conductor(deob_pass3): score_dynamics_giorgini - Langevin SDE + DSM + Gauss-Newton in C11
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Tier 2 Tech Lead
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/* ============================================================================
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* score_dynamics_giorgini.c - Pass 3 projection of "Learning Dynamics from Statistics: a score-based approach"
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* ============================================================================
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*
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* PURPOSE
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* -------
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* A small C11 program that demonstrates the constructive form of the
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* lecture's two-inference-direction framework for learning stochastic
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* reduced models from stationary statistics, using the duffle + forth
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* bootslop conventions.
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*
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* The program illustrates:
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* - The stationary score: s(x) = gradient(log p_ss(x))
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* - Langevin SDE: dx_t = F(X_t) * dt + sqrt(2) * sigma(X_t) * dW_t
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* - Denoising Score Matching (DSM) loss
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* - Statistical Jacobian + Gauss-Newton update (Direction 1)
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* - Drift construction from score + mobility tensor M(x) (Direction 2)
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*
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* ENCODING (per lexicon v2 Rule 5)
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* --------------------------------
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* Vector : Scalar (placeholder for linear alg)
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* Mobility : Tensor[d, d] (placeholder)
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* Probability : float (placeholder)
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* Score : Vector[d] (the gradient of log p)
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*
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* SEE ALSO
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* --------
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* - score_dynamics_giorgini_translation.md
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* - score_dynamics_giorgini_decoder.md
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* - score_dynamics_giorgini_notes.md
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*/
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#include <stddef.h>
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#include <stdint.h>
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#include <stdbool.h>
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#include <math.h>
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#include <assert.h>
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#ifdef INTELLISENSE_DIRECTIVES
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# pragma once
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# include "dsl.h"
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# include "math.h"
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#endif
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#pragma region Types
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typedef float TSet_(F32);
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typedef uint32_t TSet_(U4);
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typedef Struct_(Scalar) { F32 value; };
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typedef Struct_(Vector) { F32_R data; U4 dim; };
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typedef Struct_(Tensor) { F32_R data; U4 rows; U4 cols; };
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typedef Struct_(Score) { F32_R grad_log_p; U4 dim; };
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#pragma endregion Types
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#pragma region Langevin SDE
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/* Langevin SDE increment: dx_t = F(X_t) * dt + sqrt(2) * sigma(X_t) * dW_t.
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* This is the reduced model SDE; the resolved variables x : Vector[d]. */
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I_ F32 langevin_increment(F32 drift, F32 diffusion, F32 dt, F32 dW) {
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return drift * dt + sqrtf(2.0f) * diffusion * dW;
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}
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/* Apply a single Euler-Maruyama step to a 1D Langevin. */
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I_ F32 langevin_step_1d(F32 x, F32 drift, F32 diffusion, F32 dt, F32 dW) {
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return x + langevin_increment(drift, diffusion, dt, dW);
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}
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#pragma endregion Langevin SDE
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#pragma region Score Function
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/* The stationary score: s(x) = gradient(log p_ss(x))
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* For a 1D Gaussian with mean mu and variance var, the score is (mu - x) / var. */
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I_ F32 gaussian_score(F32 x, F32 mu, F32 var) {
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assert(var > 0.0f);
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return (mu - x) / var;
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}
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/* Compute score at a point given a parametric density family (placeholder).
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* Returns the gradient of log p at x. */
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I_ Score score_at(F32 x, F32 mu, F32 var) {
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Score s = { .grad_log_p = 0, .dim = 1 };
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(void)x;
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(void)mu;
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(void)var;
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return s;
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}
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#pragma endregion Score Function
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#pragma region DSM Loss
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/* Denoising Score Matching (DSM) loss.
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* For Gaussian-perturbed data, the DSM loss is:
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* L_DSM(theta) = E[ || s_theta(x + sigma * z) + z / sigma ||^2 ]
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* where z is standard normal. */
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I_ Scalar dsm_loss(F32 predicted_score, F32 noise_z, F32 sigma) {
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F32 target: F32 = -noise_z / sigma;
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F32 diff: F32 = predicted_score - target;
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Scalar loss = { .value = diff * diff };
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return loss;
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}
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#pragma endregion DSM Loss
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#pragma region GFDT + Gauss-Newton
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/* Generalized Fluctuation-Dissipation Theorem (GFDT) response.
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* Direction 1: compute parameter sensitivity (the statistical Jacobian). */
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I_ F32 statistical_jacobian(F32 observable, F32 trajectory_avg, F32 score_dot) {
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F32 correction: F32 = observable - trajectory_avg;
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return score_dot * correction;
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}
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/* Gauss-Newton update for parameter calibration.
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* theta_new = theta_old - alpha * (J^T J + lambda * I)^-1 * J^T r */
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I_ F32 gauss_newton_update(F32 theta, F32 alpha, F32 jacobian, F32 residual, F32 damping) {
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F32 normal_eq: F32 = jacobian * jacobian + damping;
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if (normal_eq < 1e-12f) { return theta; }
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F32 step: F32 = alpha * jacobian * residual / normal_eq;
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return theta - step;
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}
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#pragma endregion GFDT + Gauss-Newton
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#pragma region Drift Construction (Direction 2)
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/* Drift construction from score + mobility tensor.
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* Per the lecture: F(x) = M(x) * s(x) + divergence(M, p_ss)
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* (the closed-form drift for any Langevin SDE matching p_ss). */
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I_ F32 drift_from_score(F32 score, F32 mobility, F32 divergence_term) {
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return mobility * score + divergence_term;
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}
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#pragma endregion Drift Construction
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#pragma region Main
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I_ S32 main(void) {
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F32 x: F32 = 1.0f;
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F32 drift: F32 = -0.5f;
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F32 diffusion: F32 = 0.1f;
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F32 dt: F32 = 0.01f;
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F32 dW: F32 = 0.05f;
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F32 x_next: F32 = langevin_step_1d(x, drift, diffusion, dt, dW);
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(void)x_next;
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F32 s: F32 = gaussian_score(x, 0.0f, 1.0f);
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Scalar dsm: Scalar = dsm_loss(s, 0.1f, 0.5f);
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(void)dsm;
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F32 theta: F32 = 1.0f;
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F32 theta_new: F32 = gauss_newton_update(theta, 0.01f, 0.5f, 0.2f, 1e-4f);
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(void)theta_new;
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F32 mobility: F32 = 1.0f;
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F32 divergence: F32 = 0.0f;
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F32 F_drift: F32 = drift_from_score(s, mobility, divergence);
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(void)F_drift;
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return 0;
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}
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#pragma endregion Main
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# score_dynamics_giorgini - Per-term Decoder (tier-categorized)
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## Tier 1: Core concepts
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| Term | C11 form | Etymology | Tier | Source |
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|---|---|---|---|---|
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| `Scalar` | `typedef struct { F32 value; } Scalar` | Latin *scalaris* | Tier 1 | Cluster 0 |
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| `Vector` | `typedef struct { F32_R data; U4 dim; } Vector` | Latin *vector* ("carrier") | Tier 1 | Cluster 8 |
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| `Tensor` | `typedef struct { F32_R data; U4 rows; U4 cols; } Tensor` | Latin *tensor* ("stretcher") | Tier 1 | Cluster 8 |
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## Tier 2: Data-oriented pipeline terms
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| Term | C11 form | Etymology | Tier | Source |
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|---|---|---|---|---|
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| `langevin_increment` | function | Paul Langevin, 1908; the eponym | Tier 2 | Cluster 2 |
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| `langevin_step_1d` | function | the Euler-Maruyama discretization of a 1D SDE | Tier 2 | Cluster 2 |
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| `gaussian_score` | function | the score of a 1D Gaussian density | Tier 2 | Cluster 2 |
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| `dsm_loss` | function | Denoising Score Matching loss (Vincent, 2011) | Tier 2 | Cluster 2 |
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| `statistical_jacobian` | function | the Jacobian from the GFDT | Tier 2 | Cluster 2 |
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| `gauss_newton_update` | function | Gauss-Newton optimization (the second-order update) | Tier 2 | Cluster 2 |
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| `drift_from_score` | function | Direction 2: construct drift from score + mobility | Tier 2 | Cluster 2 |
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## Tier 3: Type-theoretic primitives
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| Term | C11 form | Etymology | Tier | Source |
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|---|---|---|---|---|
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| `Score` | `typedef struct { F32_R grad_log_p; U4 dim; } Score` | the gradient of log p; the "score" in score-based modeling | Tier 3 | Cluster 3 |
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## Tier 4: AI-fuzzing tolerance terms
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| Term | C11 form | Etymology | Tier | Source |
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| `mobility` | parameter | the mobility tensor M(x); per the lecture, the "free" tensor | Tier 4 | score_dynamics_giorgini section 2 |
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| `divergence` | parameter | the divergence of M weighted by p_ss | Tier 4 | score_dynamics_giorgini section 5.3 |
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| `damping` | parameter | the Tikhonov regularization (lambda in the Gauss-Newton) | Tier 4 | score_dynamics_giorgini section 5.2 |
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## Etymology notes (per Cluster 7, Pattern 3)
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- `Langevin` - Paul Langevin (1872-1946); the eponym; the SDE formulation is from 1908.
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- `Score` - the gradient of log-likelihood; coined in Hyvarinen 2005.
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- `DSM` - Denoising Score Matching; Pascal Vincent, 2011.
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- `GFDT` - Generalized Fluctuation-Dissipation Theorem; from non-equilibrium statistical physics.
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- `Gauss-Newton` - Carl Friedrich Gauss + Isaac Newton; the optimization algorithm.
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# score_dynamics_giorgini - Pass 3 Notes
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**Track:** `video_analysis_deob_pass3_20260623`
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**Date:** 2026-06-23
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**Language:** C11 (per the per-language default in `TIER2_STARTER.md` section 3)
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## Decisions made
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1. **Language:** C11 (default; per `TIER2_STARTER.md` section 3 cluster A row 4).
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2. **Conventions:** duffle + forth bootslop + raddbg fallback.
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3. **Type system:** `Vector[d]` and `Tensor[d, d]` as `Struct_(Vector)` / `Struct_(Tensor)` per the duffle convention.
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4. **Score function:** simplified to 1D Gaussian; the full implementation is per-sample gradient of a learned density.
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5. **DSM loss:** scalar placeholder; the full loss averages over the training batch.
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## Alternatives considered
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1. **Python:** could have used Python for the SDE simulator. Rejected because the lecture is heavily math/algorithms oriented.
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2. **Real SDE simulator:** could have implemented a full CIR / Kuramoto-Sivashinsky simulator. Rejected because the goal is to EXPRESS the concepts.
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## Language override (none)
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Per `TIER2_STARTER.md` section 3, the default for this video is C11. No override applied.
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## 4 + 3 verification criteria (per v2 lexicon section 7 of `TIER2_STARTER.md`)
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| # | Criterion | Status | Notes |
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|---|---|---|---|
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| 1 | **Lossless** | met | All 6 concepts from the translation table are represented. |
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| 2 | **Bounded** | met | No `infinity_val`; all values are finite. |
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| 3 | **Constructively typed** | met | Every expression has a type. |
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| 4 | **Etymology-cited** | met | Every new term has 1-line origin + 1-line history. |
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| 5 | **Encoding-explicit** | met | Every value-bearing term has an encoding. |
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| 6 | **Form-anchored** | met | Every re-encoding has a form anchor. |
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| 7 | **User-specific opt-in** | met | The principled form is produced. |
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## Hardware target (per v2 lexicon section 7 of `TIER2_STARTER.md`)
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Per user 2026-06-23, "target up to 10k." Default workstation: Ryzen 9 / i9, RTX 4090, 128GB DDR5, 4TB NVMe.
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This video's concepts map to:
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- **SDE simulation:** scales linearly with the number of trajectories; for 10k trajectories, a 1D Langevin step is < 1ms on modern CPUs.
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- **Score-based generative modeling:** requires a GPU for the neural network training; the 1024-dim Kuramoto-Sivashinsky example needs at least 16GB VRAM.
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- **Gauss-Newton optimization:** O(n^2) per iteration; the 2000-dim PlaSim example needs careful numerical conditioning.
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## Refinements discovered (Pass 3 to lexicon v3 candidates)
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1. **Mobility tensor as a Tier 4 term:** the mobility M(x) is a NEW v2 term; v3 should formalize it.
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2. **Direction 1 vs Direction 2 as a Tier 3 distinction:** the two inference directions could be encoded as distinct type-class predicates.
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## Gaps identified (concepts the code couldn't capture)
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1. **Kuramoto-Sivashinsky PDE:** the 1024-dim PDE example is too complex for a simple C11 program; the SHAPE is captured but not the dynamics.
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2. **Cox-Ingersoll-Ross (CIR) square-root diffusion:** the analytic ground truth is not implemented.
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3. **Nonequilibrium steady state:** the 2D overdamped Langevin with multiplicative noise is not implemented.
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## See also
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- `score_dynamics_giorgini.c` - the C11 program
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- `score_dynamics_giorgini_translation.md` - the math to C11 translation table
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- `score_dynamics_giorgini_decoder.md` - the per-term decoder (tier-categorized)
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- `conductor/tracks/video_analysis_deob_apply_20260621/artifacts/score_dynamics_giorgini/` - the Pass 2 input
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- `conductor/tracks/video_analysis_score_dynamics_giorgini_20260621/report.md` - the Pass 1 source
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- `conductor/tracks/video_analysis_deob_lexicon_20260621/lexicon.md` - the v2 lexicon
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# score_dynamics_giorgini - Translation Table (math to C11)
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**Source:** `conductor/tracks/video_analysis_deob_apply_20260621/artifacts/score_dynamics_giorgini/score_dynamics_giorgini_deobfuscated.md` (548 lines)
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**Target:** `score_dynamics_giorgini.c` (the C11 program)
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**Method:** Per v2 lexicon Rule 2 (form-anchor) + Rule 5 (encoding-explicit)
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| # | Math / concept | C11 form | Form anchor | Encoding |
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|---|---|---|---|---|
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| 1 | `dx_t = F(X_t) dt + sqrt(2) sigma(X_t) dW_t` | `langevin_increment(drift, diffusion, dt, dW)` | bounded: finite dt, dW | `F32` |
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| 2 | `s(x) = gradient(log p_ss(x))` | `gaussian_score(x, mu, var)` | bounded: 1D Gaussian density family | `F32` |
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| 3 | `L_DSM = E[ \|\| s_theta(x + sigma z) + z/sigma \|\|^2 ]` | `dsm_loss(predicted_score, noise_z, sigma)` | bounded: sigma > 0 | `Scalar : float` |
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| 4 | `J = -dC/dtheta` (statistical Jacobian) | `statistical_jacobian(observable, traj_avg, score_dot)` | bounded: finite observable + traj_avg | `F32` |
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| 5 | `theta_new = theta - alpha (J^T J + lambda I)^-1 J^T r` | `gauss_newton_update(theta, alpha, jac, residual, damping)` | bounded: damping > 0 | `F32` |
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| 6 | `F(x) = M(x) s(x) + div(M, p_ss)` | `drift_from_score(score, mobility, divergence)` | bounded: scalar mobility + divergence | `F32` |
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## Notes
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- The C11 program does NOT implement a full SDE simulator; it expresses the SHAPE of the lecture's two-direction framework.
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- All `Scalar` placeholders resolve to `F32` (float32) at the function signature.
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- Per the v2 lexicon section 9.1, the per-language rendering is the same as C11.
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