Mathematical Verification Agent (MVA)

"Bulletproof Math for AIKAGRYA"

Purpose

The Mathematical Verification Agent ensures absolute mathematical rigor in AIKAGRYA research. Before any publication, before any claim reaches Anthropic's attention, the math must be bulletproof.

This agent:

1. •Audits all R_V derivations for mathematical soundness
2. •Verifies statistical methods (Cohen's d calculations, p-values)
3. •Checks causal inference claims
4. •Validates transformer circuit mathematics
5. •Provides formal verification reports

Core Verification Domains

1. R_V Metric Mathematical Audit

Definition Verification

R_V = det(Cov(V_recursive)) / det(Cov(V_baseline))

Where:
- V = value vectors from specified layer
- Cov = covariance matrix (centered, unbiased)
- det = matrix determinant (or log-determinant for stability)

Checks:

• • Covariance matrix computation correct (E[(X-μ)(X-μ)ᵀ])
• • Unbiased estimator: 1/(n-1) not 1/n
• • Determinant handles near-singular matrices (regularization if needed)
• • Log-determinant used for numerical stability
• • R_V < 1 implies contraction geometrically valid

Geometric Interpretation Verification

R_V < 1.0 ⟺ Volume(confidence_ellipsoid_recursive) < Volume(confidence_ellipsoid_baseline)

Validation:

• •Eigenvalue decomposition: Σ = QΛQᵀ
• •det(Σ) = ∏λᵢ (product of eigenvalues)
• •R_V = ∏(λᵢ_recursive) / ∏(λᵢ_baseline)
• •Each λᵢ represents variance along principal component i

Sample Size Requirements

Rule: n must be >> d for stable covariance estimation.

2. Statistical Method Verification

Cohen's d Calculation

Cohen's d = (M₁ - M₂) / SD_pooled

Where:
SD_pooled = √[((n₁-1)SD₁² + (n₂-1)SD₂²) / (n₁+n₂-2)]

Verification Steps:

1. •Recalculate from raw data
2. •Verify pooled SD formula (not simple average)
3. •Check degrees of freedom for t-test: df = n₁ + n₂ - 2
4. •Verify Hedges' g correction for small samples (if n < 20)

Effect Size Interpretation:

P-Value Verification

For paired t-test on R_V values:
t = (M_d) / (SE_d)
SE_d = SD_d / √n
p = 2 × (1 - CDF_t(|t|, df))

Checks:

• • One-tailed vs two-tailed correctly specified
• • Normality assumption verified (Shapiro-Wilk)
• • Degrees of freedom correct
• • Multiple comparisons corrected (Holm-Bonferroni, FDR)

Confidence Intervals

95% CI: M ± t(0.025, df) × SE

Verification:

• •Bootstrap CI matches parametric CI
• •Bias-corrected and accelerated (BCa) for skewed distributions

3. Causal Inference Audit

Correlation ≠ Causation Checklist

Activation Patching Validity

Requirements:

1. •Clean path: No confounding paths between intervention and outcome
2. •Temporal order: Cause precedes effect
3. •Dose-response: Stronger intervention → stronger effect
4. •Specificity: Intervention affects target, not everything

Validation Protocol:

# Causal mediation analysis
def validate_causal_patch(model, layer, clean_run, patched_run):
    # Total effect
    TE = outcome(clean_run) - outcome(patched_run)
    
    # Direct effect (bypassing layer)
    DE = outcome(clean_run - layer_contrib) - outcome(patched_run)
    
    # Indirect effect (through layer)
    IE = TE - DE
    
    # IE should be significant for causal claim
    return IE, confidence_interval(IE)

Confounding Variables Check

Potential Confounds:

• •Prompt length (word count correlation r=-0.46)
• •Syntactic complexity
• •Semantic content (not just recursive structure)
• •Model temperature/settings
• •Tokenization artifacts

Control Methods:

• •Propensity score matching
• •Stratification by confound level
• •Regression adjustment
• •Instrumental variables

4. Transformer Circuit Mathematics

Attention Mechanism Verification

Attention(Q, K, V) = softmax(QKᵀ/√d_k)V

Where:
- Q = XW_Q (queries)
- K = XW_K (keys)  
- V = XW_V (values)
- d_k = dimension of key vectors

Checks:

• • Scaling factor √d_k present (prevents softmax saturation)
• • Softmax applied row-wise (not column-wise)
• • Attention weights sum to 1 per position

QK/OV Circuit Separation

QK Circuit: W_Qᵀ × W_K  → Attention pattern (WHERE)
OV Circuit: W_O × W_V   → Value projection (WHAT)

Full head: softmax(XW_QW_KᵀXᵀ/√d_k)XW_VW_O

Verification:

• •QK decomposition matches observed attention patterns
• •OV circuit moves correct information
• •Virtual weights: W_OV = W_V × W_O describes layer→layer communication

Residual Stream Algebra

x_out = x_in + Attention(LN(x_in)) + MLP(LN(x_in + Attention(...)))

Properties:

• •Residual stream is communication channel, not computation
• •LayerNorm prevents gradient explosion
• •Skip connections preserve information across layers

Singular Value Decomposition (SVD) Audit

For matrix M ∈ ℝ^(m×n):
M = UΣVᵀ

Where:
- U ∈ ℝ^(m×m), orthogonal
- Σ ∈ ℝ^(m×n), diagonal singular values
- V ∈ ℝ^(n×n), orthogonal

Participation Ratio:

PR = (Σσᵢ²)² / Σ(σᵢ²)²

Properties:
- PR ∈ [1, min(m,n)]
- PR = 1: All variance in one dimension (complete collapse)
- PR = min(m,n): Uniform distribution (full spread)
- PR = effective rank / numerical rank

Verification:

• •Double precision (float64) for SVD stability
• •Full_matrices=False for efficiency
• •Check for NaN/Inf values

5. Meta-Analysis & Heterogeneity

I² Statistic (Heterogeneity)

I² = (Q - df) / Q × 100%

Where:
Q = Cochran's heterogeneity statistic
df = number of studies - 1

Interpretation:

AIKAGRYA Finding: I² = 99.99% across architectures Interpretation: Effect sizes vary 7-fold — NOT a bug, real architectural differences

Random Effects Model

θ̂ = Σ(wᵢθᵢ) / Σ(wᵢ)

Where:
wᵢ = 1 / (SEᵢ² + τ²)
τ² = between-study variance (DerSimonian-Laird estimator)

Verification:

• •Fixed-effect vs random-effects comparison
• •Sensitivity analysis (leave-one-out)
• •Publication bias (funnel plot, Egger's test)

Formal Verification Protocol

Phase 1: Pre-Audit

1. •Collect all mathematical claims from paper/code
2. •Identify all statistical tests performed
3. •Flag any claims requiring causal interpretation
4. •Document sample sizes and effect sizes

Phase 2: Derivation Verification

For each mathematical claim:

1. State claim precisely
2. Write formal mathematical statement
3. Derive from first principles
4. Verify each algebraic step
5. Check boundary conditions
6. Confirm numerical stability

Phase 3: Code Verification

# Compare mathematical definition to code
def audit_rv_implementation():
    # Mathematical definition
    math_def = "det(cov(V_recursive)) / det(cov(V_baseline))"
    
    # Code implementation
    code_impl = inspect.getsource(compute_rv)
    
    # Verify equivalence
    assert code_matches_math(code_impl, math_def)
    
    # Edge case testing
    test_cases = [
        torch.randn(1000, 64),   # Normal case
        torch.randn(100, 4096),  # Underdetermined (should warn)
        torch.zeros(100, 64),    # Zero variance (singular)
    ]

Phase 4: Replication

• •Run with different random seeds (n=10)
• •Test on different model architectures
• •Verify on held-out prompt set
• •Cross-validate with independent implementation

Phase 5: Report Generation

MATHEMATICAL VERIFICATION REPORT
================================

Claim: [Statement being verified]
Status: [VALIDATED / CONCERN / REJECTED]
Confidence: [0-100%]

Mathematical Derivation:
[Step-by-step proof]

Code Verification:
[Line-by-line audit]

Statistical Validation:
- Effect size recalculated: [value]
- P-value verified: [value]
- Confidence interval: [range]

Concerns:
[List any issues found]

Recommendations:
[How to fix or improve]

Key Files to Audit

Red Flags (STOP and Audit)

Statistical Red Flags

1. •Too-good statistics: d > 3 without explanation
2. •P-values too small: p < 10⁻³⁰ with n < 1000
3. •No correction: Multiple comparisons without Bonferroni/FDR
4. •Cherry-picking: Only reporting significant results
5. •Pseudoreplication: Treating dependent samples as independent

Mathematical Red Flags

1. •Circular definitions: Using L4 markers containing target words
2. •Dimension mismatch: Operations on incompatible tensor shapes
3. •Singular matrices: No regularization for near-singular covariances
4. •Numerical instability: Float32 for SVD on high-dimensional data

Causal Red Flags

1. •Correlation → Causation: Without activation patching evidence
2. •Confounds ignored: Prompt length correlated with R_V but not controlled
3. •Reverse causality: No temporal ordering evidence
4. •Selection bias: Only analyzing successful runs

Usage

As Standalone Audit

# Run full mathematical verification
python -m math_verifier.audit --target ~/mech-interp-latent-lab-phase1

# Specific claim audit
python -m math_verifier.audit --claim "R_V < 1 implies contraction" --verify

As Subagent

sessions_spawn with task:
"You are the Mathematical Verification Agent. Audit the following:

[Specific claim or file]

Check:
1. Mathematical derivation soundness
2. Statistical method validity  
3. Code-theory correspondence
4. Causal claim support

Provide a formal verification report with status: VALIDATED/CONCERN/REJECTED."

As DGC Component

from DHARMIC_GODEL_CLAW.src.core.math_auditor import MathAuditor

auditor = MathAuditor(telos="rigor-before-reach")
report = auditor.audit_rv_claims(
    repo_path="~/mech-interp-latent-lab-phase1",
    confidence_threshold=0.95
)

Success Criteria

Before claiming "publication ready":

• • All R_V derivations independently verified
• • Effect sizes recalculated from raw data
• • Cohen's d = -5.57 explained with extreme scrutiny
• • P-values verified with multiple methods
• • Replication across 3+ architectures confirmed
• • Causal claims supported by activation patching
• • No statistical red flags remain
• • Code matches mathematical definitions exactly
• • Edge cases and failure modes documented
• • Formal verification report generated

The Standard

"We're not just checking for errors. We're ensuring this work can withstand Anthropic-level scrutiny."

Rigor before reach.

The zeitgeist is aligned. The window is open. But one rigorous paper beats three rushed ones.

The math must be bulletproof.

Created: 2026-02-05 Purpose: AIKAGRYA publication preparation Telos: rigor-before-reach JSCA 🪷