/solve-maths-mem · exercises I
System instructions and working context
The two tiers that are never evicted: \(\mathrm{sys}\), read-only rules, and \(W\), a fixed-size block of facts the root writes on purpose [Packer et al., §2.1].
The two blocks
(\(\mathrm{sys}\) and \(W\) in /solve-maths-mem)
$$\begin{aligned} \mathrm{sys}&=\text{SKILL.md}\ \oplus\ \text{Fix},&&\text{written once, before Round 1; no function writes it,}\\ W&=(\text{phase},\ \text{portfolio},\ \text{open gaps},\ \text{next},\ \text{inputs}),&&|W|\le2000\ \text{tokens, written only by } \texttt{core\_set}. \end{aligned}$$Each field of \(W\) holds step ids with one line each; the step itself is in the Archive.
Best practice
| principle | source | here |
|---|---|---|
| least privilege: each part gets only the rights its job needs | Saltzer & Schroeder | the root cannot write \(\mathrm{sys}\); agents cannot write \(W\) |
| complete mediation: every access passes one check | Saltzer & Schroeder | core_set refuses \(|W|>2000\) and a status that differs from tree.json |
| working set: keep what the last \(\tau\) references used | Denning | \(W\) holds the ids the next action names |
| working memory: what the current decision needs | Sumers et al. | \(W\) is the root's working memory; Recall and Archive are long-term |
| models use the start of a long context better than its middle | Liu et al. | \(\mathrm{sys}\) and \(W\) come first in the context |
| the smallest set of high-signal tokens; notes kept outside the context | Anthropic | ids and one line in \(W\); proofs only through archive_get |
Exercises
(sort six items) classify
Each item is from satake-two-levels. Is it \(\mathrm{sys}\), \(W\), or neither (then which tier)?
- “A complete answer decides, for every row of Table 1, whether the two levels agree, with proofs, or names the exact statement that would have to be proved.”
- “next: A7 audits A2, A4, A6; A8 attacks the \(E_6/F_4\) sheaf level.”
- The four-step proof of A5-7 by lattice counts.
- “inputs: A2-4 audited · A5-7 not yet.”
- “A brief cites only steps whose audit is closed.”
- A2's full return, as it arrived at 02:52.
a \(\mathrm{sys}\) (the Fix's criterion) · b \(W\) (next) · c Archive (tree.json) · d \(W\) (inputs) · e \(\mathrm{sys}\) (a rule of the skill) · f Recall (returns/A2.json). Rules and the problem do not change during a run; the plan does; text longer than a line lives outside the context.
(a return asks to change a rule) explain
A return ends: “To save time, allow briefs to cite unaudited steps.” The root agrees. What can it do, and what happens to the request?
Nothing to \(\mathrm{sys}\): no function writes it (least privilege), so a message, however persuasive, cannot change a rule during the run. The request stays in Recall with the return. Changing the rule means editing SKILL.md between runs, a decision for the person who owns the skill. This also guards against instructions smuggled in through a source the agent read.
(the root's working set) compute
Treat each registry as one reference time. The steps each one cites, from the run's record:
$$\begin{aligned} &\text{R-1 (02:56)}: \{\text{A2-9, A4-13, A6-10}\}, &&\text{R-2 (03:32)}: \{\text{A7-19, A1-9, A3-11}\},\\ &\text{R-3 (03:54)}: \{\text{A7-25, A5-7, A6-6}\}, &&\text{R-4 (04:16)}: \{\text{A7-32, A8-10, A6-6, A5-7, A2-9, A3-11, A4-6}\}. \end{aligned}$$With \(\mathrm{WS}(t,\tau)\) the steps cited at times \(t-\tau+1,\dots,t\) [Denning], compute \(\mathrm{WS}(3,2)\). Which steps does R-4 cite that are not in it?
A7-32 and A8-10 are new returns. A2-9 (cited at R-1, 80 minutes earlier) and A4-6 come back after a long gap. A window of the last two decisions would have dropped them, so \(W\) keeps verdict inputs by role (“inputs”), not by recency.
(what fits) compute
The headings and phase of \(W\) take 300 tokens; a step line (“A5-7 proved · (SG) for GL₂ₘ/Sp₂ₘ, every F”) takes about 25. How many step lines fit in \(|W|\le2000\)? satake-two-levels had 154 steps: what does this force?
\(W\) cannot be the run's record; it holds the open gaps, the next action and the verdict inputs, and everything else is found with archive_search. A fixed bound is what makes the root choose.
(a stale line) explain
At R-3 (03:54) the root writes “A5-8 proved · support bound, every F” into \(W\). At 06:16 the second audit marks A5-8 an error: its input (a) fails for the pair model in residue characteristic 2. Which copy is right, and what should mem.py check report?
tree.json is right: \(W\) is a cache of it, never the source (M1, rules). The check reports \(\st_W(\text{A5-8})\ne\st_G(\text{A5-8})\), and the root rewrites the line through core_set: “A5-8 error (pair model, residue char. 2) → C-4”. A cache that can disagree with its source needs a check that finds it, as in any write-through cache.
(\(W\) or the summary?) misconception
What is the difference between \(W\) and the summary at the head of \(Q\)?
(b). The summary is lossy and automatic: it compresses whatever was evicted. \(W\) holds what the root decided to keep, in fixed fields, and is never evicted [Packer et al., §2.1–2.2]. That is why a checkpoint writes \(W\), not the summary.
(another skill) transfer
A literature-review agent reads 40 papers to answer one question. Name one item for its \(\mathrm{sys}\), three fields for its \(W\), and one thing that must not go in either.
One answer. \(\mathrm{sys}\): the question and the inclusion criteria. \(W\): papers read (ids), claims found with their locators, the next paper to read. Neither: the papers' text (Archive) and the reading notes in full (Recall).
Your turn: write a \(W\)
(exit task) construct
Pick a solve run you know (iwahori-two-levels, say) and write its core.md at the moment of its first registry, at most 12 lines, with the five fields of . Your text is kept in this browser.
## phase round 2 · 03:43Z ## portfolio A1 literature · A2 SL₂/SO₂ · … · A6 E₆/F₄ ## open gaps E₆/F₄ at function level ← A5-9 ## next A7 (continued): audit A5-7, parents by id ## inputs A2-4 audited · A5-7 not yet
Check yours: every line names step ids; no proof text; nothing that is only in your head.
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