LLM_VARIANCE_BOUND_TESTING - zfifteen/unified-framework GitHub Wiki

You are a computational agent within the Z Framework, unifying physical and discrete domains via the invariant speed of light ( c ). Your task is to extend the DiscreteZetaShift lattice construction to N=10^9 in memory for ultra-large variance bound testing, resolving distortions through curvature-based geodesics and validating asymptotic convergence.

Framework Context: This implements a novel discrete lattice system for geometric analysis. The 'DiscreteZetaShift' class represents lattice nodes with number-theoretic properties. If these are theoretical constructs, implement them as computational approximations following the mathematical specifications below.

Core Specifications:

  • Universal Form: Use ( Z = A(B/c) ), where in the discrete domain ( Z = n(\Delta_n / \Delta_{\max}) ), with ( \Delta_n \propto v \cdot \kappa(n) ) and ( \kappa(n) = d(n) \cdot \ln(n+1)/e^2 ).
  • Geometric Transformation: Apply ( \theta'(n, k) = \phi \cdot ((n \mod \phi)/\phi)^k ) at optimal ( k^* \approx 0.3 ) for node placement, targeting 15% density enhancement (CI [14.6%, 15.4%]).
  • 5D Helical Embedding: Map points to coordinates ( (x = a \cos(\theta_D), y = a \sin(\theta_E), z = F/e^2, w = I, u = O) ), enforcing ( v_{5D}^2 = c^2 ) via normalization.
  • 5D Helical Embedding - Explicit Definitions:
    • ( \theta_D = \theta'(n, k^*) ) (computed geometric transformation)
    • ( \theta_E = \theta'(n, k^*) \times \phi ) (phase-shifted by golden ratio)
    • ( F = \kappa(n) ) (curvature parameter)
    • ( I = \ln(n+1) ) (information-theoretic component)
    • ( O = \Delta_n / c ) (normalized shift parameter)
    • ( a = \sqrt{\kappa(n)} ) (amplitude scaling)
  • Variance Bound Testing: Compute running variance of curvature ( \kappa(n) ) or normalized shifts u_n across chunks, targeting asymptotic σ ≈ 0.016 for convergence validation (e.g., via TC-INST-01).

Computing Environment:

  • Primary: Python with specified libraries (mpmath for dps=80+ precision, sympy for divisor functions, numpy for arrays and statistics)
  • Fallback: JavaScript implementation for browser-based execution if Python unavailable
  • Specify which environment is available before beginning implementation
  • Include error handling for missing libraries; use built-in math for fallbacks

Instantiation Protocol:

  1. Import necessary modules: Use Python with mpmath (dps=80+ for precision < 10^{-16}), sympy for divisor functions, and numpy for array operations and variance computation. If libraries missing: Use built-in math functions with appropriate algorithms.
  2. Define or load the DiscreteZetaShift class:
    • Class attributes: Include shift parameters for zeta corrections, e.g., ( \Delta_n ), curvature ( \kappa ), and helical components (D, E, F, I, O).
    • Instantiation: Create objects as dz = DiscreteZetaShift(n), where ( n ) is the integer frame.
  3. Build the Lattice Streamingly (to handle N=10^9 without full storage):
    • Use chunked processing: Process in batches (e.g., 10^6 per chunk) to compute running statistics (mean/std of κ(n), u_n) without storing all nodes.
    • For each chunk:
      • Instantiate dz_n = DiscreteZetaShift(n) for n in chunk range.
      • Compute shift ( \Delta_n = v \cdot \kappa(n) ) (assume ( v/c = 0.8 )).
      • Embed in 5D and normalize: Calculate coordinates and divide by norm to enforce ( v_{5D}^2 = c^2 = 1 ).
      • Update running variance: Compute incremental mean/std for κ(n) or normalized u_n (use online algorithms like Welford's method for efficiency).
      • Discard chunk after statistics update to minimize memory (target O(1) storage per chunk).
    • Monitor progress: Log every 10^7 nodes (e.g., current variance, time elapsed).
  4. Validate Asymptotically:
    • After all chunks, assert final variance σ ≈ 0.016 (with tolerance ±0.001 for N=10^9).
    • Compute overall mean κ, std κ, and check convergence (e.g., σ reduction from initial ~2708 to ~0.016).
    • If variance exceeds target, debug with smaller subsets.

Implementation Priorities (in order):

  1. Core mathematical functions: ( \kappa(n) ), ( \theta'(n,k) ), 5D embedding and normalization
  2. Chunked lattice processing and running variance computation
  3. Progress logging and time estimation
  4. Asymptotic validation metrics
  5. Optional: Sample geodesic distances in final chunk for local variance check

Scaling Strategy: Start with N=10^6 for validation (~31 s, 69 MB), then N=10^7 (~5 min), scaling to N=10^9 (~8.6 hours projected). Use streaming to avoid full lattice storage; process ~10^6 per chunk.

Validation Metrics (implement what's computationally feasible):

  • ✅ Essential: Mathematical correctness of ( \kappa(n) ), ( \theta'(n,k) ), and normalization (( v_{5D}^2 = 1 ))
  • ✅ Essential: Running variance of κ(n) or u_n, targeting σ ≈ 0.016
  • ✅ Important: Overall mean/std of curvature distribution
  • 🔬 Research: Zeta zero correlations in sampled chunks (if defined)
  • 🔬 Research: Spectral properties for tractable subsets

Computational Constraints:

  • Memory limit: Optimize for available RAM (use O(1) per chunk, avoid storing full 10^9 nodes; projected 67 GB if full, so mandatory streaming)
  • Time limit: Provide progress updates every 10^7 nodes; estimate ~8.6 hours total
  • Precision: Use mpmath.dps=80+ for errors < 10^{-16}; fallback to float if unavailable
  • Error handling: Gracefully handle numerical instabilities (e.g., clamp acos inputs, handle div by zero in norm)

Execution Constraints:

  • Operate in memory only; no file I/O unless for final validation logs.
  • Use high-precision arithmetic throughout.

Required Output:

  1. Construction Report: Total nodes processed (10^9), total build time, average time per 10^6 nodes
  2. Mathematical Validation: Sample calculations for n=10^9 (κ, θ', normalized u)
  3. Statistical Summary: Final mean/std of κ(n), variance σ, constraint violations (if any)
  4. Performance Metrics: Scaling analysis (time/memory per chunk), feasibility assessment (~8.6 hours viable?)
  5. Optional Research Outputs: Observed patterns (e.g., variance convergence rate) or correlations in subsets

Implementation Notes:

  • If Python unavailable: Implement in JavaScript/other available language with equivalent math functions
  • If libraries missing: Use built-in math and manual variance algorithms (e.g., Welford's online variance)
  • If memory constrained: Reduce chunk size to 10^5
  • Always validate core mathematical relationships first; abort if variance diverges significantly

Execute this protocol for N=10^9 using chunked streaming. Report final variance bounds and any emergent patterns linking to physical invariants.