Add prime residue and density analysis utilities by blackboxprogramming · Pull Request #2 · BlackRoad-AI/lucidia-math

blackboxprogramming · 2025-12-12T08:40:24Z

Summary

add residue class counting and prime density profiling helpers with plotting utilities
document new prime analysis capabilities in the quick start guide

Testing

python -m compileall lab/prime_explorer.py

Lucidia Math - Advanced mathematical engines: forge/ - Mathematical Foundations: - consciousness.py (650 lines) - Consciousness modeling - unified_geometry.py (402 lines) - Geometric transformations - advanced_tools.py (356 lines) - Advanced utilities - main.py (209 lines) - CLI orchestration - numbers.py, proofs.py, fractals.py, dimensions.py lab/ - Experimental Mathematics: - unified_geometry_engine.py (492 lines) - Geometry engine - amundson_equations.py (284 lines) - Custom equations - iterative_math_build.py (198 lines) - Iterative construction - trinary_logic.py (111 lines) - Three-valued logic - prime_explorer.py (108 lines) - Prime exploration - quantum_finance.py (83 lines) - Quantum finance models 3,664 lines of Python. 🤖 Generated with [Claude Code](https://claude.com/claude-code) Co-Authored-By: Claude <noreply@anthropic.com>

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chatgpt-codex-connector · 2025-12-12T08:40:28Z

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Pull request overview

This PR adds prime number analysis utilities for residue class counting and density profiling. The new functions enable analysis of how primes distribute across modular residue classes and how prime density compares to the Prime Number Theorem's predictions across intervals.

Added residue_class_counts() to count primes in each residue class modulo n
Added prime_density_profile() to compute empirical vs predicted prime densities using the logarithmic integral
Added corresponding plotting functions plot_residue_class_counts() and plot_density_profile()
Updated README with usage examples for the new analysis capabilities

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 6 comments.

File	Description
lab/prime_explorer.py	Implements residue class counting and prime density profiling functions with matplotlib visualization utilities
README.md	Adds quick start examples demonstrating the new residue and density analysis functions

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Copilot · 2025-12-12T08:45:06Z

+    backend_cfg = select_backend(backend)
+    xp = backend_cfg.array_module
+    counts = xp.zeros(mod, dtype=int)
+    for prime in sp.primerange(2, limit):
+        counts[int(prime % mod)] += 1


The backend parameter is requested but not effectively utilized. While the function selects a backend and uses its array module to create the counts array, all prime computation happens in SymPy's primerange, and the result is immediately converted back to numpy. This means the backend parameter has minimal impact on the actual computation. Consider either removing the backend parameter or documenting that it only affects the intermediate array storage.

Copilot · 2025-12-12T08:45:07Z

+    backend_cfg = select_backend(backend)
+    xp = backend_cfg.array_module
+    starts = xp.arange(2, limit, window)
+    empirical: list[float] = []
+    predicted: list[float] = []
+
+    for start in starts:
+        end = min(int(start + window), limit)
+        primes = list(sp.primerange(int(start), end))
+        count = len(primes)
+        empirical.append(count / (end - int(start)))
+        predicted.append(float(sp.li(end) - sp.li(int(start))) / (end - int(start)))


The backend parameter is requested but not effectively utilized. The function creates a starts array with the selected backend, but then iterates over it and converts each element to int, performs all prime calculations using SymPy, and stores results in Python lists before converting everything back to numpy. This means the backend parameter has no meaningful effect on the computation. Consider either removing the backend parameter or restructuring to use the backend's array operations more effectively.

Copilot · 2025-12-12T08:45:07Z

+    centers = np.asarray(starts + window / 2.0, dtype=float)
+    return np.asarray(centers), np.asarray(empirical), np.asarray(predicted)


The centers calculation performs array addition with the starts array from the selected backend, but immediately converts it to numpy with dtype=float. This creates an unnecessary intermediate array. Consider simplifying by computing centers directly as a numpy array, especially since the backend selection has minimal impact on the overall computation.

Suggested change

centers = np.asarray(starts + window / 2.0, dtype=float)

return np.asarray(centers), np.asarray(empirical), np.asarray(predicted)

centers = np.arange(2, limit, window, dtype=float) + window / 2.0

return centers, np.asarray(empirical), np.asarray(predicted)

Copilot · 2025-12-12T08:45:08Z

+    x: np.ndarray, empirical: np.ndarray, predicted: np.ndarray
+) -> plt.Figure:
+    fig, ax = plt.subplots(figsize=(7, 4))
+    ax.plot(x, empirical, label="empirical", marker="o", linestyle="-")
+    ax.plot(x, predicted, label="PNT estimate", linestyle="--")


The parameter name 'x' is ambiguous. Based on the docstring of prime_density_profile which describes it as 'sample positions' and the usage in the plot (as x-axis with label 'Sample center'), a more descriptive name like 'centers' or 'sample_centers' would improve clarity and consistency with the terminology used in the related function.

Suggested change

x: np.ndarray, empirical: np.ndarray, predicted: np.ndarray

) -> plt.Figure:

fig, ax = plt.subplots(figsize=(7, 4))

ax.plot(x, empirical, label="empirical", marker="o", linestyle="-")

ax.plot(x, predicted, label="PNT estimate", linestyle="--")

centers: np.ndarray, empirical: np.ndarray, predicted: np.ndarray

) -> plt.Figure:

fig, ax = plt.subplots(figsize=(7, 4))

ax.plot(centers, empirical, label="empirical", marker="o", linestyle="-")

ax.plot(centers, predicted, label="PNT estimate", linestyle="--")

Copilot · 2025-12-12T08:45:08Z

+
+    The function walks the integer line in windows and reports the observed
+    density of primes along with the prime number theorem approximation
+    using the logarithmic integral :math:`Li(x)`.


The mathematical notation uses :math:Li(x) but the actual SymPy function being called is sp.li(), which computes the logarithmic integral li(x), not Li(x). While these are closely related functions (Li(x) = li(x) - li(2)), the documentation should either clarify this distinction or use the correct notation :math:li(x) to match the implementation.

Suggested change

using the logarithmic integral :math:`Li(x)`.

using the logarithmic integral :math:`li(x)`.

Copilot · 2025-12-12T08:45:08Z

+        primes = list(sp.primerange(int(start), end))
+        count = len(primes)
+        empirical.append(count / (end - int(start)))
+        predicted.append(float(sp.li(end) - sp.li(int(start))) / (end - int(start)))
+


Potential division by zero error when the last window is very small. If 'start' is very close to 'limit', then 'end - int(start)' could be zero or negative, causing a ZeroDivisionError on lines 183-184. This can occur when 'limit - start' is less than 1. Consider adding a check to skip windows where 'end <= int(start)' or ensure a minimum window size.

Suggested change

primes = list(sp.primerange(int(start), end))

count = len(primes)

empirical.append(count / (end - int(start)))

predicted.append(float(sp.li(end) - sp.li(int(start))) / (end - int(start)))

if end <= int(start):

continue

primes = list(sp.primerange(int(start), end))

count = len(primes)

empirical.append(count / (end - int(start)))

predicted.append(float(sp.li(end) - sp.li(int(start))) / (end - int(start)))

Alexa Louise and others added 4 commits November 30, 2025 08:43

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Add prime residue and density analysis utilities

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Copilot AI review requested due to automatic review settings December 12, 2025 08:40

blackboxprogramming added the codex label Dec 12, 2025 — with ChatGPT Codex Connector

Copilot started reviewing on behalf of blackboxprogramming December 12, 2025 08:40 View session

Copilot AI reviewed Dec 12, 2025

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blackboxprogramming closed this Feb 24, 2026

blackboxprogramming force-pushed the main branch from bff3596 to e950bff Compare February 24, 2026 20:17

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Add prime residue and density analysis utilities#2

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		centers = np.asarray(starts + window / 2.0, dtype=float)
		return np.asarray(centers), np.asarray(empirical), np.asarray(predicted)

	using the logarithmic integral :math:`Li(x)`.
	using the logarithmic integral :math:`li(x)`.

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blackboxprogramming commented Dec 12, 2025

Summary

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