Render technical report equations as vector mathematics

TECHNICAL_REPORT.md

@@ -26,9 +26,9 @@ The value of a decision model lies in interpreting a changing problem, rather th
 
 This representation preserves a central strength of language models: task meaning is communicated through the input. A common model can process evidence-based questions, natural-language principles, workflow states, and browser actions without requiring a separate output classifier for every business category. Compact output labels identify candidates; their meaning comes from the descriptions supplied for the current request. They are an interface to the decision space, not permanent class identities.
 
-For candidates `A(x) = {a₁, …, aₖ}`, an execution depth `d` produces logits `z_d(x, A)`. The decision distribution is:
+For candidates <i>A</i>(<i>x</i>) = {<i>a</i><sub>1</sub>, …, <i>a</i><sub>K</sub>}, an execution depth <i>d</i> produces logits <i>z</i><sub>d</sub>(<i>x</i>, <i>A</i>). The decision distribution is:
 
-`p_d(aᵢ | x, A) = exp(z_d,i) / Σⱼ exp(z_d,j)`.
+![Decision distribution: softmax normalized over the K supplied candidates.](assets/equations/decision-distribution.svg)
 
 Normalization takes place within the legal candidate set. A deterministic application layer can map the selected label into its structured response. This removes the need to generate JSON punctuation or repeatedly decode a known answer schema. It guarantees membership in the supplied candidate set, while semantic correctness remains a learned capability that must be evaluated. Open text is retained where the task genuinely requires it, such as entering a value during a browser workflow.
 
@@ -51,11 +51,9 @@ The important training question is whether an intermediate representation has ac
 
 ### 3.1 Joint supervision and cross-depth knowledge transfer
 
-For a supervised decision with target distribution `q`, the delivered objective is:
+For a supervised decision with target distribution <i>q</i>, the delivered objective is:
 
-`L_decision = 0.5 CE(q, p_low) + 0.5 CE(q, p_high)`
-
-`             + 0.1 KL(stopgrad(p_high) || p_low)`.
+![Joint-depth objective: equally weighted low and high cross-entropy plus 0.1 times KL from the detached high distribution to the low distribution.](assets/equations/joint-depth-objective.svg)
 
 Both exits receive direct task supervision. The deeper exit additionally acts as a distributional teacher for the earlier exit. Its detached probabilities convey relative candidate preferences beyond the identity of the correct answer, while stop-gradient keeps the distillation target from being changed through that term. The deeper pathway continues to learn through its own supervised objective.
 
@@ -107,17 +105,15 @@ A useful next step is to distinguish three properties that accuracy alone cannot
 
 Its candidate-space objective combines proper-scoring components, reference preservation, and decision utility:
 
-`L_DCRL = mean_d [NLL(q, p_d) + λ_B Brier(q, p_d)]`
-
-`         + λ_ref KL(p_high || p_reference)`
+![Jet-DCRL research objective: mean NLL and Brier across depths, reference KL, and low-path utility.](assets/equations/dcrl-objective.svg)
 
-`         + λ_U L_utility(p_low)`.
+Here <i>D</i> denotes the set of supervised decision depths.
 
 The NLL and Brier terms supervise probabilities at both exits. The reference term anchors the deeper distribution to a frozen reference. The utility term targets the earlier exit, where better decisions have particular value for low-cost execution. Candidate utilities are supplied as verified training information; they are not inferred merely from the model's confidence.
 
 For bounded candidate sets, the exact utility objective is tractable:
 
-`L_utility = −Σ_a p_low(a) u(a)`.
+![Expected-utility loss: negative candidate-probability-weighted utility.](assets/equations/expected-utility.svg)
 
 A sampled alternative draws actions from the current shallow policy and uses an expected-utility baseline in its policy-gradient estimator. Comparing exact and sampled optimization is valuable here: when the action space is small, sampling should earn its complexity through evidence rather than being assumed necessary.
 
@@ -135,7 +131,7 @@ Our performance target is the operating curve between useful decisions and deplo
 
 Depth controls how many backbone transformations construct the decision representation. Output width controls how many language-output rows are projected from that representation. APUS-OpenJev's native low-effort runtime combines these axes: an earlier trained exit supplies the representation, and a candidate-aware readout restricts the output projection to the labels required by the current request. Candidate descriptions remain part of the language input; the output uses validated, unique single-token labels rather than a fixed business-category classifier.
 
-Let `h_d` be the normalized readout at depth `d`, `W` the language output matrix with `V` vocabulary rows and hidden width `H`, and `C` the request's `K` candidate token IDs. The two paths are:
+Let <i>h</i><sub>d</sub> be the normalized readout at depth <i>d</i>, <i>W</i> the language output matrix with <i>V</i> vocabulary rows and hidden width <i>H</i>, and <i>C</i> the request's <i>K</i> candidate token IDs. The two paths are:
 
 ```text
 Native low:
@@ -148,9 +144,17 @@ Native high / current vLLM decision path:
         → candidate normalization
 ```
 
-The candidate projection computes `z_C = W[C] h_d + b[C]`, instead of constructing `z = W h_d + b` and then selecting `z[C]`. For the same representation, weights, bias and candidate set, these are algebraically identical. Floating-point kernels may differ numerically, so deployment equivalence still requires validation. This identity does not imply that low and high exits have the same predictions: they use different representations and executed depths.
+The candidate projection selects the output weight and bias rows before the matrix multiplication:
+
+![Candidate projection identity: selected weight rows times the depth representation plus selected bias equals selected entries of the full projection.](assets/equations/candidate-projection.svg)
+
+This avoids constructing the full vocabulary score vector before selecting its candidate entries. For the same representation, weights, bias and candidate set, these are algebraically identical. Floating-point kernels may differ numerically, so deployment equivalence still requires validation. This identity does not imply that low and high exits have the same predictions: they use different representations and executed depths.
+
+For one decision position, the output-head arithmetic scales as:
+
+![Output-head cost: full vocabulary O of H times V; selected candidate rows O of H times K.](assets/equations/projection-cost.svg)
 
-For one decision position, the dense output projection has arithmetic cost proportional to `H × V`; the selected-row projection scales with `H × K`. This is an output-head cost model, not an end-to-end speedup claim. Backbone computation, memory traffic, row gathering and serving overhead remain. Candidate softmax expresses preference within the supplied set; it is not automatically calibrated confidence or the full-vocabulary probability mass.
+This is an output-head cost model, not an end-to-end speedup claim. Backbone computation, memory traffic, row gathering and serving overhead remain. Candidate softmax expresses preference within the supplied set; it is not automatically calibrated confidence or the full-vocabulary probability mass.
 
 ### 6.2 Implemented paths and serving scope
 

TECHNICAL_REPORT.pdf

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equations-source.json

@@ -0,0 +1,26 @@
+{
+  "renderer": "matplotlib mathtext, STIX fonts, vector paths",
+  "formulas": {
+    "decision-distribution": [
+      "p_d(a_i\\mid x,A)=\\frac{\\exp(z_{d,i})}{\\sum_{j=1}^{K}\\exp(z_{d,j})}"
+    ],
+    "joint-depth-objective": [
+      "\\mathcal{L}_{\\mathrm{decision}}=0.5\\,\\mathrm{CE}(q,p_{\\mathrm{low}})+0.5\\,\\mathrm{CE}(q,p_{\\mathrm{high}})",
+      "\\qquad +0.1\\,\\mathrm{KL}\\!\\left(\\mathrm{stopgrad}(p_{\\mathrm{high}})\\,\\Vert\\,p_{\\mathrm{low}}\\right)"
+    ],
+    "dcrl-objective": [
+      "\\mathcal{L}_{\\mathrm{DCRL}}=\\frac{1}{|\\mathcal{D}|}\\sum_{d\\in\\mathcal{D}}\\left[\\mathrm{NLL}(q,p_d)+\\lambda_B\\,\\mathrm{Brier}(q,p_d)\\right]",
+      "\\qquad +\\lambda_{\\mathrm{ref}}\\,\\mathrm{KL}(p_{\\mathrm{high}}\\,\\Vert\\,p_{\\mathrm{reference}})",
+      "\\qquad +\\lambda_U\\,\\mathcal{L}_{\\mathrm{utility}}(p_{\\mathrm{low}})"
+    ],
+    "expected-utility": [
+      "\\mathcal{L}_{\\mathrm{utility}}=-\\sum_{a\\in A(x)}p_{\\mathrm{low}}(a)\\,u(a)"
+    ],
+    "candidate-projection": [
+      "z_C=W[C]h_d+b[C]=(Wh_d+b)[C]"
+    ],
+    "projection-cost": [
+      "\\mathrm{Full\\ vocabulary}:\\ O(HV)\\qquad\\mathrm{Candidate\\ rows}:\\ O(HK)"
+    ]
+  }
+}