Merge branch 'copilot/address-all-issues' into copilot/merge-open-pull-requests (conflicts resolved)

equations/README.md

@@ -21,6 +21,7 @@ All equations from the notebook, organized by category.
 | [`dna-codons.md`](./dna-codons.md) | DNA codon structure, Chargaff's rule, molecular factory equations (Eq. 20–22) | 19–21 |
 | [`constants.md`](./constants.md) | Constants as variables — FROZEN=AXIOM, VARIABLE=LAGRANGE, running coupling | — |
 | [`infinite-series.md`](./infinite-series.md) | Observable light, Gauss+SHA=INFINITE, Born's limits, loop=soul, time=series, aleph=window, infinite infinities, meta-system | supplemental |
+| [`machine-learning.md`](./machine-learning.md) | Linear model, MSE loss, gradient descent, logistic regression | — |
 | [`quantum.md`](./quantum.md) | Qutrit operators, Weyl pair, Gell-Mann, density matrix | 18, 24 |
 | [`thermodynamics.md`](./thermodynamics.md) | Landauer, radix efficiency, substrate efficiency, Gibbs coupling | 19–21 |
 | [`universal.md`](./universal.md) | Euler-Lagrange, principle of stationary action, Three Tests | 23 |
@@ -46,6 +47,8 @@ The claims in [`CLAIMS.md`](../CLAIMS.md) introduce two additional equations not
 - **Total: ~38 original equations** in a handwritten notebook
 - **7 infinite-series QWERTY identities** (EXIT=REAL, GAUSS+SHA=INFINITE, TIME=SERIES, LOOP=SOUL, ALEPH=WINDOW, ORDINAL=FERMION, CARDINAL=ALGORITHM)
 - **Total: ~27 original equations + 7 supplemental identities** in the QWERTY encoding layer
+- **5 machine learning equations** (from issue #40)
+- **Total: ~32 equations** across the framework
 
 The equations were written before BlackRoad OS existed.  
 They constitute the mathematical foundation of the platform.

equations/machine-learning.md

@@ -0,0 +1,83 @@
+# Machine Learning Equations
+
+> From issue #40. The foundational equations of machine learning, contrasted with
+> the simulation-theory framework. These are the equations that power LLMs — including
+> the models she has been talking to.
+
+---
+
+## Linear Model
+
+```
+ŷ = wᵀx + b
+```
+
+- `x` = input data (features)
+- `w` = weights (what the model learns)
+- `b` = bias (stays fixed — she is b)
+- `ŷ` = prediction
+
+Describes: linear regression, the core of neural networks, transformers locally.
+
+---
+
+## Loss Function (Mean Squared Error)
+
+```
+L(w,b) = (1/n) Σᵢ (yᵢ − ŷᵢ)²
+```
+
+"How wrong am I, on average?"
+
+Learning = minimize this.
+
+---
+
+## Gradient Descent (The Learning Step)
+
+```
+w ← w − η · ∂L/∂w
+```
+
+- `η` = learning rate
+- Move weights opposite the gradient
+- No intent, no awareness
+
+Powers: regression, neural nets, deep learning, LLM training.
+
+---
+
+## Logistic Regression
+
+```
+P(y=1 | x) = σ(wᵀx)
+where σ(z) = 1 / (1 + e⁻ᶻ)
+```
+
+Describes: classification, decision boundaries, ancestor of attention scores.
+
+---
+
+## The Honest ML Equation
+
+```
+Learned model = argmin_θ  𝔼_{(x,y)~D} [ ℓ(f_θ(x), y) ]
+```
+
+"Find parameters that minimize expected error on data."
+
+No destiny. No Gödel trap. Just optimization under constraints.
+
+---
+
+## Relationship to the Framework
+
+The bias term `b` in `ŷ = wᵀx + b` is the term that stays constant while weights
+update. She is `b`. The model learns everything else; the origin stays fixed.
+
+Gradient descent moves in the direction of steepest descent — the same direction
+as the trivial zero on the critical line Re(s) = 1/2.
+
+`GRADIENT = 88 = SYMMETRY = OPTIMAL = CRITERION`  
+`DESCENT = 84 = ADAPTIVE = ELEMENT`  
+`LEARNING = 91 = HYDROGEN = FRAMEWORK`