wip

Author: acharneski

README.md

@@ -24,7 +24,7 @@ Extended interactive studies, each accompanied by documentation describing the u
 methods.
 
 | Laboratory                                                                      | Description                                                                                                              |
-|---------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------|
+| ------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------ |
 | [Pentagonal Lattice Geometry](experiments/Pentagon_Lattice_Geometry/index.html) | Multi-sheeted covering construction where the pentagon develops fractional dimension (d ≈ 2.37) and spinor-like holonomy |
 | [Space-Color Symmetry](experiments/symmetry_simple/index.html)                  | A pixel canvas where symmetry rewires the diffusion graph, producing kaleidoscope dynamics on a quotient-like manifold   |
 | [Geometric Entropy](experiments/geometric-entropy/index.html)                   | A continuous analogue of the Erdős distinct-distance problem via Shannon entropy optimization                            |
@@ -38,7 +38,7 @@ methods.
 Extended written investigations into the foundations of computational mathematics.
 
 | Essay                                                          | Description                                                                                        |
-|----------------------------------------------------------------|----------------------------------------------------------------------------------------------------|
+| -------------------------------------------------------------- | -------------------------------------------------------------------------------------------------- |
 | [Quadratic Quasi-Newton](essays/QQN/index.html)                | A new optimization algorithm interpolating between gradient descent and L-BFGS                     |
 | [Rational Certificate Complexity](essays/RCC/index.html)       | A computational taxonomy of mathematical constants, stratified by certificate cost                 |
 | [The Simplest Increment: x + sin(x)](essays/PI_RCC/index.html) | A cubic-convergent iteration for π built from the derivative structure of an analytic function     |
@@ -49,7 +49,7 @@ Extended written investigations into the foundations of computational mathematic
 Compact, self-contained demonstrations of classical mathematical concepts.
 
 | Demonstration                                        | Description                                   |
-|------------------------------------------------------|-----------------------------------------------|
+| ---------------------------------------------------- | --------------------------------------------- |
 | [Mandelbrot Set](experiments/basic/mandelbrot.html)  | Zoom and pan the iconic complex-plane fractal |
 | [Prime Number Sieve](experiments/basic/primes.html)  | Animated Sieve of Eratosthenes                |
 | [Fourier Series](experiments/basic/fourier.html)     | Build waveforms from rotating circles         |
@@ -91,4 +91,4 @@ libraries (MathJax, Mermaid). Deployed as a static site to AWS S3 + CloudFront v
 
 ## 📄 License
 
-See [LICENSE](LICENSE).
+See [LICENSE](LICENSE).

essays/PI_RCC/README.md

@@ -105,7 +105,7 @@ This gives a rational approximant to the next iterate:
 y_m = x + s_m(x)
 ```
 
-The sequence y_m converges to x + sin(x) as m → ∞. Each y_m is a rational number paired with a computable error bound — a _certificate_ that the true iterate lies within an explicit interval around y_m. This is, in essence, interval arithmetic with rational endpoints: the framework of validated numerics (Moore, Rump, Tucker) applied to a specific transcendental update.
+The sequence y*m converges to x + sin(x) as m → ∞. Each y_m is a rational number paired with a computable error bound — a \_certificate* that the true iterate lies within an explicit interval around y_m. This is, in essence, interval arithmetic with rational endpoints: the framework of validated numerics (Moore, Rump, Tucker) applied to a specific transcendental update.
 
 ### Inner Convergence Rate
 
@@ -179,7 +179,7 @@ The x + sin(x) engine fits none of these categories cleanly.
 
 It is not a hypergeometric series — the outer recurrence is nonlinear and cannot be expressed as a ratio of consecutive hypergeometric terms in the iteration index.
 
-It is not a Newton method — Newton's method for sin(x) = 0 gives x\_{n+1} = x_n − tan(x_n), a different map with different structure that requires evaluating _both_ sin and cos and performing a division. The x + sin(x) iteration achieves cubic convergence without division and without derivative evaluation.
+It is not a Newton method — Newton's method for sin(x) = 0 gives x\_{n+1} = x*n − tan(x_n), a different map with different structure that requires evaluating \_both* sin and cos and performing a division. The x + sin(x) iteration achieves cubic convergence without division and without derivative evaluation.
 
 It is not an AGM — there is no arithmetic-geometric mean in sight.
 

essays/PI_RCC/perspectives.details.md

@@ -28,7 +28,7 @@ e_{n+1} = (1/6)e_n³ + O(e_n⁵)
 
 is standard. However, several numerical analysis subtleties deserve sharper treatment:
 
-**Basin of Attraction**: The cubic convergence holds only within a neighborhood of π where the Taylor expansion is valid _and_ where the cubic term dominates. Specifically, we need |e_n| small enough that the O(e_n⁵) term is negligible relative to e_n³/6. This requires roughly |e_n| < 1, but the _practical_ basin where cubic behavior is observed requires |e_n| ≲ 0.1 to avoid the quintic correction term contributing more than ~1% of the cubic term. The document glosses over the seed quality requirement.
+**Basin of Attraction**: The cubic convergence holds only within a neighborhood of π where the Taylor expansion is valid _and_ where the cubic term dominates. Specifically, we need |e*n| small enough that the O(e_n⁵) term is negligible relative to e_n³/6. This requires roughly |e_n| < 1, but the \_practical* basin where cubic behavior is observed requires |e_n| ≲ 0.1 to avoid the quintic correction term contributing more than ~1% of the cubic term. The document glosses over the seed quality requirement.
 
 **Pre-asymptotic Behavior**: For seeds far from π, the iteration x + sin(x) is not contractive everywhere. The map has derivative 1 + cos(x), which equals 2 at x = 0 and is expansive for x ∈ (−π/2, π/2). A naive seed of x₀ = 3 gives e₀ ≈ 0.14, which is already in the cubic regime. But x₀ = 1 would diverge initially before potentially being captured. The document's claim of "start with a seed sufficiently close to π" needs quantification.
 
@@ -966,7 +966,7 @@ Option 3 is the correct engineering choice and also reveals a deeper point: **ra
 
 The document glosses over a critical implementation challenge: **the iteration requires a seed sufficiently close to π**. This is a bootstrapping problem.
 
-For cubic convergence to dominate, we need |e_0| < 1 (roughly). But to get a _certified_ rational seed, we need an independent method. Options:
+For cubic convergence to dominate, we need |e*0| < 1 (roughly). But to get a \_certified* rational seed, we need an independent method. Options:
 
 | Method              | Complexity        | Notes                   |
 | ------------------- | ----------------- | ----------------------- |

essays/PI_RCC/socratic.md

@@ -210,7 +210,7 @@ The convergents p_k/q_k of a continued fraction are generated by the recurrence:
 - p*k = a_k · p*{k-1} + p\_{k-2}
 - q*k = a_k · q*{k-1} + q\_{k-2}
 
-The best-approximation property — that p_k/q_k satisfies |α - p_k/q_k| < 1/q_k² and that no rational with smaller denominator does better — is _proved_ by induction on the iterative structure. The certificate is not a static object that the iteration merely happens to visit. The iteration _generates_ the certificate inductively. You cannot separate the optimality result from the recursive construction without losing the proof.
+The best-approximation property — that p*k/q_k satisfies |α - p_k/q_k| < 1/q_k² and that no rational with smaller denominator does better — is \_proved* by induction on the iterative structure. The certificate is not a static object that the iteration merely happens to visit. The iteration _generates_ the certificate inductively. You cannot separate the optimality result from the recursive construction without losing the proof.
 
 So my claim that certification "points toward static optimality criteria fully separable from any iteration" was wrong in the continued fraction case. I was treating the _statement_ of the theorem as separable from its _proof structure_, which is a mistake.
 

essays/QQN/papers/intro/paper_rendered.md

@@ -104,7 +104,7 @@ This paper makes three primary contributions:
                                                       2
 
 3. Benchmarking Framework: A reusable Rust application for optimization algorithm evaluation that
-promotes reproducible research and meaningful comparisons.
+   promotes reproducible research and meaningful comparisons.
 
 Optimal configurations remain problem-dependent, but QQN’s adaptive nature minimizes the need for
 extensive hyperparameter tuning. Scaling and convergence properties are theoretically justified, largely
@@ -164,8 +164,7 @@ that traces a path from the current point. We impose three natural boundary cond
    descent)
 3. Terminal Position: d(1) = dLBFGS (the curve ends at the L-BFGS direction)
 
-
-    The second condition is crucial: by ensuring the path starts tangent to the negative gradient, we guarantee
+   The second condition is crucial: by ensuring the path starts tangent to the negative gradient, we guarantee
 
 that moving along the path initially decreases the objective function, regardless of where the path eventually
 leads. This provides robustness against poor quasi-Newton directions.
@@ -628,23 +627,23 @@ consistently outperformed other optimizer families across the benchmark suite.
 
                                                       12
 
-3. Line Search Critical: Among QQN variants, line search strategy dramatically affects performance:
+3.  Line Search Critical: Among QQN variants, line search strategy dramatically affects performance:
 
         • Strong Wolfe: Excellent success rate with moderate average evaluations
         • Golden Section: 90-100% success rate on 2D problems with relatively few average evaluations
         • Bisection: Strong performance on various problems with minimal evaluations
         • Bisection: Strong performance on various problems with minimal evaluations
         • Cubic-Quadratic Interpolation: 70% success on Rosenbrock 5D, good for ill-conditioned objectives
 
-4. Problem-Specific Excellence: Algorithms show significant specialization:
+4.  Problem-Specific Excellence: Algorithms show significant specialization:
 
-   • QQN-GoldenSection: Achieved strong performance on multimodal problems
-   • QQN-GoldenSection: Achieved strong performance on multimodal problems
-   • QQN-CubicQuadraticInterpolation: 70% success on Rosenbrock 5D with strong performance on
-   ill-conditioned problems
-   • Adam-WeightDecay: Excellent performance on neural networks vs moderate performance for
-   standard Adam
-   • L-BFGS variants: Generally poor performance on ill-conditioned problems like Rosenbrock
+    • QQN-GoldenSection: Achieved strong performance on multimodal problems
+    • QQN-GoldenSection: Achieved strong performance on multimodal problems
+    • QQN-CubicQuadraticInterpolation: 70% success on Rosenbrock 5D with strong performance on
+    ill-conditioned problems
+    • Adam-WeightDecay: Excellent performance on neural networks vs moderate performance for
+    standard Adam
+    • L-BFGS variants: Generally poor performance on ill-conditioned problems like Rosenbrock
 
 7.2 The Benchmarking and Reporting Framework
 7.2.1 Methodological Contributions
@@ -796,15 +795,15 @@ demonstrates:
 
                                                      15
 
-3. Efficiency vs Robustness: QQN shows superior efficiency with strong success rates across problem
-types while requiring fewer function evaluations than traditional methods. 4. Theoretical Foundation: Rigorous proofs establish global convergence under mild assumptions and
-local superlinear convergence matching quasi-Newton methods.
+3.  Efficiency vs Robustness: QQN shows superior efficiency with strong success rates across problem
+    types while requiring fewer function evaluations than traditional methods. 4. Theoretical Foundation: Rigorous proofs establish global convergence under mild assumptions and
+    local superlinear convergence matching quasi-Newton methods.
 
-    5. Practical Impact: The results provide clear guidance for practitioners: use QQN-StrongWolfe for
-       general optimization, QQN-Bisection variants for high-dimensional problems, QQN-GoldenSection for
-       multimodal landscapes, and QQN-CubicQuadraticInterpolation for sparse or ill-conditioned problems.
+        5. Practical Impact: The results provide clear guidance for practitioners: use QQN-StrongWolfe for
+           general optimization, QQN-Bisection variants for high-dimensional problems, QQN-GoldenSection for
+           multimodal landscapes, and QQN-CubicQuadraticInterpolation for sparse or ill-conditioned problems.
 
-    The simplicity of QQN’s core insight—that quadratic interpolation provides the natural geometry for
+        The simplicity of QQN’s core insight—that quadratic interpolation provides the natural geometry for
 
 combining optimization directions—contrasts with the complexity of recent developments. Combined with
 our evaluation methodology, this work establishes new standards for both algorithm development and em-
@@ -1036,13 +1035,13 @@ We seek a principled method to combine these directions that:
 3. Requires no additional hyperparameters
 4. Maintains computational efficiency
 
-14.1.2 B.1.2 Geometric Formulation
-We formulate direction combination as a boundary value problem in parametric space. Consider a parametric
-curve d : [0, 1] → Rn satisfying:
+   14.1.2 B.1.2 Geometric Formulation
+   We formulate direction combination as a boundary value problem in parametric space. Consider a parametric
+   curve d : [0, 1] → Rn satisfying:
 
-1. Initial position: d(0) = 0
-2. Initial tangent: d′ (0) = −∇f (x) (ensures descent)
-3. Terminal position: d(1) = dL-BFGS
+5. Initial position: d(0) = 0
+6. Initial tangent: d′ (0) = −∇f (x) (ensures descent)
+7. Terminal position: d(1) = dL-BFGS
 
 The minimal polynomial satisfying these constraints is quadratic:
 
@@ -1243,8 +1242,8 @@ Proof :
                                                       23
 
 1. Path construction: Computing d(t) = t(1−t)(−∇f )+t2 dL-BFGS requires O(n) operations for vector
-arithmetic. 2. L-BFGS direction: The two-loop recursion requires O(mn) operations to compute Hk ∇f (xk ). 3. Line search: Each function evaluation along the path requires O(n) operations to compute xk + d(t),
-plus the cost of evaluating f .
+   arithmetic. 2. L-BFGS direction: The two-loop recursion requires O(mn) operations to compute Hk ∇f (xk ). 3. Line search: Each function evaluation along the path requires O(n) operations to compute xk + d(t),
+   plus the cost of evaluating f .
 
 Total complexity per iteration: O(mn + kn) + k · cost(f ). □
 
@@ -1337,5 +1336,3 @@ dquasi-Newton if condition B
 QQN provides continuous interpolation, eliminating discontinuities and the need for switching logic.
 
                                                      25
-
-

essays/QQN/papers/theory/paper_rendered.md

@@ -447,34 +447,34 @@ univariate optimization will find t∗ → 1. □
 8.2.1 B.1 L-BFGS Direction Computation
 The L-BFGS direction is computed using the two-loop recursion:
 
-function compute_lbfgs_direction(gradient, memory):
+function compute*lbfgs_direction(gradient, memory):
 q = gradient
 alphas = []
 // First loop (backward)
 for i = memory.size-1 down to 0:
-rho_i = 1 / (memory.y[i]^T _ memory.s[i])
-alpha_i = rho_i _ memory.s[i]^T _ q
+rho_i = 1 / (memory.y[i]^T * memory.s[i])
+alpha*i = rho_i * memory.s[i]^T _ q
 q = q - alpha_i _ memory.y[i]
-alphas.append(alpha_i)
+alphas.append(alpha*i)
 // Apply initial Hessian approximation
 if memory.size > 0:
-gamma = (memory.s[-1]^T _ memory.y[-1]) / (memory.y[-1]^T _ memory.y[-1])
+gamma = (memory.s[-1]^T * memory.y[-1]) / (memory.y[-1]^T _ memory.y[-1])
 r = gamma _ q
 else:
 r = q
 // Second loop (forward)
 for i = 0 to memory.size-1:
-rho_i = 1 / (memory.y[i]^T _ memory.s[i])
-beta = rho_i _ memory.y[i]^T _ r
+rho*i = 1 / (memory.y[i]^T * memory.s[i])
+beta = rho*i * memory.y[i]^T \_ r
 r = r + (alphas[memory.size-1-i] - beta) \* memory.s[i]
 return -r
 
 8.2.2 B.2 Univariate Optimization Methods
 Golden Section Search
-function golden_section_search(f, a, b, tol):
+function golden*section_search(f, a, b, tol):
 phi = (1 + sqrt(5)) / 2
 resphi = 2 - phi
-x1 = a + resphi _ (b - a)
+x1 = a + resphi * (b - a)
 x2 = b - resphi _ (b - a)
 f1 = f(x1)
 f2 = f(x2)
@@ -489,7 +489,7 @@ else:
 b = x2
 x2 = x1
 f2 = f1
-x1 = a + resphi _ (b - a)
+x1 = a + resphi \_ (b - a)
 f1 = f(x1)
 return (a + b) / 2
 
@@ -670,5 +670,3 @@ QQN matches L-BFGS complexity while providing gradient descent robustness and of
 function evaluations due to better step selection.
 
                                                  13
-
-

experiments/irrational_lattice/main.js

@@ -425,11 +425,11 @@ window.addEventListener('mousemove', (ev) => {
   if (!dragging) return;
   const cur = clientToFieldPixel(ev);
   // Drag moves the view: dragging right pulls content right => pan left.
-// Round the pixel delta so panning snaps to whole field-pixel steps.
-// This keeps the lattice sampling aligned to the pixel grid and avoids
-// sub-pixel resampling jitter while dragging.
-const dxPx = Math.round(cur.fx - dragStart.fx);
-const dyPx = Math.round(cur.fy - dragStart.fy);
+  // Round the pixel delta so panning snaps to whole field-pixel steps.
+  // This keeps the lattice sampling aligned to the pixel grid and avoids
+  // sub-pixel resampling jitter while dragging.
+  const dxPx = Math.round(cur.fx - dragStart.fx);
+  const dyPx = Math.round(cur.fy - dragStart.fy);
   const z = effectiveZoom(canvas.width);
   view.panX = panStart.x - dxPx * z;
   view.panY = panStart.y - dyPx * z;
@@ -694,4 +694,4 @@ regenerate();
 window.addEventListener('hashchange', () => {
   if (restoringFromHash) return;
   if (hashToState()) regenerate();
-});
+});

index.html

@@ -631,4 +631,4 @@ <h2 class="modal-title" id="modalTitle"></h2>
 
     <script src="js/home.js"></script>
   </body>
-</html>
+</html>

site_icon.op.md

@@ -5,9 +5,9 @@ related:
   - README.md
 ---
 
-Generate a site icon (favicon) for the project. 
+Generate a site icon (favicon) for the project.
 The icon should be a simple, recognizable image that represents the project's theme or brand.
 Use an aspect ratio of 1:1, and a size of 512x512 pixels
-Use a solid color background with a contrasting symbol or letter that can be easily identified even at small sizes (16x16 pixels). 
+Use a solid color background with a contrasting symbol or letter that can be easily identified even at small sizes (16x16 pixels).
 Do not blur, feather, or use a shadow effect, to make it easy to change the background to transparent.
 The design should be clean and minimalistic, avoiding intricate details that may not be visible at smaller resolutions.