Add Chronos–EntropyDepth theory definitions, lower bound, non-amplification lemma, example SAT distribution, and ED simulation script

Author: inaciovasquez2020

docs/theory/chronos_entropy_bridge.md

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+# Chronos–Entropy Bridge
+
+Let
+
+ΔI_t = I(X ; Y_t | Y_{<t})
+
+Then
+
+Σ ΔI_t ≤ T * C
+
+where C is the transcript capacity constant.
+
+Therefore
+
+T ≥ H(X) / C.
+
+This inequality is the Chronos refinement bound.

docs/theory/chronos_lower_bound.md

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+# Chronos Lower Bound
+
+Let a refinement process satisfy
+
+1. Locality constraint radius r
+2. Bounded transcript capacity
+3. FO^k definability of local operations
+
+Then
+
+I_t ≤ O(1)
+
+for every step.
+
+Therefore
+
+ED ≥ Ω(H(X))
+
+In SAT distributions with
+
+H(X) = Θ(n)
+
+we obtain
+
+ED ≥ Ω(n).

docs/theory/configuration_non_amplification.md

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+# Configuration Non-Amplification Lemma
+
+In FO^k refinement systems operating on bounded-degree graphs,
+
+the conditional mutual information per refinement step
+is bounded by a constant.
+
+Proof sketch:
+
+1. FO^k operations inspect radius-r neighborhoods.
+2. The number of local types is finite.
+3. Transcript outputs lie in a finite alphabet.
+4. Therefore
+
+I(X ; Y_t | history) ≤ log |Alphabet| = O(1).

docs/theory/entropy_depth_definition.md

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+# Entropy Depth (ED)
+
+Let X be the hidden solution variable with entropy H(X).
+
+Let Y_1, …, Y_T be the transcript produced by an algorithm.
+
+Define the per-step information gain
+
+I_t = I(X ; Y_t | Y_{<t})
+
+The **Entropy Depth** of the process is
+
+ED = min T such that
+Σ_{t=1}^T I_t ≥ H(X).
+
+Interpretation:
+
+ED measures the minimal number of sequential refinement
+steps required to extract all solution entropy.

examples/sat_entropy_instance.md

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+# Hard SAT Distribution
+
+Construct random XOR-SAT instances:
+
+m = Θ(n)
+variables = n
+
+Solution entropy
+
+H(X) = Θ(n)
+
+Since
+
+I_t ≤ O(1)
+
+any FO^k refinement algorithm must satisfy
+
+T ≥ Ω(n).

scripts/entropy_depth_demo.py

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+import numpy as np
+
+def simulate_entropy_depth(n, info_per_step=1):
+    H = n
+    steps = 0
+    gained = 0
+
+    while gained < H:
+        gained += info_per_step
+        steps += 1
+
+    return steps
+
+for n in [100, 500, 1000]:
+    print(n, simulate_entropy_depth(n))