Finite horizon and retained-context cost [ftip-00DH]
✍️source

    Counting exposure to retained tokens makes the finite-horizon cost comparison precise. A context cap alone does not guarantee information preservation.

    Definition 1. Finite rollout token budget [ftip-00DI]

      A rollout has a finite token budget \(T\in \mathbb N_{\geq 1}\) when its history contains at most \(T\) generated positions after the prompt. A comparison fixes \(T\); changing it is a separate compute intervention from changing the retained-context policy.

      Definition 2. Context exposure count [ftip-00DJ]

        For a retained-context policy \(\kappa \) and generated history \(h_t\), define its exposure count through budget \(T\) by

        \[ E_T(\kappa )=\sum _{t=0}^{T-1}\left |\kappa _t(h_t)\right |. \]

        This is a finite attention-input proxy. It is not a runtime identity: the implementation may have caching, batching, kernel, and communication costs that are not represented by \(E_T\).

        Lemma 3. Prefix-window exposure is uniformly capped [ftip-00DK]

          Let \(p,w\in \mathbb N_0\) and let \(\kappa ^{p,w}\) be the prefix-window policy of Definition [ftip-00DD]. For every history and every \(t\),

          \[ \left |\kappa ^{p,w}_t(h_t)\right |\leq p+w. \]

          Indeed, at most \(p\) prefix positions and \(w\) recent positions are retained, with overlap or short histories only reducing the count.

          Theorem 4. Capped retained context bounds finite exposure work [ftip-00DL]

            Under the hypotheses of Lemma 3, every rollout with budget \(T\) satisfies

            \[ E_T(\kappa ^{p,w})\leq T(p+w). \]

            Proof. Apply the pointwise bound in Lemma 3 to each of the \(T\) nonnegative summands in Definition 2, then sum. This bounds token exposure, not wall-clock runtime or evaluation quality.

            Example 5. Warm-up and sliding-window arithmetic [ftip-00DM]

              For \(p=3\), \(w=2\), and \(T=6\), the first steps grow the retained set until the cap \(p+w=5\) is reached. The exposure bound is therefore \(E_T\leq 6\cdot 5=30\); the exact count depends on the prompt and stopping convention.

              The arithmetic is an FTIP finite consequence, not a runtime measurement from the Prefix Sliding experiments.

              Remark 6. A memory cap does not preserve information [ftip-00DN]

                The bound in Theorem 4 controls the number of retained token positions. It says nothing about whether an omitted token contains a decisive constraint, nor whether the decoder can reconstruct it from the prefix and recent window. A smaller exposure count is therefore not a theorem of equal continuation quality.

                Context

                  Dynamic compute, retained context, and rollout approximation [ftip-00DA]

                    In a joint inference and training intervention, the retained-context rule, rollout horizon, runtime budget, and gradient estimator may all change together. Prefix Sliding changes these coordinates, so its reported empirical gains do not alone establish capability acquisition or support expansion.

                    Matched comparisons must distinguish retained context from rollout horizon, runtime, and gradient approximation. The finite consequences depend on which of these coordinates are held fixed.

                    Inference intervention coordinates [ftip-00DB]

                      A context intervention is a change to what a fixed model can attend to while it generates a rollout. The intervention is meaningful only after the prompt law, decoding rule, token budget, and retained-state rule are named.

                      Definition. Retained-context policy [ftip-00DC]

                        Fix a token history \(h_t=(x_0,\ldots ,x_{t-1})\). A retained-context policy is a deterministic map \(\kappa _t(h_t)\) that selects an ordered subsequence of the history, together with a decoder that conditions its next token on that subsequence. The policy, rather than the model weights, names which prior tokens remain available at step \(t\).

                        A comparison of two policies is conditional on the same model, prompt law, decoding law, stopping rule, and budget unless the comparison includes a second declared intervention coordinate.

                        Definition. Prefix-window retained state [ftip-00DD]

                          For prefix and window lengths \(p,w\in \mathbb N_0\), define the prefix-window policy by retaining every token in positions \(0,\ldots ,p-1\) and the most recent \(w\) tokens after that prefix. If the history is shorter than either region, the available positions are retained without padding.

                          The policy is a context rule only. It does not assert that omitted tokens are irrelevant, nor that the resulting continuation distribution equals the full-history distribution.

                          Remark. Prefix Sliding is a source-reported intervention [ftip-00DE]

                            Muennighoff et al. describe Prefix Sliding in Sections 2--5, Figures 6--9, and the Limitations section of [muennighoff2026prefixsliding, Sections 2--5, Figures 6--9, and Limitations]. Their method preserves a prefix while sliding a recent attention window during test-time scaling. The reported experiments change retained context and runtime, and may also change rollout horizon and the training gradient approximation. These are source observations, not a theorem of universal speedup or capability.

                            The source studies named model and task configurations. It does not by itself establish support expansion, latent capability acquisition, or a deployment-shift guarantee.

                            Example. One trace under full and prefix-window attention [ftip-00DF]

                              This schematic keeps the model and token history fixed while changing only the retained-context policy. It is a local illustration, not a source figure.

                              The two outputs can differ even though the represented weights and visible prompt are identical. Any claim about improvement must therefore name the retained-context intervention and its evaluation law.

                              Remark. Retained context is not weight learning [ftip-00DG]

                                Changing \(\kappa _t\) changes the information supplied to a fixed decoder; it does not update represented weights. A successful continuation can therefore be an elicitation effect, a context effect, or both. Weights, optimizer and feedback, inference state, and evaluation remain distinct intervention coordinates, as in § [ftip-0001].

                                Finite horizon and retained-context cost [ftip-00DH]
                                ✍️source

                                  Counting exposure to retained tokens makes the finite-horizon cost comparison precise. A context cap alone does not guarantee information preservation.

                                  Definition. Finite rollout token budget [ftip-00DI]

                                    A rollout has a finite token budget \(T\in \mathbb N_{\geq 1}\) when its history contains at most \(T\) generated positions after the prompt. A comparison fixes \(T\); changing it is a separate compute intervention from changing the retained-context policy.

                                    Definition. Context exposure count [ftip-00DJ]

                                      For a retained-context policy \(\kappa \) and generated history \(h_t\), define its exposure count through budget \(T\) by

                                      \[ E_T(\kappa )=\sum _{t=0}^{T-1}\left |\kappa _t(h_t)\right |. \]

                                      This is a finite attention-input proxy. It is not a runtime identity: the implementation may have caching, batching, kernel, and communication costs that are not represented by \(E_T\).

                                      Lemma. Prefix-window exposure is uniformly capped [ftip-00DK]

                                        Let \(p,w\in \mathbb N_0\) and let \(\kappa ^{p,w}\) be the prefix-window policy of Definition [ftip-00DD]. For every history and every \(t\),

                                        \[ \left |\kappa ^{p,w}_t(h_t)\right |\leq p+w. \]

                                        Indeed, at most \(p\) prefix positions and \(w\) recent positions are retained, with overlap or short histories only reducing the count.

                                        Theorem. Capped retained context bounds finite exposure work [ftip-00DL]

                                          Under the hypotheses of Lemma [ftip-00DK], every rollout with budget \(T\) satisfies

                                          \[ E_T(\kappa ^{p,w})\leq T(p+w). \]

                                          Proof. Apply the pointwise bound in Lemma [ftip-00DK] to each of the \(T\) nonnegative summands in Definition [ftip-00DJ], then sum. This bounds token exposure, not wall-clock runtime or evaluation quality.

                                          Example. Warm-up and sliding-window arithmetic [ftip-00DM]

                                            For \(p=3\), \(w=2\), and \(T=6\), the first steps grow the retained set until the cap \(p+w=5\) is reached. The exposure bound is therefore \(E_T\leq 6\cdot 5=30\); the exact count depends on the prompt and stopping convention.

                                            The arithmetic is an FTIP finite consequence, not a runtime measurement from the Prefix Sliding experiments.

                                            Remark. A memory cap does not preserve information [ftip-00DN]

                                              The bound in Theorem [ftip-00DL] controls the number of retained token positions. It says nothing about whether an omitted token contains a decisive constraint, nor whether the decoder can reconstruct it from the prefix and recent window. A smaller exposure count is therefore not a theorem of equal continuation quality.

                                              Training and evaluation confounds [ftip-00DO]

                                                A retained-context comparison can alter both inference and training. Attribution to either mechanism depends on the joint intervention and the coordinates held fixed by a matched evaluation.

                                                Definition. Joint intervention cell [ftip-00DP]

                                                  A joint intervention cell is a tuple \(I=(M,\mu ,\pi ,T,\kappa ,\widehat g)\) consisting of fixed model weights \(M\), prompt law \(\mu \), decoding law \(\pi \), token budget \(T\), retained-context policy \(\kappa \), and training or evaluation estimator \(\widehat g\). Two cells differ in a declared coordinate only when all other coordinates are held fixed.

                                                  Definition. Estimator-changing training coordinate [ftip-00DQ]

                                                    An estimator-changing coordinate is a change in \(\widehat g\), the map used to turn sampled histories and rewards into an update. A context mask can change this coordinate when it changes which tokens contribute to the sampled loss or gradient. The notation does not assume a particular optimizer.

                                                    Remark. Training masks change more than attention cost [ftip-00DR]

                                                      If a retained-context rule is used during training, it can change the attention inputs, sampled continuation, loss support, and gradient estimator at once. A lower value of Definition [ftip-00DJ] is therefore not an isolated compute intervention unless the training estimator and all other coordinates are matched explicitly.

                                                      Example. Equal terminal score, different trajectory exposure [ftip-00DS]

                                                        Two cells can reach the same terminal score while exposing different numbers of context positions. This finite counterexample blocks an inference from equal endpoint score to equal trajectory cost.

                                                        The score equality is compatible with unequal intermediate histories and unequal exposure counts.

                                                        Remark. What a retained-context comparison can identify [ftip-00DT]

                                                          With matched weights, prompt law, decoding, budget, estimator, and evaluation interface, a comparison can identify a conditional contrast between retained-context policies. Without those controls it identifies only the joint cell, not a causal effect of context alone.

                                                          Definition. Retained-context evaluation protocol [ftip-00DU]

                                                            A retained-context evaluation protocol fixes a model \(M\), prompt law \(\mu \), decoder \(\pi \), budget \(T\), evaluator \(\mathsf E\), and two policies \(\kappa _1,\kappa _2\). It reports the paired outcomes under the same sampled prompts and declared randomization coupling, together with exposure counts from Definition [ftip-00DJ].

                                                            Lemma. Matched protocols isolate a conditional policy contrast [ftip-00DV]

                                                              Let \(m\in \mathbb N_{\geq 1}\) and let \(Y_{1j},Y_{2j}\in \mathbb R\) for \(j=1,\ldots ,m\) be the paired scalar evaluator outcomes generated by the protocol of Definition [ftip-00DU]. Then the finite paired contrast

                                                              \[ \widehat \Delta =\frac 1m\sum _{j=1}^{m}(Y_{1j}-Y_{2j}) \]

                                                              is a statistic of the declared policy contrast under that common protocol. It is not an unconditional capability effect and does not identify what would happen after changing any held-fixed coordinate.

                                                              Remark. Separating horizon, cache, and estimator effects [ftip-00DW]

                                                                Prefix Sliding is useful as a named intervention for retained context and test-time scaling, but its reported configurations do not establish a universal runtime law, gradient theorem, support expansion, or capability acquisition. The next questions are to measure horizon, cache, estimator, and evaluation effects separately, while preserving the controls in Definition [ftip-00DU].

                                                                The finite conclusions depend on the declared protocols. They do not describe a full optimizer or establish a result for all language models.