# Reading notes and page index

Self-Improving Intelligence — Visual Guide v1.0.0

These notes accompany the diagrams. Full hypotheses and proofs remain in the source manuscript.

## 01. Self-Improving Intelligence: A Visual Guide

A standalone visual companion.

**Interpretation.** Four different objects improve across the finite chain.

**Boundary.** No claim of general intelligence or indefinite recursive acceleration.

Source: [S] First-edition scope, pp. ii-iii

## 02. The route through this guide

Read the worked world first. Then follow the discovery procedure outward, keeping the supplied machinery in view.

**Interpretation.** Every plate identifies its source section. Manuscript page numbers are the printed numbers, not the PDF viewer index.

**Boundary.** A solid arrow is a specified transition or dependency. Dashed paths are explicitly labeled as recurrence or a qualified extension. Counts retain their original units.

Source: [S] Scope guide; §§1-3 | manuscript pp. iii, 1-4

## 03. A better system is not always a better improver

Changing a library and changing the procedure that uses it answer different questions. A crossed comparison keeps them separate.

**Interpretation.** Holding the library fixed isolates a change in the improvement procedure. The combined descendant may improve for a different reason.

**Boundary.** Use comparable starting solvers, task laws, installed information, and budgets. A stronger base model or an extra library is another changed component.

Source: [S] §3; Counterexample 3.2 | manuscript pp. 3-4

## 04. One defect. Eight possible cells.

The procedure cannot see the defect index. It receives only the reports of the questions it is allowed to ask.

**Interpretation.** Retain an executable way to find a justified repair in a later episode. Do not install the next episode’s hidden answer.

**Boundary.** The binary sensors and repair meanings come from a declared finite model. The 88 query-table entries are supplied, not learned through executed calibration.

Source: [S] §4; §32 | manuscript pp. 4, 49-51

## 05. Two investigations, the same installed sensors

Equality-first testing and coordinate queries use the same declared language. Their paths have different depths, but their complete trees have the same size.

**Interpretation.** Equality-first: seven questions, then repair. Coordinate queries: three questions, then repair. Total costs are eight and four calls.

**Boundary.** Both full binary trees have eight repair leaves and seven query nodes: fifteen records. These are literal tree records, not bytes or minimal program sizes.

Source: [S] §4; Theorem 32.1 | manuscript pp. 4, 49-50

## 06. Follow one episode: the defect is at cell 5

This page follows the canonical coordinate policy. The procedure knows the reports, not the hidden index shown to the reader.

**Interpretation.** The three reports select a repair for this episode. The reusable artifact is the branching procedure that works for every allowed starting cell.

**Boundary.** The shipped compiler can use an equality query at the last two-case branch. It has the same cost. This plate deliberately shows the canonical all-coordinate tree.

Source: [S] §4; §32; Appendix A | manuscript pp. 4, 49-54

## 07. The path is not the stored program

A single successful run touches four records. Its complete reusable controller still contains all fifteen.

**Interpretation.** Execution follows one branch. Literal storage sums every branch. The highlighted path for reports 1, 0, 1 reaches repair 5.

**Boundary.** Immutable cases, binary observations, a different successful repair for each case, and one record per node. Shared code or state-reset operations change the model.

Source: [S] §6; Theorem 11.1; §28 | manuscript pp. 6-7, 14-15, 44

## 08. A residual is a question, not a diagnosis

The same mismatch can be consistent with several causes. A useful next experiment separates repair-relevant explanations.

**Interpretation.** Retain competing explanations. Seek an admitted observation whose possible outcomes change which correction is justified.

**Boundary.** A representation cannot recover a distinction unavailable in its complete transcript. With disjoint repairs and identical transcript laws, equal-prior success is at most one half.

Source: [S] §5; Proposition 5.1 | manuscript pp. 5-6

## 09. The best question need not identify the most

A repair may work for several hidden cases. The right stopping condition is a common successful action, not complete knowledge of the state.

**Interpretation.** The 2/2 question has more information about four equally likely cases, but one branch still mixes incompatible repairs.

**Boundary.** Both questions cost one unit and preserve the state. The 3/1 question supports a correct repair after one observation in every case.

Source: [S] Counterexample 6.3; §5.1 | manuscript pp. 6-8

## 10. One tree must satisfy both resource limits

The synthesis combines complete child policies. Worst-case execution takes a maximum. Stored controller size takes a sum.

**Interpretation.** This illustrative root has cost 1. Its children cost (2 calls, 3 records) and (4 calls, 7 records). The combined tree costs (5 calls, 11 records).

**Boundary.** This is an illustration of the recurrence, not a new optimality result. A real child must be valid on the actual post-experiment support, and every possible branch needs a witness.

Source: [S] Theorem 6.2 | manuscript pp. 6-7

## 11. An unknown sensor needs a reference

A correct procedure for each known model need not give a common procedure when the model itself is unknown.

**Interpretation.** The report o = x ⊕ θ cannot distinguish (x,θ) = (0,0) from (1,1). They require different repairs. Repeating that same reading adds no information.

**Boundary.** A reference reports θ at cost 2. An ordinary query costs 1, then repair costs 1. Total work is 4, and the complete binary controller stores 7 records.

Source: [S] Theorem 7.1; Counterexample 7.2 | manuscript pp. 8-10

## 12. Transport the investigation, not only the repair

A translated answer can be correct while the observation needed to obtain it is unavailable in the target system.

**Interpretation.** Experiment permissions, report labels, state effects, successful-repair meanings, and costs. Correspondences are supplied in this finite theorem.

**Boundary.** Copying the two possible repairs into a world with no bit-revealing observation does not transport the policy that chooses between them.

Source: [S] Theorem 9.1; Counterexample 9.2 | manuscript pp. 12-13

## 13. Now improve the procedure that builds the tree

A checked candidate can end an exhaustive search only when it is valid and attains the relevant global lower bounds.

**Interpretation.** For n immutable cases with binary unit-cost queries and singleton repairs, a valid tree attaining both 1 + ceil(log₂ n) calls and 2n − 1 records closes the frontier.

**Boundary.** An invalid scope, missing candidate, or unattained bound sends control to the exact parent. That preliminary work still counts.

Source: [S] Theorem 11.1; §11.2 | manuscript pp. 14-15

## 14. What the shortcut actually reads

On the sixteen-case model, both compilers return an optimal controller with 5 execution calls and 31 stored records.

**Interpretation.** The recorded compilation count is 576,128 reads for the parent and 1,728 for the shortcut. Both include completed witness checking.

**Boundary.** Each also pays 320 common input-validation reads. Selection among all candidates costs 21,124 development reads. Table accesses are not total CPU instructions.

Source: [S] Proposition 12.1; §13.3 | manuscript pp. 16, 19

## 15. A fallback preserves meaning, not free speed

A preliminary shortcut can fail to prove an optimum and then spend extra work before invoking the same exact parent.

**Interpretation.** With four cells and only singleton queries, the shortcut falls back. Both return the same optimum. The wrapper uses 472 reads instead of 402.

**Boundary.** A successful witness, a proved finite-horizon impossibility, and an interrupted search are different results. A timeout cannot certify infeasibility.

Source: [S] Counterexample 13.1; §16.2 | manuscript pp. 18, 24-25

## 16. Sometimes the search state is too coarse

No longer formula can recover a feature the learner was never given. Here, the missing distinction is remaining branch capacity.

**Interpretation.** Both contexts have (a,b,m) = (0,3,4). With capacity 2 the root must fail. With capacity 4, a completion with branch sizes (1,3) is possible.

**Boundary.** Capacity is exposed from a supplied feature registry. This repairs an internal search representation, not a physical sensor model or the definition of depth.

Source: [S] Proposition 14.1; Counterexample 14.2 | manuscript pp. 20-21

## 17. Four proposals, three counterexamples

The learner composes an arithmetic predicate. Retained counterexamples prevent it from proposing the same rejected behavior again.

**Interpretation.** The final five-node expression is max(a,b) > C. The reference learner visits 190 of 3,960 candidates and makes four teacher calls.

**Boundary.** The teacher checks 198 developmental contexts. That certifies this finite agreement, not validity at every larger capacity. Installation needs a separate scope argument.

Source: [S] Theorem 15.1; §15.2; rule trace | manuscript pp. 21-23

## 18. Stop reading when no suffix can rescue the root

Eight cases must fit into two branches of capacity four. After five equal reports, the outcome is already decided.

**Interpretation.** Counts cannot decrease. Once one count exceeds C, no unread report can bring that branch back within capacity. The suffix cannot change rejection.

**Boundary.** Passing the capacity test does not prove that the remaining query library can separate either child. The compiler must still search and verify complete subtrees.

Source: [S] Proposition 16.1; Theorem 16.3 | manuscript pp. 23-25

## 19. A training-perfect rule can prune the answer

A dangerous shortcut is not made safe by a final answer checker: it may delete the only successful candidate before that checker sees it.

**Interpretation.** At (a,b,m,C) = (0,5,16,8), the bounded-cap formula rejects. But the unread cases can complete both branches to eight, so capacity remains feasible.

**Boundary.** Examples help select a formula. A proof with applicable premises authorizes deleting possibilities. The supplied structural gate accepts only the supported rule form.

Source: [S] Counterexample 16.2 | manuscript pp. 23-24

## 20. Fewer model reads can still mean more work

The same ordered compiler returns the same literal controller. Streaming changes when impossible roots are abandoned.

**Interpretation.** At 64 cases, streaming avoids 8,287 model rows. The returned controller still costs 7 online calls and 127 stored records.

**Boundary.** Streaming also adds 14,209 comparisons in the conservative account. At equal prices for reads and comparisons, that isolated modification costs more.

Source: [S] §18; Theorem 16.3 | manuscript pp. 24-28

## 21. Count repair distinctions, not hidden states

If one repair fixes a pair of cases, a diagnostic policy need not identify which member of the pair occurred.

**Interpretation.** Put nonnegative weights on cases so every repair-success set has total at most 1. A successful leaf then carries at most 1 unit. Binary depth h supports at most 2ʰ leaves.

**Boundary.** Adding a universal repair makes the old total weight of 4 violate its success-set constraint. Recheck the weights before using the old pruning rule.

Source: [S] Propositions 17.1, 22.1; Theorem 22.2 | manuscript pp. 25, 33-35

## 22. A useful equation begins as a proposal

The next acquired object is a task-independent library of equations. Evaluations suggest candidates; derivations decide which can be used.

**Interpretation.** The finite proposer constructs 1,068 equations from equal probe vectors. The limited proof procedure admits 336 with replayable arithmetic derivations.

**Boundary.** The remaining 732 may be false, valid only on a narrower scope, or simply beyond this proof search. The primitive arithmetic axioms and checking machinery are supplied.

Source: [S] §§19-20 | manuscript pp. 29-30

## 23. Keep the earliest checked representative

Equivalent expressions can share a search representative without changing the original program submitted to the teacher.

**Interpretation.** The exact proposed program trees, retained counterexamples, final result, and exhaustion outcome. Removed later equivalents would already fail a retained counterexample.

**Boundary.** Pure total arithmetic, sound equations, fixed order, an extensional deterministic teacher, and no forgotten counterexamples. It does not preserve every possible cost-sensitive objective.

Source: [S] Theorems 21.1-21.2 | manuscript pp. 30-32

## 24. Less repetition. The same corrective evidence.

The quotient changes which redundant candidates are evaluated. It does not change the task teacher or smuggle in the target rule.

**Interpretation.** Both learners begin the new task with no task-specific counterexamples. They submit the same four programs and receive the same three counterexamples.

**Boundary.** Equation discovery, proof admission, quotient preparation, dispatch, and storage must be charged separately. The ten targets are finite tasks in one arithmetic language.

Source: [S] §21.1; Theorem 21.2 | manuscript pp. 30-32

## 25. One chain, not multiplied headline gains

The representation that speeds up rule acquisition feeds a separately certified compiler change. Each link keeps its own comparison.

**Interpretation.** If the quotient preserves rule acquisition and the acquired rule passes the unchanged compiler gate, the complete chain returns the same rule and controller.

**Boundary.** Let K be extra preparation. The composite saves work in a common additive unit only if (A − A′) + (D − D′) > K. Do not multiply ratios from different baselines or units.

Source: [S] Proposition 23.1; §25 | manuscript pp. 35-40

## 26. A certificate has a dependency address

When a premise changes, mark the affected claims and their descendants. Reuse unaffected witnesses only under a complete dependency declaration.

**Interpretation.** Arithmetic equations and a repair-weight witness belong to different branches. A change in one branch need not invalidate everything.

**Boundary.** This is not automatic discovery of every dependency in arbitrary code. An omitted premise can leave a registry green while its claim becomes false. Hashes do not repair missing semantics.

Source: [S] Theorem 24.1; §24.2 | manuscript pp. 37-38

## 27. A stronger repair can invalidate an old proof

The conclusion “this controller is valid” and the conclusion “this change saves work” do not automatically survive together.

**Interpretation.** For the 32-case paired model, adding a universal repair invalidates the old weight constraints. Rebuilt weights permit a one-step, one-record controller.

**Boundary.** The earlier assessment referred to another controller and compilation. The replacement needs a new comparison. The unaffected arithmetic equation library remains applicable.

Source: [S] Counterexample 22.3; §24.3 | manuscript pp. 35, 38-39

## 28. The same controller, opposite cost conclusions

Before claiming improvement, choose a common ledger and state the prices used to combine its event counts.

**Interpretation.** These are old-minus-new search events on the same 32-case weighted problem. Scope and witness checks are equal and cancel only in this isolated difference.

**Boundary.** Weight preparation, equation acquisition, installed programs, storage, and revision management do not become free. Event counts alone do not determine processor time or energy.

Source: [S] Proposition 25.1; Counterexample 25.2 | manuscript pp. 39-40

## 29. Do not turn selection evidence into a fresh audit

What the improver has already inspected belongs to its developmental history, even when the inner solver never saw it.

**Interpretation.** A proof supports a conditional statement. A checker validates a finite witness. A fresh performance audit supports a comparison on a declared task law.

**Boundary.** Repeatedly reading a “holdout” exposes it to the outer loop. A correct proof does not establish its physical premises. Finite algorithm counters are not independent neural trials.

Source: [S] §26; Proposition 26.1 | manuscript pp. 41-42

## 30. An audit can stop early without betting in hindsight

Choose the evidence update before seeing its paired score. Stopping after observing evidence is a different operation.

**Interpretation.** Freeze the comparison, choose the next bet from past information, observe the next paired score, then update the evidence process.

**Boundary.** With four fair ± scores, betting only after seeing a positive doubles the evidence on positives. The nominal threshold 4 is crossed on 11 of 16 paths, not at most 1/4.

Source: [S] Theorem 27.1; Counterexample 27.2 | manuscript pp. 42-43

## 31. Safe repetition is not guaranteed progress

An authorized attempt must be selected before its outcome is known. One policy must meet the continuation requirement on every allowed branch.

**Interpretation.** Given fully observed lineage states, a finite attempt catalogue, and separately justified progress marks, the recurrence decides whether one policy guarantees k gains within h attempts.

**Boundary.** If every attempt can reject and return to the same safe state, any finite horizon admits an all-rejection path. Safety can hold without a bounded sure gain.

Source: [S] Theorem 29.1; Counterexample 29.2 | manuscript pp. 45-46

## 32. Positive gains need not accelerate

Recursion, continued improvement, and increasing productivity per unit resource are separate claims.

**Interpretation.** The gains 1/2, 1/4, 1/8, … are all positive. Their accumulated total remains below 1 after every finite number of steps.

**Boundary.** Faster improvement requires a claim about gain divided by complete discovery-and-acceptance cost. More headroom or a successful feedback loop does not establish it.

Source: [S] Proposition 30.1; §30 | manuscript pp. 46-47

## 33. The learned semantic bridge is still a research target

A commuting square specifies meaningful reuse. It does not discover the representation, learn the operations, or demonstrate their use by a neural system.

**Interpretation.** If the domain is invariant, every admitted operation commutes with q, and q preserves the required output, the relation extends to every finite operation sequence.

**Boundary.** Learning encoders, operations, experiment selectors, and proposers is a separate problem. The complete learned Neuralese–OSL bridge is not implemented in this edition.

Source: [S] §31; Proposition 31.1; §34 | manuscript pp. 48-49, 52-53

## 34. Locate the recursive achievement exactly

The finite chain acquires objects at several levels. It does not erase the supplied foundations of the next outer level.

**Interpretation.** A generated equation library changes the search representation used to acquire later discovery rules, while preserving the specified evidence and final programs.

**Boundary.** General intelligence, learned physical semantics, autonomous invention of the primitive proof system, and indefinite improvement of the equation learner remain open.

Source: [S] First-edition scope; §§23, 34 | manuscript pp. iii, 35-36, 52-53

## 35. A small vocabulary for reading the proofs

Use these distinctions to avoid turning several different achievements into one word such as “intelligence” or “success.”

**Interpretation.** The repair changes a target. The compiler discovers a policy. The learner can change a compiler rule. The equation library can change the learner’s candidate representation.

**Boundary.** A witness is checked under a model. A proof has hypotheses. A resource claim has a ledger. A generalization claim needs evidence beyond semantic relabeling.

Source: [S] Definitions 3.1, 6.1; §§5, 24-25 | manuscript pp. 3, 5-7, 37-40

## 36. Return to the source at the right level

This companion prioritizes mechanisms and counterexamples. The first-edition manuscript is the authority for complete hypotheses and proofs.

**Interpretation.** All substantive claims are restatements or worked illustrations of [S]. Source statistics are attributed to the supplied first-edition records, not presented as new learning runs.

**Boundary.** The guide’s examples and numerical transcriptions were checked separately. The full original 336,867-assertion suite was not rerun to make this visual companion.

Source: [S] Komorebi P., Self-Improving Intelligence, v1.0.0, 8 October 2026.