernel C_hat_proj(z) modifies the sub-horizon growth source through

mu(z) = G_eff/G = 1 + eta * C_hat_proj(z)

with C_hat_proj normalised over 0 <= z <= 3. This makes eta the maximum fractional growth-coupling deformation in the benchmark interval.

Core results:

Quantity	Result
Growth comparison points	`13`
Eta scan	`eta in [0, 0.08]`, `41` grid points
Stable branches	`41/41`
Best eta	`0.0000`
LCDM chi2	`12.2021`
Representative eta	`0.0200`
Max `	Delta D/D
Max `	Delta f sigma_8 / f sigma_8
Max `	mu - 1
Representative eta chi2	`12.5737`

Key finding:

The compact growth comparison does not prefer a nonzero MAAT coupling: the best value is the LCDM limit eta=0. However, nonzero perturbative branches remain stable and produce only small, structured deviations. Paper 43 therefore establishes a controlled bridge from projection observables to explicit linear growth dynamics.

Scientific status: This is an effective sub-horizon linear-growth benchmark, not a Boltzmann-code calculation, not a CMB or weak-lensing likelihood, and not evidence for modified gravity.

Scripts and reproducibility:

Folder	Role
`experiments/maat_linear_perturbations_paper43/`	Paper 43 linear perturbation benchmark, eta scan, compact `f sigma_8` comparison, output CSV/JSON files, and figures

Data attribution and license note: The Planck-normalised reference parameters and compact f sigma_8 comparison points are external scientific data and should be cited to the original publications/collaborations. The repository CSV/PNG files are derived reproducibility artifacts only. No endorsement by the Planck Collaboration, survey collaborations, or original data authors is implied.

Documentation PDF: documentation/43_Linear_Perturbations_and_Structure_Growth.pdf

Paper 44 — Active Structural Significance

Active Structural Significance: A Toy Extension of the MAAT v1.2.1 Robustness Closure
MAAT Structural Selection Series --- Paper 44 / Diagnostic Note

Core idea: Extends the v1.2.1 interpretation without returning to a fifth primitive R sector. The paper treats R as an emergent hierarchy over the four primary support sectors H, B, S, V:

R_resp = (H B V)^(1/3)
R_rob  = min(R_resp, (H B S_eff V)^(1/4))
R_sig  = R_resp^(1-alpha) S_eff^alpha

Here S_eff is controlled activity in an optimal window, not raw activity:

S_eff(A) = exp[-0.5 ((A - A_star) / sigma_A)^2]

Core results:

Quantity	Result
Ensemble size	`6000`
Activity optimum	`A_star = 1.0`
Activity width	`sigma_A = 0.28`
Significance exponent	`alpha = 0.45`
Max `R_sig` location	`A = 0.9977`
Max `R_sig`	`0.9918`
Dormant coherent `R_resp` / `R_sig`	`0.6361 / 0.0980`
Stable-star `R_resp` / `R_sig`	`0.9850 / 0.9917`
Chaotic-burst `R_resp` / `R_sig`	`0.4451 / 0.0806`
High `R_resp` but low `R_sig` fraction	`13.07%`

Key finding:

MAAT v1.2.1 should be read as a robustness closure, not a complete theory of structural significance. Active significance peaks only when passive adherence is combined with controlled activity in the optimal window.

Scientific status: This is a diagnostic note and toy conceptual simulation, not a physical stellar model, not a cosmological fit, and not a new primitive-field proposal. The use of S_eff in the robustness boundary is part of this toy extension and does not retroactively modify the v1.2.1 papers.

Scripts and reproducibility:

Folder	Role
`experiments/active_respect_significance/`	Paper 44 active significance toy simulation, archetype comparison, ensemble CSV/JSON outputs, and figures

Documentation PDF: documentation/44_Active_Structural_Significance.pdf

Paper 45 — Effective Metric Response in MAAT Structural Cosmology

Effective Metric Response in MAAT Structural Cosmology: Gravitational Slip, Weyl-Potential Proxies, and Growth-Lensing Consistency

Core idea: Extends the Paper-43 linear-growth benchmark from a growth-only response channel to a minimal metric-response benchmark. The bounded MAAT projection kernel C_hat_proj(z) now sources both the effective Newtonian growth coupling and a gravitational-slip channel:

mu(z)       = G_eff/G = 1 + eta_g * C_hat_proj(z)
eta_slip(z)= Phi/Psi = 1 + beta_s * C_hat_proj(z)
Sigma(z)   = mu(z) * [1 + eta_slip(z)] / 2

Here Sigma parameterizes the Weyl/lensing response channel. A nonzero eta_slip is equivalent, at the effective perturbation level, to introducing a metric anisotropic-stress channel.

Two diagnostic consistency proxies are reported:

Weyl_proxy(z) = Sigma(z) * D_MAAT(z) / D_LCDM(z)
EG_proxy(z)   = Sigma(z) * f_LCDM(z) / f_MAAT(z)

Core results:

Quantity	Result
`eta_g` scan	`[0.00, 0.08]`, `41` points
`beta_s` scan	`[-0.06, 0.06]`, `61` points
Total metric-response branches	`2501`
Stable / positive branches	`2501 / 2501`
Growth-only best `eta_g`	`0.0000`
Growth-only `beta_s`	unconstrained without lensing data
Representative branch	`eta_g = 0.02`, `beta_s = 0.03`
Representative max `	mu - 1
Representative max `	eta_slip - 1
Representative max `	Sigma - 1
Representative Weyl-proxy deviation	`2.7314%`
Representative `E_G`-proxy deviation	`2.3019%`

Key finding:

MAAT projection structure can be coupled to matter growth and metric potentials as two distinct bounded response channels. Growth-only data still prefer the LCDM limit for eta_g and cannot constrain beta_s, because slip changes the metric potentials rather than the matter-growth source term; lensing or metric-potential data are required for the next step.

Scientific status: This is an effective metric-response benchmark, not a Boltzmann-code implementation, not a weak-lensing likelihood, not a CMB anisotropy calculation, and not evidence for modified gravity.

Scripts and reproducibility:

Folder	Role
`experiments/maat_metric_response_paper45/`	Paper 45 metric-response benchmark, gravitational-slip scan, Weyl/`E_G` proxy outputs, CSV/JSON files, and figures

Data attribution and license note: The Planck-normalised reference parameters and compact f sigma_8 comparison points are external scientific data and should be cited to the original publications/collaborations. The repository CSV/PNG files are derived reproducibility artifacts only. No endorsement by the Planck Collaboration, survey collaborations, or original data authors is implied.

Documentation PDF: documentation/45_Effective_Metric_Response_in_MAAT_Structural_Cosmology.pdf

Paper 46 — Environmental Screening in MAAT Structural Cosmology

Environmental Screening in MAAT Structural Cosmology: Emergent Suppression of Metric Response in High-Coherence Regions

Core idea: Extends Paper 45 by making the bounded MAAT projection response environment-dependent. The same projection kernel that sources growth and metric response is multiplied by an environmental suppression factor:

C_env(z, Delta, Sigma_env) = C_hat_proj(z) * S_env(Delta, Sigma_env)

S_env = [1 + alpha_rho Delta_+^n + alpha_sigma Sigma_env^m]^(-1)
Delta_+ = max(Delta - 1, 0)

The screened response channels are:

mu_env(z)        = 1 + eta_g  C_env(z)
eta_slip_env(z)  = 1 + beta_s C_env(z)
Sigma_lens_env   = mu_env(z) * [1 + eta_slip_env(z)] / 2

Because 0 <= C_hat_proj <= 1 and 0 <= S_env <= 1, the environmental response is bounded by construction. The GR-recovery limit is explicit:

S_env -> 0  =>  mu_env, eta_slip_env, Sigma_lens_env -> 1

Core results:

Quantity	Result
Representative response	`eta_g = 0.02`, `beta_s = 0.03`
Screening parameters	`alpha_rho = 0.15`, `alpha_sigma = 1.0`, `n = 0.75`, `m = 2.0`
Synthetic environments	`void`, `sheet`, `field`, `filament`, `cluster`, `local_dense`
Stable environments	`6 / 6`
`S_env(void)`	`0.9975`
`S_env(cluster)`	`0.1555`
`S_env(local_dense)`	`0.0002107`
Void max `	Sigma_lens - 1
Cluster max `	Sigma_lens - 1
Local-dense max `	Sigma_lens - 1
Screening transition at `Sigma_env=0.2`	`S_env ~= 0.5` at `Delta ~= 12.35`

Key finding:

MAAT metric response can remain percent-level in void-like regions while being strongly suppressed in dense or highly organized environments. The construction provides an effective structural screening layer, not a microscopic chameleon, Vainshtein, or symmetron derivation.

Scientific status: This is an effective environmental-screening benchmark, not a local-gravity test, not an N-body or halo-model analysis, and not a first-principles derivation of a screening mechanism.

Scripts and reproducibility:

Folder	Role
`experiments/maat_environmental_screening_paper46/`	Paper 46 environmental screening benchmark, environment archetypes, phase-space grid, CSV/JSON outputs, and figures

Data attribution and license note: The Planck-normalised reference parameters and compact f sigma_8 comparison points are external scientific data and should be cited to the original publications/collaborations. The repository CSV/PNG files are derived reproducibility artifacts only. No endorsement by the Planck Collaboration, survey collaborations, or original data authors is implied.

Documentation PDF: documentation/46_Environmental_Screening_in_MAAT_Structural_Cosmology.pdf

Paper 47 — From Environmental Screening to Observable Void Signatures

From Environmental Screening to Observable Void Signatures: A Halo-Environment Interpretation of MAAT Metric Response

Core idea: Addresses the main limitation of Paper 46 by translating the environmental screening axis into large-scale-structure language. Instead of using only an abstract Sigma_env, Paper 47 uses overdensity Delta = rho / rho_bar and a bounded tidal/shear proxy:

Sigma_env(q) = q^2 / (q^2 + q_star^2)

The screened kernel becomes:

C_env(z, Delta, q) = C_hat_proj(z) * S_env(Delta, q)

S_env(Delta, q)
  = [1 + alpha_rho Delta_+^n + alpha_sigma Sigma_env(q)^m]^(-1)

Delta_+ = max(Delta - 1, 0)

The observable target is the lensing-to-growth response ratio:

R_LG(z) = Sigma_lens_env(z) / mu_env(z)
        = [1 + eta_slip_env(z)] / 2

Core results:

Quantity	Result
Synthetic populations	`void`, `field`, `filament`, `cluster`, `local_dense`
Samples per population	`6000`
Random seed	`47`
`mean S_env(void)`	`0.998821`
`mean S_env(cluster)`	`0.160656`
`mean S_env(local_dense)`	`0.000213`
`mean max R_LG - 1` in voids	`1.498231%`
`mean max R_LG - 1` in clusters	`0.240984%`
`mean max R_LG - 1` in local-dense proxy	`0.000320%`
Void--cluster `R_LG` contrast	`1.257248 percentage points`
Void--local `R_LG` contrast	`1.497912 percentage points`
Minimum stable fraction	`1.0`
Screening sensitivity cases	`11`
Positive void--cluster contrast under sensitivity	`11 / 11`
Void--cluster contrast range under sensitivity	`0.571843` to `1.400886` percentage points

Key finding:

If the MAAT metric response is environmentally screened, the residual lensing-to-growth response should be largest in void-like environments, strongly suppressed in clusters, and essentially absent in high-density local-test proxies.

Scientific status: This is a phenomenological bridge benchmark. It is not a full halo model, not a weak-lensing likelihood, not a Boltzmann-code calculation, and not a microscopic derivation of a screening mechanism.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/maat_void_environment_paper47

Folder	Role
`experiments/maat_void_environment_paper47/`	Paper 47 void/cluster environment benchmark, Monte-Carlo samples, tidal phase-space grid, CSV/JSON outputs, and figures

Data attribution and license note: The benchmark uses Planck-normalised reference cosmological parameters as external literature values and synthetic environment populations generated by the script. No external survey catalogue is redistributed. Repository CSV/PNG files are derived reproducibility artifacts only. No endorsement by the Planck Collaboration, survey collaborations, or cited authors is implied.

Documentation PDF: documentation/47_From_Environmental_Screening_to_Observable_Void_Signatures.pdf

Paper 48 — Maximum-Entropy Derivation of Universal Structural Selection

A Maximum-Entropy Derivation of Universal Structural Selection: From Constraint Defects to Exponential Structural Measures

Core idea: Provides the formal closure behind the structural-selection measure. If admissible configurations are selected under finite information about structural defect means, Jaynes maximum entropy implies an exponential selection measure:

P[X] = Z^{-1} exp[-F_struct[X]]

F_struct[X] = sum_a lambda_a d_a[X]

The MAAT support form used in later papers is recovered as a logarithmic multiplicative-support representation:

F_MAAT[X] = - sum_a lambda_a log(epsilon + Gamma_a[X])
Gamma_a[X] = 1 / (1 + d_a[X])

Core result:

Quantity	Result
Formal status	Conditional MaxEnt theorem
Required inputs	admissible space `X_adm`, reference measure `mu0`, defect observables `d_a`
Derived measure	`P[X] proportional to exp[-F_struct[X]]`
Linear form	Gibbs--Jaynes structural cost
Support form	bounded multiplicative viability representation
Weight interpretation	Lagrange multipliers / covariance-response pressures
No new numerical solver	formal derivation; empirical tests live in existing experiment folders

Key finding:

The exponential MAAT/structural-selection form is not an arbitrary scoring ansatz. It is the least-biased probability measure generated by fixed structural defect constraints.

Scientific status: This paper proves the form of the measure conditional on the chosen defects, reference measure, and constraints. It does not prove that the MAAT defect basis is unique or fundamental. Universality still requires recurrence across domains and falsification against simpler alternatives such as energy-only, entropy-only, stability-only, random, anthropic, or complexity-only rankings.

Reproducibility: This is a formal derivation with no standalone code. The empirical evidence ladder is reproduced by the existing benchmark folders in experiments/, including cosmological CCI, growth/projection, string-landscape, SAT/constraint, and environmental-response tests.

Documentation PDF: documentation/48_Maximum_Entropy_Derivation_of_Universal_Structural_Selection.pdf

Paper 49 — Universal Structural Selection Benchmarks

Universal Structural Selection Benchmarks: Testing MAAT Against Energy, Stability, Random, and Domain-Specific Baselines

Core idea: Turns the universality claim following Paper 48 into an explicit falsification matrix. The paper audits existing experiments across fields, string landscapes, cosmology, SAT/constraint systems, AI/safety boundary calibration, SM-like constants, and quantum measurement.

Each domain is checked against competitor classes:

energy-only
entropy/activity-only
stability-only
random or shuffled nulls
domain-specific baselines
cross-domain / holdout transfer

Core results:

Quantity	Result
Domains audited	`7`
Complete domain evidence	`2`
Partial domain evidence	`4`
Missing domain evidence	`1`
Complete matrix cells	`8`
Partial matrix cells	`13`
Missing matrix cells	`17`
Benchmark-readiness index	`0.3816`
Strongest current successes	field fixed-energy and string/path energy-only baselines
Main open stress tests	SAT and quantum measurement
Immediate roadmap	SAT Structural Hardness II, Quantum Pointer-State Selection, Cross-Domain Transfer

Key finding:

Energy-only baselines are already defeated in field and string benchmarks, but the universal MAAT claim is not yet earned. Entropy/activity-only, stability-only, shuffled-null, SAT, and quantum tests remain the decisive next layer. In fixed-energy fields the structural score changes by nearly 8x at sub-permille energy variation, while in the KKLT path graph the top-20 structural and energy transitions have 0/20 overlap.

Scientific status: This is a meta-benchmark registry and audit paper. It does not introduce a new solver and does not claim universal proof. It turns the post-MaxEnt universality claim into a reproducible benchmark programme. The low readiness index is interpreted as an audit warning, not as a success metric. SAT is treated explicitly as an Achilles test: the current SAT result is poor and motivates new graph-local defect sectors and future cross-solver validation.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/universal_structural_selection_benchmarks_paper49

Folder	Role
`experiments/universal_structural_selection_benchmarks_paper49/`	Paper 49 meta-benchmark aggregator, evidence registry, readiness matrix, JSON summary, and figures

Data attribution and license note: The script aggregates derived outputs already present in the repository. External scientific data used by the underlying experiments remain attributed in their respective experiment folders and papers. No new external dataset is redistributed here.

Documentation PDF: documentation/49_Universal_Structural_Selection_Benchmarks.pdf

Paper 50 — SAT Structural Hardness II

SAT Structural Hardness II: Graph-Local Defects and Structural Hardness in Random 3-SAT

Core idea: Directly addresses the SAT weakness identified in Paper 49. The older SAT validation used coarse global fields and performed poorly. Paper 50 generates a deterministic synthetic random-3-SAT ensemble and tests new graph-local defects against density-only, standard graph, stability-only, scalar MAAT, MAAT graph-local, and shuffled-null baselines.

New structural diagnostics:

constraint-graph expansion
clause-variable interaction cycles
backdoor-density proxy
local covariance conditioning
unit-propagation depth
contradiction rate
literal and degree balance

Core results:

Quantity	Result
Synthetic random-3-SAT instances	`520`
Variable range	`24` to `36`
Clause-density range	`3.17 <= alpha <= 5.22`
SAT fraction	`0.6346`
Timeout fraction	`0.0000`
Density-only CV `R2`	`0.1886`
Standard graph CV `R2`	`0.2469`
Stability-only CV `R2`	`-0.0145`
Scalar `F_MAAT` CV `R2`	`-0.0145`
MAAT graph-local CV `R2`	`0.1469`
MAAT graph-local + density CV `R2`	`0.2347`
MAAT+density gain over density-only	`+0.0461`
MAAT+density gap to standard graph	`-0.0122`
Shuffled-null 95% Spearman	`0.0623`
MAAT graph-local Spearman	`0.3914`

Key finding:

SAT hardness is graph-local and phase-aware. The new MAAT graph-local diagnostics clearly beat shuffled-null structure and improve over density-only once the SAT phase-density channel is retained, but they do not yet outperform a simple standard graph baseline. The key quantitative gain is Delta R2 = +0.0461 over density-only, indicating nontrivial structural information beyond phase-transition proximity alone.

Scientific status: This is a partial success and useful failure. It does not solve SAT hardness, prove P != NP, or establish MAAT universality. It narrows the SAT problem: scalar F_MAAT and stability-only scores fail, while local covariance conditioning and graph-local diagnostics carry real but still insufficient hardness signal. The graph-local diagnostics are solver-independent inputs, but the target is a deterministic DPLL node-count proxy; solver-independent hardness remains a future cross-solver validation task.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/sat_structural_hardness_paper50

Folder	Role
`experiments/sat_structural_hardness_paper50/`	Paper 50 synthetic SAT benchmark, graph-local features, DPLL target, model comparisons, JSON/CSV outputs, and figures

Data attribution and license note: The benchmark generates synthetic random-3-SAT formulas with a fixed random seed. No external SAT benchmark dataset is redistributed. Repository CSV/PNG files are derived reproducibility artifacts generated by the script.

Documentation PDF: documentation/50_SAT_Structural_Hardness_II.pdf

Paper 50b — SAT Structural Hardness IIb

SAT Structural Hardness IIb: Local Frustration Fields and the Failure of Scalar Compression

Core idea: Companion test to Paper 50. Paper 50 showed that scalar F_MAAT and global robustness are weak for SAT hardness. Paper 50b tests the sharper thesis that SAT hardness is damaged by aggressive global scalar compression and is better represented by local frustration fields, rare-event tails, and hotspot geometry.

Scientific status: This is not the Paper 51 roadmap item announced in Paper 49. It is a Paper-50 companion experiment. It does not solve SAT hardness, prove P != NP, or provide a modern CDCL benchmark. The target is still a deterministic DPLL node-count proxy.

Core results:

Quantity	Result
Synthetic SAT instances	`550`
Formula families	random, planted, modular
Variable range	`24` to `38`
Clause-density range	`3.26 <= alpha <= 5.18`
SAT fraction	`0.7018`
Timeout fraction	`0.0000`
RF density-only CV `R2`	`0.0362`
RF density + formula type CV `R2`	`0.1834`
RF standard graph CV `R2`	`0.2959`
RF global MAAT scalar/supports CV `R2`	`0.1375`
RF single `F_MAAT^multi` scalar CV `R2`	`-0.0806`
RF `F_MAAT^multi` components CV `R2`	`-0.0224`
RF `F_MAAT^multi` components + density/type CV `R2`	`0.2833`
RF local frustration + density/type CV `R2`	`0.2946`
RF raw graph + frustration CV `R2`	`0.3152`
Local frustration + density/type gain over global MAAT	`+0.1571`
Multi components + density/type gain over global MAAT	`+0.1458`
Raw graph + frustration gain over standard graph	`+0.0194`

Key finding:

SAT hardness is not well described by global robustness or scalar F_MAAT. The useful signal lives in graph-local, phase-aware, multi-scale frustration geometry. Local frustration fields only become useful when the density/formula-family channel is retained. The Paper-50c test confirms that F_MAAT^multi should not be re-compressed into one scalar; its components carry the useful signal when context is retained.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/sat_frustration_fields_paper50b

Folder	Role
`experiments/sat_frustration_fields_paper50b/`	Paper 50b SAT frustration-field compression test, synthetic instances, model ladder, feature importance, JSON/CSV outputs, and figures

Data attribution and license note: The benchmark generates synthetic SAT formulas with a fixed random seed. No external SAT benchmark dataset is redistributed. Repository CSV/PNG files are derived reproducibility artifacts generated by the script.

Documentation PDF: documentation/50b_SAT_Frustration_Field_Compression_Test.pdf

Paper 50c — SAT Structural Hardness IIc

SAT Structural Hardness IIc: From Global Robustness to Multi-Scale Defect-Field Structural Selection

Core idea: Theoretical companion note to Papers 50 and 50b. It does not replace the old MAAT formula, but reinterprets it as a low-frequency macro projection of a richer defect-field functional. SAT motivates the update because scalar F_MAAT and R_rob lose rare local frustration, tails, hotspots, and cluster structure.

Updated master form:

F_MAAT^multi = F_mean + F_tail + F_cluster + F_scale

with:

F_mean    = sum_a lambda_a^(0) <d_a(x)>_G
F_tail    = sum_a lambda_a^tail q_p(d_a)
F_cluster = sum_a lambda_a^cl C_max^(a)/|G|
F_scale   = sum_{a,k} lambda_{a,k} d_a^(k)

Key finding:

Global MAAT is the macro projection; full MAAT is the defect-field functional. The old scalar formula remains useful for smooth or self-averaging systems, while SAT-like domains require local, tail, cluster, and scale-resolved defect structure. The companion code test shows the same lesson numerically: a single F_MAAT^multi scalar is weak, but its components plus density/type reach R2 = 0.2833, far above global MAAT compression.

Scientific status: Effective-theory update, not a microscopic derivation and not a new benchmark. It is the theoretical synthesis of the Paper 50/50b SAT failure analysis.

Documentation PDF: documentation/50c_Multi_Scale_Defect_Field_Structural_Selection.pdf

Paper 51 — Quantum Pointer-State Selection

Quantum Pointer-State Selection: A MAAT v1.4 Defect-Field Benchmark in a Non-Commuting Lindblad System

Core idea: Tests the MAAT v1.4 defect-field thesis in a minimal open quantum system. A qubit evolves under non-commuting Lindblad channels: z-dephasing, x-dephasing, and amplitude damping. The target is initial state survival fidelity after Lindblad evolution. All features are computed from the initial state and generator parameters, not from the evolved final state.

Core results:

Model	CV `R2`
MAAT scalar/supports	`0.5383 ± 0.0151`
MAAT v1.4 field features	`0.7424 ± 0.0057`
Rates only	`0.0542 ± 0.0547`
Basis geometry only	`-0.1134 ± 0.0331`
Hamiltonian only	`-0.0406 ± 0.0241`
Domain combo	`0.1362 ± 0.0291`
v1.4 gain over scalar MAAT	`+0.2041`
v1.4 gain over domain combo	`+0.6062`

Key finding:

Pointer-state robustness is not captured by rates alone. Scalar MAAT retains moderate signal, but the v1.4 field-feature extension substantially improves prediction by combining environment alignment, channel conflict, damping pressure, coherence exposure, and robustness closure.

Scientific status: Toy benchmark only. This is not a derivation of quantum measurement, not a Born-rule derivation, not an experimental fit, and not a full decoherence theory.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/quantum_pointer_state_selection_paper51

Folder	Role
`experiments/quantum_pointer_state_selection_paper51/`	Paper 51 quantum pointer-state toy benchmark, generated ensemble, model comparison, feature importance, JSON/CSV outputs, and figures

Data attribution and license note: The benchmark generates synthetic quantum toy-model data with a fixed random seed. No external experimental dataset is redistributed. Repository CSV/PNG files are derived reproducibility artifacts generated by the script.

Documentation PDF: documentation/51_Quantum_Pointer_State_Selection.pdf

Paper 52 — Cross-Domain Transfer of Structural Selection

Cross-Domain Transfer of Structural Selection: A Frozen-Weight Two-Domain Pilot Study

Core idea: Tests whether a MAAT defect architecture calibrated in one domain transfers to a second internal benchmark domain without target-domain retuning. The common architecture uses only the shared supports H,B,S,V,R_rob. Source-domain weights are learned once, frozen, and then evaluated in the target domain with a predeclared score direction and shuffled-defect nulls.

Core results:

Transfer	Frozen Spearman	Equal/scalar Spearman	Interpretation
SAT -> Quantum	`0.6034`	`0.3771`	strong asymmetric transfer above shuffled-defect null
Quantum -> SAT	`-0.0408`	`-0.2683`	weak/failed positive transfer, but less bad than scalar baselines

Key finding:

Transfer is partial and asymmetric. SAT-trained connectivity transfers strongly to quantum pointer robustness, while quantum-trained structure does not solve SAT hardness. The dominant transferred signal is V connectivity, not a balanced multi-sector architecture. The conservative interpretation is that the two internal benchmarks share a connectivity-sensitive structural mode; this is not yet independent evidence for universal cross-domain transfer. The result supports the v1.4/v1.4.1 reading that universality, if present, is more likely to live in partially transferable structural modes than in one compressed scalar score.

Scientific status: Two-domain internal transfer benchmark, not a universality proof and not independent external validation. The protocol is intentionally adversarial: frozen transfer is allowed to fail. The result is deliberately reported as mixed: one direction succeeds strongly, the reverse direction remains a failure mode for global scalar transfer. The SAT side uses DPLL node count as a transparent controlled proxy, not modern CDCL conflict statistics.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/cross_domain_transfer_paper52

Folder	Role
`experiments/cross_domain_transfer_paper52/`	Paper 52 frozen cross-domain transfer runner, shuffled-null validation, CSV/JSON outputs, and figures

Data attribution and license note: This experiment reuses repository-internal synthetic/derived artifacts from Paper 50b and Paper 51. No new external dataset is introduced. Repository CSV/PNG files are reproducibility artifacts generated by the script.

Documentation PDF: documentation/52_Cross_Domain_Transfer_Structural_Selection.pdf

Paper 53 — Structural Mode Transfer Robustness

Robustness of Structural Mode Transfer in MAAT v1.4: Alternative Frozen Fits, Regularisation Tests, and Multi-Domain Null Validation

Core idea: Tests whether the Paper 52 V/connectivity result is stable under alternative source-only fit rules, regularisation schemes, and additional internal benchmark domains. This is an internal robustness audit, not an external cross-domain validation. The protocol keeps the frozen-transfer rule: source weights are learned once, frozen, and transferred without target-domain retuning or sign flips.

Domains and fit rules:

Category	Included
Domains	SAT-Frustration, Quantum-Pointer, Active-Significance, Societal-CCI
Fit rules	equal, NNLS, positive ridge, positive Lasso, response-top20, V-only
Nulls	shuffled-defect nulls for cross-domain transfers

Core results:

Result	Value
SAT -> Quantum, NNLS	`rho = 0.6034`, `lambda_V = 1.0000`
SAT -> Quantum, ridge_pos	`rho = 0.5981`, `lambda_V = 0.8394`
SAT -> Quantum, lasso_pos	`rho = 0.6034`, `lambda_V = 1.0000`
SAT -> Quantum, response_top20	`rho = 0.5975`, `lambda_V = 0.8264`
Mean cross-domain Spearman, equal	`0.5156`
Overall V-dominant frozen architectures	`~58%`

Key finding:

The Paper 52 SAT -> Quantum connectivity result is not merely an NNLS artifact: it survives ridge, Lasso, covariance-response closure, and a V-only baseline. But V is not universal. Across all internal domains, broad equal support averaging performs best on support-composite toy targets, and SAT remains resistant to simple incoming transfer.

The equal-weight result is an important negative control rather than a cosmetic detail: in the present partly support-composite domain set, simple broad averaging remains competitive with more sophisticated source-specific fitting. Thus Paper 53 supports conditional structural portability, not universal transfer.

Scientific status: Robustness/ablation paper, not external validation. The added Active-Significance and Societal-CCI domains are internal toy domains whose targets are partly support-composite, so aggregate cross-domain averages must be interpreted cautiously. The Societal-CCI ensemble is deliberately synthetic and is not a real social dataset, political ranking, or independent external validation target.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/structural_mode_transfer_paper53

Folder	Role
`experiments/structural_mode_transfer_paper53/`	Paper 53 multi-fit/multi-domain transfer robustness runner, shuffled-null validation, CSV/JSON outputs, and figures

Data attribution and license note: This experiment reuses repository-internal synthetic/derived artifacts from earlier papers. No new external dataset is introduced. Repository CSV/PNG files are reproducibility artifacts generated by the script.

Documentation PDF: documentation/53_Structural_Mode_Transfer_Robustness.pdf

Paper 54 — External Validation of Structural Modes

External Validation of Structural Modes:
MAAT Defect Features on Public Datasets Accessed via scikit-learn

Core idea: Breaks the internal-artifact loop of Papers 52--53 by testing MAAT structural supports on public external machine-learning datasets loaded via scikit-learn. Sample-level hardness is predeclared as the fraction of repeated cross-validation runs in which a sample is misclassified by a fixed Random Forest classifier.

External datasets:

Dataset	Samples	Features	Classes
`breast_cancer`	569	30	2
`wine`	178	13	3
`digits`	1797	64	10

Core results:

Model / score	Mean Spearman	Mean AUROC
external geometry feature set	`0.3889`	`0.6685`
combined geometry feature set	`0.3824`	`0.6666`
MAAT supports	`0.3241`	`0.6393`
MAAT v1.4 field features	`0.2795`	`0.6191`
label disagreement scalar	`0.4299`	`0.6719`
MAAT robustness-loss scalar	`0.3301`	`0.6708`
MAAT scalar cost	`0.3226`	`0.6706`

Key finding:

MAAT scalar cost and robustness loss carry positive external sample-hardness signal on all three public datasets, but simple external geometry and local label-disagreement baselines remain stronger in this first public-data benchmark.

Scientific status: First external validation benchmark, not a proof of universal transfer. The MAAT v1.4 field features beat shuffled-defect nulls on breast cancer and digits, but not on wine. The result is therefore mixed and useful: it establishes measurable external signal while preventing internal consistency from being mistaken for external universality.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/external_ml_hardness_paper54

Folder	Role
`experiments/external_ml_hardness_paper54/`	Paper 54 external ML sample-hardness benchmark, scikit-learn dataset loader, CV hardness target, CSV/JSON outputs, and figures

Data attribution and license note: The experiment accesses public datasets via scikit-learn. No raw external dataset files are redistributed in the experiment folder. Generated CSV/JSON/PNG files are derived reproducibility artifacts. Cite scikit-learn and the original dataset sources as appropriate. No endorsement by scikit-learn maintainers, UCI dataset curators, or original dataset authors is implied.

Documentation PDF: documentation/54_External_Validation_Structural_Modes.pdf

Paper 55 — CDCL Hardness Prediction on Standard SAT Families

CDCL Hardness Prediction on Standard SAT Families:
A Benchmark for Graph-Local and Defect-Field Features

Core idea: Replaces the earlier transparent DPLL SAT-hardness target with real CDCL solver statistics from PySAT implementations of Glucose3 and MiniSat22. The benchmark uses deterministically generated standard SAT families rather than previous repository-internal artifacts: random 3-SAT, planted 3-SAT, modular 3-SAT, 3-XOR CNF, graph coloring, and pigeonhole formulas.

Target:

y_CDCL = log(1 + mean_conflicts + 0.10 mean_decisions + 0.01 mean_propagations)

Core results:

Feature set	RF R2	RF Spearman
Standard graph	`0.2742`	`0.4254`
Multi-scale defect field	`0.2621`	`0.3984`
Graph-local structural	`0.2573`	`0.4170`
Scalar structural	`0.1062`	`0.3400`
Shuffled defect null	`-0.0609`	`-0.0070`
Density-only	`-0.0894`	`0.2503`

Key finding:

Multi-scale defect-field features carry real CDCL-hardness signal and strongly beat shuffled-defect nulls and scalar compression, but they do not beat a strong standard graph baseline. The decisive negative result is leave-family-out transfer: all feature sets fail to extrapolate cleanly across canonical SAT families.

Scientific status: This is a standard-family CDCL benchmark, not a SAT Competition or industrial SAT benchmark. It strengthens the local defect-field direction while weakening broad universality claims.

Scripts and reproducibility:

Folder	Role
`experiments/sat_cdcl_external_paper55/`	canonical SAT-family generator, PySAT/CDCL solver statistics, feature extraction, shuffled-null tests, model comparisons, figures

Data attribution and license note: No external CNF dataset is redistributed. Instances are generated from standard SAT benchmark families by the script. Solver statistics are produced locally through PySAT-compatible Glucose3 and MiniSat22 solvers. Generated CSV/JSON/PNG files are derived reproducibility artifacts.

Documentation PDF: documentation/55_External_SAT_CDCL_Hardness_Validation.pdf

Paper 56 — From Scalar Scores to Defect-Field Geometry

From Scalar Scores to Defect-Field Geometry:
Formal Definitions for Multi-Scale Structural Selection

Core idea: Consolidates the operator language needed by MAAT v1.4.1: local defect fields, multi-scale coarse-graining, defect-field dynamics, observation projections, and no-retuning universality tests. The paper is a formal definition layer rather than a new numerical benchmark.

Key definitions:

d_a(x) = D_a(Phi|N(x), C|N(x), partial N(x), mu|N(x)) >= 0
d_a^(ell) = Pi_ell[d_a]
partial_logell d_a^(ell) = beta_a[d^(ell)]
partial_t d_a = F_a[d] + xi_a
Pi_obs: Omega_full -> Omega_obs
U(theta) = E_{D_test notin D_train}[1{Delta I > 0}]

v1.4.1 anchor: The paper explicitly uses the v1.4.1 lesson that structural sectors must be operationally separated rather than merely renamed correlated summaries. In the quantum two-level anchor case:

B = 1 - |p0 - p1|
S = |rho_01| / sqrt(rho_00 rho_11)

This keeps balance and activity distinct: balanced-inactive, balanced-coherent, imbalanced-incoherent, and imbalanced-coherent states are not collapsed into the same structural mode.

Scientific status: This is a consolidation and definition paper. It does not claim a microscopic derivation, a complete RG theorem, or universal cross-domain transfer. It defines the formal objects needed to make those claims testable.

Documentation PDF: documentation/56_Defect_Field_Geometry_Projection_and_Universality.pdf

Paper 57 — Where Does SAT Hardness Live?

Where Does SAT Hardness Live? Mean, Tail, Cluster, and Scale Projections of Defect-Field Geometry

Core idea: Tests the formal projection language of Paper 56 on the Paper-55 SAT/CDCL benchmark. Instead of asking whether one global structural scalar predicts hardness, the experiment decomposes the defect field into mean, tail, cluster, and scale projections and asks which component carries the CDCL hardness signal.

Core results:

Feature set / projection	5-fold RF R2	Spearman
Standard graph baseline	`0.2169`	`0.4234`
All defect projections	`0.1996`	`0.3751`
Macro scalar	`0.1898`	`0.4193`
Tail + scale	`0.1725`	`0.3793`
Scale projection	`0.1381`	`0.3304`
Tail projection	`0.0937`	`0.2982`
Mean projection	`0.0575`	`0.2944`
Shuffled all-projection null	`-0.0666`	`-0.0134`

Projection importance inside the all-defect model:

Projection	Grouped R2 drop
Tail	`0.4044`
Scale	`0.3414`
Mean	`0.1567`
Cluster	`0.0229`

Key finding:

SAT/CDCL hardness signal is not primarily localized in mean scalar compression. In this benchmark, tail and scale projections carry most of the internal defect-field signal, while leave-family-out transfer remains weak.

Scientific status: This is a projection-decomposition benchmark, not a solution of SAT hardness and not evidence for universal transfer. The standard graph baseline remains slightly stronger in ordinary cross-validation. The positive result is signal localization; the negative result is continued family-transfer failure.

Scripts and reproducibility:

Folder	Role
`experiments/sat_projection_decomposition_paper57/`	Paper-57 projection ablation, Paper-55 CSV reader, mean/tail/cluster/scale tests, shuffled-null output, figures

Data attribution and license note: No external CNF files are redistributed. The experiment uses derived reproducibility artifacts generated by Paper 55 from deterministic canonical SAT-family generators and PySAT-compatible solver statistics.

Documentation PDF: documentation/57_Where_Does_SAT_Hardness_Live.pdf

Paper 58 — Stationarity Balance Beats Population Balance

Stationarity Balance Beats Population Balance:
A Defect-Field Benchmark on Open Quantum Pointer States

Core idea: Tests structural defect-field diagnostics on a fully classical one-qubit Lindblad simulation. No quantum computer, Qiskit runtime, cloud hardware, or paid backend access is required. The benchmark asks whether stationarity-sensitive balance predicts robust pointer-like states better than raw population balance.

Simulation setup:

Component	Description
System	one-qubit density matrix
Dynamics	Lindblad master equation integrated by RK4
Channel families	z-dephasing, x-dephasing, amplitude damping, thermal relaxation, mixed z/x, depolarizing
Instances	`3600` sampled trajectories reused from a `5200`-trajectory generated ensemble
Target	`0.50 fidelity_retention + 0.35 probability_stability + 0.15 purity_final`

Key definitions:

B_pop  = 1 - |p0 - p1|
B_stat = 1 / (1 + 2 |dp/dt|_0 + 0.35 |d Tr(rho^2)/dt|_0)

Core results:

Feature set	5-fold RF R2	Spearman
Standard quantum baseline	`0.9011`	`0.9241`
Defect-field stationary	`0.8527`	`0.8856`
Scalar stationarity	`0.6457`	`0.7719`
Scalar population balance	`0.4561`	`0.6221`
Rates only	`0.1641`	`0.4646`
State geometry only	`0.0008`	`0.2945`
Shuffled defect null	`-0.0261`	`-0.0003`

Leave-channel-family-out results:

Feature set	LFO R2	LFO Spearman
Defect-field stationary	`0.7075`	`0.8286`
Standard quantum baseline	`0.6637`	`0.8105`
Scalar stationarity	`0.4928`	`0.7284`
Scalar population balance	`0.3108`	`0.5604`

Key finding:

Pointer robustness is better captured by stationarity-sensitive balance than by raw population equality. The scalar stationarity model improves over the scalar population-balance model by Delta R2 = +0.1896, and the full stationarity-sensitive defect-field model performs best under leave-channel-family-out transfer.

Scientific status: This is a reproducible classical open-system simulation. It is not a proof of quantum measurement, not a collapse theory, and not a hardware experiment. The result supports a narrower refinement: quantum balance should track stationarity under the generator, not mere population equality.

Scripts and reproducibility:

Folder	Role
`experiments/open_quantum_pointer_paper58/`	Paper-58 Lindblad simulation, pointer-robustness target, B_pop/B_stat comparison, model outputs, figures

Data attribution and license note: No external quantum dataset is redistributed. All CSV/JSON/PNG files are derived reproducibility artifacts generated locally by the script.

Documentation PDF: documentation/58_Open_Quantum_Pointer_Stability.pdf

Extra Mathematical Diagnostic Paper — Riemann Zeta Critical-Line Diagnostic

A Structural Selection Diagnostic on the Riemann Zeta Critical Line
A purely diagnostic, non-proof toy experiment with controls, ablations, and pair-correlation tests

Core idea: Applies a simple heuristic structural diagnostic to known nontrivial Riemann zeta zeros. The test asks whether known zero ordinates have lower diagnostic cost on the critical line than under small off-line shifts, and how much of that behaviour is built into the scoring rule itself. This is a mathematical toy benchmark, not evidence for or a proof of the Riemann Hypothesis. The pair-correlation component compares normalised zeta-zero spacings with a GUE spacing proxy and Poisson/uniform controls.

Diagnostic terms:

Term	Definition
`H`	`1 / (1 +
`B`	`1 / (1 + alpha
`R`	symmetric real-direction zeta-response robustness

Core results:

Set	Config	Best mean delta	Mean F at best
known zeros	HBR	`0.000`	`0.00818`
known zeros	HR, no explicit B	`0.000`	`0.00818`
known zeros	BR, no H	`0.000`	`0.00818`
jittered zeros	HBR	`0.000`	`0.16544`
random critical-line points	HBR	`0.000`	`0.86054`
random critical-line points	HR, no explicit B	`0.050`	`0.83426`

Pair-correlation diagnostic:

Series	Spacings	L1 to GUE	V_pair	F_pair
jittered zeta spacings	199	`0.2059`	`0.9844`	`0.0157`
zeta normalised spacings	199	`0.2938`	`0.9797`	`0.0205`
uniform-points control	199	`0.8103`	`0.9573`	`0.0437`
poisson control	199	`0.8225`	`0.9559`	`0.0451`

Key finding:

Known zeta-zero locations exhibit a low-cost minimum on the critical line under the proposed diagnostic score. However, the score partly builds in critical-line preference through the balance term and directly rewards known zeros through the zeta-null term. The result is therefore a diagnostic benchmark, not an RH result.

Scientific status: Mathematical toy diagnostic with controls and ablations. The full HBR and no-balance HR diagnostics select delta=0 for known zeros across the tested parameter sweep, while random critical-line controls reveal how much of the critical-line preference is score-induced. The pair-correlation diagnostic is conceptually closer to spectral-universality tests than to direct zero-location tests, but it remains finite-sample and uses a GUE/Wigner-surmise proxy rather than a theorem.

Scripts and reproducibility:

Folder	Role
`experiments/zeta_critical_line_selection/`	zeta critical-line diagnostic script, zero/control tables, CSV/JSON outputs, and figures

Data attribution and license note: No external dataset file is redistributed. The script computes zeta zeros using mpmath.zetazero and evaluates the zeta function directly. Generated CSV/JSON/PNG files are derived reproducibility artifacts.

Documentation PDF: documentation/Structural_Selection_on_the_Critical_Line.pdf

Extra Experiment Paper — Societal Critical Coherence Index

Societal Critical Coherence Index: A Toy Framework for Structural Stress and Constructive Transformation

Core idea: Transfers the CCI/MAAT diagnostic logic to a deliberately synthetic social-system toy model. The goal is not to rank real people, parties, countries, organisations, or communities. The paper tests whether a structural index can separate raw activity, passive coherence, robustness, and constructive transformation.

Master definitions:

S_eff(A)    = exp[-0.5 ((A_raw - A_star) / sigma_A)^2]
R_resp,soc  = (H B V)^(1/3)
R_rob,soc   = min(R_resp,soc, (H B S_eff V)^(1/4))
R_sig,soc   = R_resp,soc^(1-alpha) S_eff^alpha
CCI_soc     = A_raw / (H + B + V + R_rob,soc + epsilon)
ASI_soc     = R_sig,soc / (1 + CCI_soc)

Core results:

Quantity	Result
Synthetic archetypes	`6`
Random ensemble size	`5000`
Fixed seed	`44`
Activity optimum	`A_star = 1.0`
Activity width	`sigma_A = 0.30`
Significance exponent	`alpha = 0.45`
Best `ASI_soc` archetype	`creative_democratic_renewal`
Best `ASI_soc`	`0.6940`
Highest `R_sig,soc` archetype	`creative_democratic_renewal`
Highest `R_sig,soc`	`0.9055`
Highest `CCI_soc` archetype	`polarized_mobilization`
Highest `CCI_soc`	`1.0177`
Ensemble mean `CCI_soc`	`0.4530`
Ensemble mean `ASI_soc`	`0.2813`

Key finding:

Low structural stress can still be stagnant, and high social activity can still be polarising or fragmenting. Constructive transformation appears only when controlled activity is supported by coherence, balance, and connectedness.

Ethical status: This is a conceptual diagnostics toy framework. It is not an instrument for political scoring, population ranking, surveillance, persuasion, or normative judgement of real communities.

Scripts and reproducibility:

Folder	Role
`experiments/societal_cci/`	Synthetic societal CCI script, archetype table, random ensemble, JSON/CSV outputs, and figures

Documentation PDF: documentation/Societal_Critical_Coherence_Index.pdf

Extra Phenomenological Paper — Structural Selection in SO(10)-Motivated Unified Field Theories

Structural Selection in SO(10)-Motivated Unified Field Theories: A Phenomenological MAAT Layer for Gauge and Yukawa Benchmarks

Core idea: Treats SO(10)-motivated grand unification as the dynamical base and MAAT v1.2.1 as a structural-selection layer over gauge and Yukawa parameter regions. The paper does not construct a complete SO(10) model, prove precision unification, or derive the Standard Model spectrum from first principles. It tests whether structurally coherent GUT-scale regions can be ranked reproducibly.

Core results:

Quantity	Result
Gauge benchmark	one-loop SM running, no threshold corrections
Best `alpha_GUT`	`0.022676468338`
Best `M_GUT`	`6.639198e15 GeV`
Gauge `SM chi2`	`100.789861`
Gauge `R_rob`	`0.60301490`
Yukawa benchmark	SO(10)-motivated `b-tau` boundary
Yukawa `M_GUT`	`1.858006e16 GeV`
`Delta_b`	`0.0506451697`
`chi2_yukawa`	`0.00024341`
Yukawa `R_rob`	`0.99916912`

Key finding:

The one-loop gauge benchmark locates a plausible GUT-scale region but fails precision unification, while the third-generation Yukawa benchmark shows that an SO(10)-motivated b-tau boundary can be compatible with low-energy effective masses using a modest bottom-threshold correction. The result is a structural compatibility benchmark, not a first-principles derivation.

Scientific status: This is an extra phenomenological bridge paper. It is not a complete SO(10) construction and not a precision GUT fit.

Scripts and reproducibility:

Folder	Role
`experiments/maat_so10_structural_selection/`	SO(10)-motivated gauge and Yukawa benchmarks, JSON summaries, and selection-landscape figure

Data attribution and license note: The electroweak-scale inputs use standard reference values for M_Z, gauge couplings, and third-generation fermion masses cited in the paper to the Particle Data Group. Repository JSON/PNG/PDF artifacts are derived analysis outputs only. No endorsement by the Particle Data Group or any external collaboration is implied.

Documentation PDF: documentation/Structural_Selection_in_SO10_Unified_Field_Theories.pdf

Extra Phenomenological Paper — Structural Selection in the String Landscape

Structural Selection in the String Landscape: A MAAT-Based Phenomenological Framework for Vacuum Ranking

Core idea: Treats string theory as the dynamical base and MAAT as a structural ranking layer over candidate string backgrounds. The paper does not replace string theory, derive a unique string measure, construct a Standard Model vacuum, or solve the landscape problem. It proposes a reproducible multi-sector ranking architecture over admissible or approximately admissible backgrounds.

Master formula:

F_MAAT^landscape[X] =
- sum_a lambda_a log(epsilon + Gamma_a[X]),
Gamma_a[X] = 1 / (1 + d_a[X])

Benchmark evidence: Existing string_landscape_selection/ scripts test 10D-inspired tadpole closure, KKLT-style bridges, period-controlled flux backgrounds, and backreaction / Standard-Model-sector proxies. Across these benchmarks, structural ranking differs from energy-only ordering and can reduce tadpole, stability, or phenomenological obstruction in the selected subsets.

Scripts and reproducibility:

Folder	Role
`experiments/string_landscape_selection/`	string-landscape toy and bridge benchmarks, result JSON files, and generated plots

Documentation PDF: documentation/Structural_Selection_in_the_String_Landscape_MAAT_Framework.pdf

Extra Phenomenological Paper — MAAT String Selection II

MAAT String Selection II: Structural Path Selection in a Reduced KKLT Landscape

Core idea: Extends the string-landscape vacuum-ranking benchmark from static endpoint selection to transition-path selection. Candidate KKLT bridge vacua become nodes in a reduced landscape graph, and near-neighbour transitions become directed edges with a structural path action.

Path action:

A_ij =
  B_ij
  + lambda_F max(F_MAAT[B_j] - F_MAAT[B_i], 0)
  + lambda_c C_ij
  + lambda_Q Q_ij
  + lambda_R R_loss,ij

P(B_i -> B_j) ∝ Gamma_ij exp[-A_ij],
Gamma_ij = exp[-B_ij]

Core results:

Quantity	Result
Minima nodes	`77`
Directed near-neighbour edges	`462`
AdS minima	`45`
dS minima	`32`
Spearman `abs_energy` vs `F_bridge` among minima	`0.1271`
Spearman `Delta_Q_10` vs `F_bridge` among minima	`0.9486`
Fraction structurally downhill edges	`0.539`
Top-20 structural/energy edge overlap	`0 / 20`
Best-energy to best-structural path	`60 -> 22 -> 30`
Structural path cost	`0.907889`
Energy-action cost to same target	`1.719058`

Key finding:

MAAT extends static string vacuum ranking into structural landscape flow: preferred endpoints and preferred transition paths need not be determined by energy alone.

Scientific status: This is a phenomenological transition-graph benchmark. It is not a Coleman-De Luccia tunnelling calculation, not a microscopic string decay-rate derivation, and not a complete landscape dynamics model.

Scripts and reproducibility:

Repository URL:

https://github.com/Chris4081/structural-selection-principle/tree/main/experiments/string_landscape_path_selection

Folder	Role
`experiments/string_landscape_path_selection/`	MAAT String Selection II path graph, transition edges, selected paths, CSV/JSON outputs, and figures

Data attribution and license note: This experiment reuses derived synthetic KKLT-bridge candidates generated by the repository's string_landscape_selection benchmark scripts. No external survey data or proprietary dataset is redistributed. The benchmark is not endorsed by the authors of the original string-theory literature cited in the paper.

Documentation PDF: documentation/MAAT_String_Selection_II_Structural_Path_Selection.pdf

Repository Structure

structural-selection-principle/
│
├── structural_selection_flrw_v4.py        ← Paper 01: FLRW benchmark
├── maat_structural_selection_study_v2.py  ← Paper 02: 1D kink sweep
├── maat_structural_selection_2d_sstar.py  ← Paper 03: 2D s★ sweep
├── cci_entropy_test.py                    ← Paper 05: CCI vs Ṡ/I
├── cci_continuous_ensemble.py             ← Paper 06: 160-run ensemble
├── cci_alpha_scaling.py                   ← Paper 07: α-scan + log-fit
├── cci_phase_diagram.py                   ← Paper 07: phase diagrams
├── cci_entropy_scaling_2d.py              ← Paper 09: 2D α-scan + log-fit
├── figure_dimensional_trend.py            ← Paper 10: Figure 1 (α trend)
├── figure_scaling_comparison.py           ← Paper 10: Figure 2 (1D vs 2D)
├── cci_entropy_scaling_3d.py              ← Paper 11: 3D ensemble + figures
├── Collapse-Plot.py                       ← Paper 12: cross-dimensional collapse
├── multiparameter_fit.py                  ← Paper 13: multi-parameter scaling fit
├── cci_entropy_scaling_3d_v2.py           ← Paper 14: 3D with directional MI
├── manifold_geometry_plot.py              ← Paper 14: PCA + log-space figures
├── xi_aniso_full_test.py                  ← Paper 14: ξ_aniso quartile analysis
├── plateau_degeneracy_exact.py            ← Paper 16: plateau degeneracy measures
├── paper17_analysis.py                    ← Paper 17: CCI regime classifier + CV
├── coherence_simulation.py                ← Paper 18: coherence-field dynamics
├── active_control_phi4.py                 ← Paper 19: active structural control
├── uni_stability_test.py                  ← Paper 20: cross-dimensional stability test
├── structural_selection_phi4_protocol.py  ← Paper 22: static φ⁴ validation benchmark
├── structural_selection_phi4_plots.py     ← Paper 22: figure generation from benchmark JSON
│
├── experiments/
│   ├── natural_constants_selection/       ← Paper 26: natural-constants v1--v13 benchmark
│   ├── standard_model_bridge/             ← Paper 26: SM-like RG bridge and v11 holdout
│   ├── boundary_aware_lambda_calibration/ ← Paper 27: fused boundary-aware λ calibration
│   ├── lambda_response_closure/           ← addendum: response-theoretic λ closure
│   ├── cosmological_cci/                  ← Paper 28: cosmological CCI observable
│   ├── cosmological_cci_v03/              ← Paper 29: growth connectivity + robustness CCI
│   ├── maat_dynamic_fields_v05_v09/       ← Paper 31: response/local/gravity/FLRW tests
│   ├── maat_observable_predictions_v10/   ← Paper 31: observable projection layer
│   ├── maat_hz_chi2_paper32/              ← Paper 32: H(z) chronometer comparison and χ² scan
│   ├── maat_cci_projection_paper33/       ← Paper 33: CCI projection observable + sensitivity scan
│   ├── maat_projection_growth_paper34/    ← Paper 34: projection observable vs growth-data diagnostic
│   ├── maat_growth_perturbation_paper35/  ← Paper 35: linear growth embedding + perturbation stability
│   ├── maat_v121_observables_stability_paper37/ ← Paper 37: v1.2.1 observable proxy + stability landscape
│   ├── maat_paper38_v121_robustness_closure/ ← Paper 38: v1.2.1 closure in linear growth
│   ├── maat_paper39_observable_growth_signature/ ← Paper 39: v1.2.1 observable growth signature proxy
│   ├── maat_paper40_structural_signature_test/ ← Paper 40: v1.2.1 CCI residual signature test
│   ├── maat_paper42_blind_projection_test/ ← Paper 42: response-derived blind projection test
│   ├── maat_so10_structural_selection/   ← extra paper: SO(10)-motivated gauge and Yukawa benchmarks
│   └── string_landscape_selection/        ← extra paper: MAAT string-landscape ranking
│
├── documentation/                         ← PDFs of all papers
└── README.md

Run the Code

pip install numpy pandas matplotlib scipy scikit-learn

Script	Paper	What it does
`structural_selection_flrw_v4.py`	01	FLRW 3-case MaxEnt benchmark
`maat_structural_selection_study_v2.py`	02	1D kink dominance, 81 settings
`maat_structural_selection_2d_sstar.py`	03	2D s★ phase diagram
`cci_entropy_test.py`	05	CCI vs Ṡ/I, discrete classes
`cci_continuous_ensemble.py`	06	CCI vs F_struct, 160 runs
`cci_alpha_scaling.py`	07	α-scan, log-fit, scatter plots
`cci_phase_diagram.py`	07	9×9 phase diagram heatmaps
`cci_entropy_scaling_2d.py`	09	2D α-scan + log-fit
`figure_dimensional_trend.py`	10	α(d) and r_s(d) trend figures
`figure_scaling_comparison.py`	10	1D vs 2D scaling side-by-side
`cci_entropy_scaling_3d.py`	11	3D ensemble, α-scan, dimension comparison
`Collapse-Plot.py`	12	cross-dimensional collapse (requires all 3 CSVs)
`multiparameter_fit.py`	13	two-parameter fit a,b across 1D–3D
`cci_entropy_scaling_3d_v2.py`	14	3D sim with directional MI (I_x, I_y, I_z)
`manifold_geometry_plot.py`	14	PCA + log-space manifold figures
`xi_aniso_full_test.py`	14	ξ_aniso quartile-split + regression
`plateau_degeneracy_exact.py`	16	exact plateau degeneracy D_width, D_norm, D_flat, D
`paper17_analysis.py`	17	CCI threshold classifier, F_struct comparison, 5-fold CV
`coherence_simulation.py`	18	coherence-field dynamics (0D + 1D, domain formation, energy decay)
`active_control_phi4.py`	19	active structural control, parameter sweep, heatmaps
`uni_stability_test.py`	20	cross-dimensional stability test, drift analysis, figure
`structural_selection_phi4_protocol.py`	22	static φ⁴ benchmark, vacua vs saddle, kink vs distortions, writes result JSON
`structural_selection_phi4_plots.py`	22	generates Paper 22 validation plots from `structural_selection_phi4_results.json`
`experiments/natural_constants_selection/naturkonstante_v7_landscape_heatmap.py`	26	natural-constants basin visualisation
`experiments/natural_constants_selection/naturkonstante_v8_gradient_flow.py`	26	gradient-flow attractor test in constant space
`experiments/natural_constants_selection/naturkonstante_v12_maxent_lambda.py`	26	maximum-entropy calibration of sector weights
`experiments/natural_constants_selection/naturkonstante_v13_maxent_sm_bridge.py`	26	MaxEnt-weighted constants bridge using calibrated sector weights
`experiments/standard_model_bridge/standard_model_rg_maat_bridge.py`	26	one-loop SM-like RG bridge from UV parameters to IR effective observables
`experiments/standard_model_bridge/standard_model_rg_maat_summary_figure.py`	26	generates the four-panel Paper 26 summary figure
`experiments/standard_model_bridge/standard_model_rg_maat_v11_holdout.py`	26	direct-term holdout benchmark for cross-sector predictivity
`experiments/boundary_aware_lambda_calibration/fit_closed_maat_lambda_v1.py`	27	closed boundary-aware MAAT lambda calibration over fused defect data
`experiments/boundary_aware_lambda_calibration/plot_closed_maat_lambda_v2.py`	27	generates Paper 27 lambda and defect-comparison figures
`experiments/lambda_response_closure/lambda_response_closure.py`	addendum	derives effective lambda weights from defect covariance and target-response geometry
`experiments/cosmological_cci/maat_cci_cosmology_v02.py`	28	generates the cosmological CCI model grid, chronometer projection, and plots
`experiments/cosmological_cci_v03/maat_cci_cosmology_v03_growth.py`	29	adds f sigma_8 growth connectivity, robustness margins, MaxEnt companion weights, and curvature transition proxy
`experiments/maat_dynamic_fields_v05_v09/v05_dynamic_lambda_flow/lambda_dynamic_flow_v05.py`	31	covariance-driven response-field dynamics
`experiments/maat_dynamic_fields_v05_v09/v06_local_selection_fields/local_selection_phi4_v06.py`	31	local selection-pressure fields in a perturbed 1D phi4 benchmark
`experiments/maat_dynamic_fields_v05_v09/v09_flrw_stability_scan/maat_flrw_stability_scan_v09.py`	31	toy scalar FLRW stability scan
`experiments/maat_observable_predictions_v10/maat_observable_predictions_v10.py`	31	observable projection: H(z), Delta H/H, w(z), Omega_MAAT, and f sigma_8 proxy
`experiments/maat_hz_chi2_paper32/maat_hz_data_comparison_v01.py`	32	fixed v0.10 MAAT comparison against Cosmic Chronometer H(z) data
`experiments/maat_hz_chi2_paper32/maat_hz_chi2_fit_v01.py`	32	two-parameter MAAT H(z) chi-square scan
`experiments/maat_v121_observables_stability_paper37/paper37_observables_emergent_robustness.py`	37	v1.2.1 baseline observable proxy with derived respect and emergent robustness
`experiments/maat_v121_observables_stability_paper37/paper37_stability_landscape.py`	37	two-parameter v1.2.1 proxy stability landscape scan
`experiments/maat_v121_observables_stability_paper37/sat_validation/maat_v121_sat_validation.py`	37	companion SAT correlation validation for the empirical discussion
`experiments/maat_paper38_v121_robustness_closure/maat_paper38_v121_robustness_closure.py`	38	v1.2.1 robustness closure in a linear-growth benchmark plus selection-field perturbation tests
`experiments/universal_structural_selection_benchmarks_paper49/universal_structural_selection_benchmarks_paper49.py`	49	aggregates existing benchmark outputs into a universality evidence registry and competitor-readiness matrix
`experiments/sat_structural_hardness_paper50/sat_structural_hardness_paper50.py`	50	synthetic random-3-SAT graph-local hardness benchmark with density, graph, MAAT, and shuffled-null baselines

Key Results at a Glance

Paper 01 (SSP):      slow-roll preferred        E = 0.889 < E_deSitter
Paper 02 (1D):       kink wins 81/81            weight 99.96%
Paper 03 (2D):       selection transition        s★_c ∈ (0.20, 0.42)
Paper 05 (ground):   CCI vs Ṡ/I                Spearman r = 0.760
Paper 06 (ensemb):   CCI ≈ F_struct             Spearman r = 0.878
Paper 07 (1D sc.):   CCI ∝ Ṡ/I^α              r_s = 0.734, α ≈ 2.5
Paper 08 (Landau):   O ~ I_nn, μ ~ CCI          theoretical proposal
Paper 09 (2D sc.):   scaling persists in 2D     r_s = 0.870, α ≈ 3.0
Paper 10 (dim.):     α increases with d         r_s: 0.734 → 0.870
Paper 11 (3D):       plateau in 3D               r_s = 0.850, α ∈ [2.8, 3.5]
Paper 12 (collapse): partial collapse 1D–3D      shared structure, no universal exponent
Paper 13 (multi-p):  sign flip a: -1.6→+0.3     dimension-dependent scaling family
Paper 14 (manifold): scaling = projection         r_s(ξ_aniso,R_α) = -0.721, p=0.0001
Paper 16 (degeneracy):D_norm non-monotonic        D(2D)<D(1D)<D(3D), transition at 2D→3D
Paper 17 (predict.): CCI perfect classifier      Acc=1.00, CV Ā=0.992±0.019
Paper 18 (dynamics): coherence field emerges     symmetry breaking, domains, F ↓ monotonic
Paper 19 (control):  active coherence steering   CCI ↓6.7%, I_nn ↑154%, two control regimes
Paper 20 (robust.):  CCI/F_struct stable           drift<2 vs ratio drift≈2000, projection vs intrinsic
Paper 22 (validate): struct. beats energy rank     Δ_vac=2.583852, p_K=1.000, gain at 20% percentile
Paper 26 (constants): basin-level SM compatibility, MaxEnt weights R>V≈S>B>H, v11/v13 test predictivity
Paper 27 (boundary): R dominates closed λ fit       R share=0.3917, λ_R=8.078, N=3400 fused samples
Lambda closure:       λ from Cov[d] response         safe target: R share≈0.833, low-defect target: R share≈0.297
Paper 28 (cosmo CCI): structural-stress history     CCI_norm(z=10)≈803.8, chronometer RMS≈1.01
Paper 29 (cosmo v03): V/R measured + λ + z_c        z_c≈1.114, λ:S>H>R>V, CCI_v03(z=2)≈15.42
Paper 31 (dynamic):  diagnostics -> observables      track err=9.91e-7, residual=0.1648, max |DeltaH/H|=0.0399
Paper 32 (H(z)):     first chronometer chi2 test      LCDM χ²=14.8759, MAAT χ²=15.3661, χ²ν(best)=0.5299
Paper 33 (CCI proj): breadth-depth projection stress z_tr=0.8499, scan median=0.8889, N=6930
Paper 34 (proj/growth): distinct info layer           z_tr=1.0436, C_proj χ²ν=42.77 vs fσ8, scan N=625
Paper 35 (growth):  forward perturbation embedding    max |ΔD/D|=0.0591%, max |Δfσ8/fσ8|=0.4570%
Paper 36 (closure): R becomes derived robustness       R_resp=(HBV)^(1/3), R_rob=min(R_resp,(HBSV)^(1/4))
Paper 37 (v1.2.1): observable proxy + stability scan   max |ΔH/H|=1.70%, max |Δfσ8/fσ8|=1.83%, stable=1089/1089
Paper 38 (growth closure): v1.2.1 in linear growth      <R_rob>=0.9313, max |ΔD/D|=0.0591%, λ positivity passed
Paper 39 (growth signature): projection-modulated fσ8   ε_best=-0.0100, Δχ²=-0.0601, max |Δfσ8/fσ8|=0.9891%
Paper 40 (residual signature): CCI vs residual stress    ρS(CCI_diag, |rσ|)=0.5934, p=0.0338, null p≈0.036
Paper 41 (variable closure): definitions + measurement map H,B,S,V primary; R_rob emergent; no standalone code
Paper 42 (blind projection): response-derived projection  ρS(CCI_diag, |rσ|)=0.4286, p=0.1456, no shape tuning
Paper 48 (MaxEnt):   structural measure theorem           P[X]∝exp[-F_struct], support form = bounded viability
Paper 49 (benchmarks): universality audit matrix           complete/partial/missing domains=2/4/1, readiness=0.3816
Paper 50 (SAT II): graph-local SAT hardness benchmark      MAAT+density R²=0.2347 vs density 0.1886, graph 0.2469
SO(10) extra: gauge one-loop + Yukawa bridge           M_GUT≈1.86e16 GeV, Δb≈0.0506, Yukawa R_rob≈0.999

www.maat-research.com

License

Code: MIT License — free to use, modify, and share with attribution.

Name		Name	Last commit message	Last commit date
Latest commit History 209 Commits
documentation		documentation
experiments		experiments
Collapse-Plot.py		Collapse-Plot.py
LICENSE		LICENSE
README.md		README.md
active_control_phi4.py		active_control_phi4.py
cci_alpha_scaling.py		cci_alpha_scaling.py
cci_entropy_scaling_2d.py		cci_entropy_scaling_2d.py
cci_entropy_scaling_3d.py		cci_entropy_scaling_3d.py
cci_entropy_scaling_3d_v2.py		cci_entropy_scaling_3d_v2.py
cci_entropy_test.py		cci_entropy_test.py
cci_phase_diagram.py		cci_phase_diagram.py
cci_struct_test.py		cci_struct_test.py
coherence_simulation.py		coherence_simulation.py
figure_dimensional_trend.py		figure_dimensional_trend.py
figure_scaling_comparison.py		figure_scaling_comparison.py
legal.md		legal.md
maat_structural_selection_2d_sstar_sweep.py		maat_structural_selection_2d_sstar_sweep.py
maat_structural_selection_study_v2.py		maat_structural_selection_study_v2.py
manifold_geometry_plot.py		manifold_geometry_plot.py
multiparameter_fit.py		multiparameter_fit.py
paper17_analysis.py		paper17_analysis.py
paper17_analysis_cv.py		paper17_analysis_cv.py
paper19_active_control_test.py		paper19_active_control_test.py
plateau_degeneracy_exact.py		plateau_degeneracy_exact.py
requirements.txt		requirements.txt
structural_selection_flrw_v4.py		structural_selection_flrw_v4.py
structural_selection_iib_kklt_plots.py		structural_selection_iib_kklt_plots.py
structural_selection_iib_kklt_scan.py		structural_selection_iib_kklt_scan.py
structural_selection_phi4_plots.py		structural_selection_phi4_plots.py
structural_selection_phi4_protocol.py		structural_selection_phi4_protocol.py
uni_stability_test.py		uni_stability_test.py
xi_aniso_full_test.py		xi_aniso_full_test.py
xi_proxy_scan.py		xi_proxy_scan.py

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Structural Selection Principle

Start Here — Core Formalism Primer

Papers in this Series

Paper 01 — General Framework

Paper 02/03 — 1D and 2D Topological Selection

Paper 04 — CCI Framework

Paper 05 — Physical Grounding of the CCI

Paper 06 — Continuous Ensemble Study

Paper 07 — Entropy–Information Scaling (1D)

Paper 08 — Landau-Type Theory

Paper 09 — Entropy–Information Scaling (2D)

Paper 10 — Dimensional Dependence of Scaling

Paper 11 — Entropy–Information Scaling (3D)

Paper 12 — Cross-Dimensional Collapse Analysis

Paper 13 — Multi-Parameter Scaling Analysis

Paper 14 — Scaling Manifold Framework

Paper 16 — Plateau Degeneracy Measure

Paper 17 — Predictive Power of the CCI

Paper 18 — Dynamical Origin of Structural Selection

Paper 19 — Active Structural Control

Paper 20 — Universal Stability Test

Paper 22 — Numerical Validation of Structural Selection

Paper 26 — Structural Selection of Effective Constants

Paper 27 — Boundary-Aware Calibration of MAAT Structural Weights

Paper 28 — Cosmological Critical Coherence Index

Paper 29 — Cosmological CCI with Growth Connectivity and Robustness

Paper 30 — Response-Based Derivation of MAAT Structural Weights

Paper 31 — Dynamic Structural Selection

Paper 32 — First H(z) Data Comparison

Paper 33 — CCI as Projection Observable

Paper 34 — Projection Observable vs Growth and Expansion Data

Paper 35 — Linear Growth Embedding

Paper 36 — The Maturation of MAAT

Paper 37 — MAAT v1.2.1 Observable Proxy Predictions

Paper 38 — MAAT v1.2.1 Robustness Closure in Linear Growth

Paper 39 — MAAT v1.2.1 Observable Growth Signature Proxy

Paper 40 — MAAT v1.2.1 Structural Signature Test in Growth Data

Paper 41 — MAAT v1.2.1 Variable Closure and Measurement Definitions

Paper 42 — Blind Projection Test

Paper 43 — Linear Perturbations and Structure Growth

Paper 44 — Active Structural Significance

Paper 45 — Effective Metric Response in MAAT Structural Cosmology

Paper 46 — Environmental Screening in MAAT Structural Cosmology

Paper 47 — From Environmental Screening to Observable Void Signatures

Paper 48 — Maximum-Entropy Derivation of Universal Structural Selection

Paper 49 — Universal Structural Selection Benchmarks

Paper 50 — SAT Structural Hardness II

Paper 50b — SAT Structural Hardness IIb

Paper 50c — SAT Structural Hardness IIc

Paper 51 — Quantum Pointer-State Selection

Paper 52 — Cross-Domain Transfer of Structural Selection

Paper 53 — Structural Mode Transfer Robustness

Paper 54 — External Validation of Structural Modes

Paper 55 — CDCL Hardness Prediction on Standard SAT Families

Paper 56 — From Scalar Scores to Defect-Field Geometry

Paper 57 — Where Does SAT Hardness Live?

Paper 58 — Stationarity Balance Beats Population Balance

Extra Mathematical Diagnostic Paper — Riemann Zeta Critical-Line Diagnostic

Extra Experiment Paper — Societal Critical Coherence Index

Extra Phenomenological Paper — Structural Selection in SO(10)-Motivated Unified Field Theories

Extra Phenomenological Paper — Structural Selection in the String Landscape

Extra Phenomenological Paper — MAAT String Selection II

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