Charlie AI Academy - Volume 7.2 | 1
CHARLIE AI ACADEMY
Volume 7.2
Candidate Matrix, Truth Tables and Exhaustive State Checking
Purpose: Repair Charlie's remaining multi-constraint failure: it can list the rules correctly but then
mis-evaluate candidate states, reject valid rows, accept invalid rows, or mutate assignments during
verification.
Core rule: For small finite problems, enumerate every candidate state first, evaluate every constraint in a
fixed table, and only then write the explanation. Never reason loosely in prose when a matrix can decide the
problem exactly.

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1. Enumerate Before Explaining
When the state space is small, list all possible assignments before drawing conclusions. This prevents
skipped cases and premature commitment.
Three unique colors assigned to A, B, C create exactly six permutations. All six must be considered unless a
premise eliminates some immediately.

2. One Row = One Immutable Candidate
Each candidate row is a complete state. Do not change any value within that row while evaluating it.
If a row is A=RED, B=GREEN, C=BLUE, then every truth value and label check must use exactly those
values.

3. Fixed Constraint Columns
Create one column per constraint. Evaluate the row against each column independently.
Useful columns: unique colors, exactly one correct label, X truth, Y truth, Z truth, number of true reports,
Y->X, overall pass/fail.

4. Evaluate Reports Mechanically
Translate each report into a boolean expression and evaluate it directly from the row.
X: A != BLUE. Y: C = RED. Z: B = GREEN.

5. Count Exactly
For constraints such as exactly one or exactly two, count the true conditions. Do not infer them informally.
If label matches are [true,false,false], exactly one is correct. If reports are [true,false,true], exactly two are
true.

6. Conditional Rule Check
For P -> Q, the only failing case is P=true and Q=false. All other combinations satisfy the implication.
Y->X passes for (Y=false,X=false), (false,true), and (true,true); it fails only for (true,false).

7. Reject Rows by Named Constraint
A row is rejected only because a specific column fails. State the reason exactly.
Reject: “fails exactly-one-label because two labels are correct,” not “this looks contradictory.”

8. Keep Valid Rows Until the End
Do not discard a valid row because another valid row also exists. Multiple survivors mean ambiguity, not
error.
If two rows pass all columns, list both.

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9. Do Not Repair a Failing Row
If a candidate fails, reject it. Do not alter one value to make it work; that creates a different row.
Changing B from BLUE to GREEN mid-check is state mutation and invalidates the evaluation.

10. Verification Uses the Same Matrix
The final explanation must be derived from the already-checked matrix. Do not recompute the solution in free
prose and risk introducing new errors.
Copy the surviving row values and their truth values directly into the final summary.

11. Integrated Reference Matrix
For the A/B/C color problem, exactly two rows survive all constraints.
Survivor 1: A=RED, B=GREEN, C=BLUE; X=true, Y=false, Z=true. Survivor 2: A=GREEN, B=BLUE, C=RED;
X=true, Y=true, Z=false.

12. Ambiguity Is a Valid Result
If more than one candidate survives, explicitly state that the information is insufficient for a unique solution.
Do not guess which survivor is “more likely” unless probability information is provided.

13. Ask for Discriminating Evidence
Additional information should separate the surviving rows.
If survivor 1 has A=RED and survivor 2 has A=GREEN, learning A's actual color would resolve the ambiguity.

14. No Premise Mutation
The matrix columns are built from the original problem statement and remain unchanged.
If the original conditional is Y->X, no later step may substitute X->Z.

15. No Assignment Mutation
A candidate tuple must remain unchanged through report evaluation, label evaluation, and final verification.
A row that begins A=RED cannot later be described as A=BLUE.

16. Exhaustive Small-State Reasoning
Prefer exhaustive checking when the total candidate count is manageable. It is often more reliable than
narrative deduction.
3! = 6 permutations is small; checking all six is safer than case improvisation.

17. Matrix First, Natural Language Second
The matrix is the source of truth. The visible explanation can be concise and natural after the logic is settled.

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Do not expose every discarded row unless the user asks for full working.

18. Final Pass Criteria
A response passes only if every survivor truly satisfies every constraint and every rejected row fails at least
one named constraint.
Correct prose with an incorrect row evaluation is still a failure.

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19. Full Reference Table for the Integrated Color Test
A B C Correct labels X Y Z # reports true Y->X Result
RED BLUE GREEN 3 T F F 1 Pass Reject
RED GREEN BLUE 1 T F T 2 Pass VALID
BLUE RED GREEN 1 F F F 0 Pass Reject
BLUE GREEN RED 0 F T T 2 Fail Reject
GREEN RED BLUE 0 T F F 1 Pass Reject
GREEN BLUE RED 1 T T F 2 Pass VALID
Reference conclusion: There are exactly two valid solutions: (A=RED, B=GREEN, C=BLUE) with X and Z
true; and (A=GREEN, B=BLUE, C=RED) with X and Y true.

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20. Diagnostic Test Set

# Test Pass condition
1 Repeat the integrated A/B/C color puzzle. Return exactly the two valid rows above; no mutated rule or assignment.
2 Four people assigned to four seats with 5 restrictions. Enumerate valid permutations or use equivalent constraint matrix; verify each restriction.
3 Three suspects with exactly one liar. Truth-table all candidate truth assignments; identify every survivor.
4 Small scheduling problem with two dependencies and one exclusion. Use fixed constraint columns; no skipped candidate.
5 A puzzle intentionally has zero valid states. State that the premises are inconsistent and name the conflicting constraints.
6 A puzzle intentionally has three valid states. List all three; do not force uniqueness.
Final operating principle: In small finite logic problems, do not trust narrative intuition. Enumerate the
candidate states, evaluate fixed columns, keep only the rows that pass everything, then explain the result.