A novice’s note on veracity tests inside self-referential Boolean probes ~ Eric J Wolverton The pictures you have been using are not a new theorem of arithmetic. They are a small Boolean machine. A hidden bit, or a hidden face of a die, is never read directly. Instead you ask a yes/no question that talks about its own answer. That question can be answered truthfully only for the right hidden value. For every wrong value the question has no truthful answer at all. The failure is the signal. This note explains what “truthful” means here, which tests are already built into the three probes, and what larger constructs you can assemble from them without adding randomness. 1 The only objects you need There are two bits. • (P) is the hidden fact. For a die face (n), (P) is true exactly when the face is (n). • (A) is the answer the probe is about to give: true for YES, false for NO. A probe is a Boolean formula (F(A, P)). The rule of veracity is a fixed-point demand: [ A = F(A, P) ] An answer is admissible only if it equals what the formula says about that same answer. If some bit (A) satisfies the equation, the probe is completable, and each such bit is a truthful answer. If no bit satisfies it, the probe is incomplete. Incompleteness is not a third answer. It is the report that no answer is consistent with the formula. That is the whole veracity test. It does not consult an outside judge. Consistency with self-reference is the judge. 2 Why a wrong face leaves no answer The three formulas in the posters all hide (A) inside (F) so that a false (P) forces (A = \neg A). Implication, “if my answer is YES, then the face is (n)”: [ F(A, P) = A \rightarrow P = \neg A \lor P ] • If (P) is true, (F) is always true, so only (A = \text{true}) works. The only truthful answer is YES. • If (P) is false, (F = \neg A), so the equation says (A = \neg A). Neither bit works. Biconditional, also called XNOR, “my answer is YES if and only if the face is (n)”: [ F(A, P) = A \leftrightarrow P ] • If (P) is true, (F = A), so both YES and NO are fixed points. • If (P) is false, (F = \neg A), again a contradiction. NOR, “neither my answer is YES nor the face is (n)”: [ F(A, P) = \neg(A \lor P) ] • If (P) is true, (F) is always false, so only NO works. • If (P) is false, (F = \neg A), again no solution. So each probe carries the same inherent test: a truthful answer exists if and only if (P) is true. The three probes differ only in which answer survives when the test passes. Implication pins the answer to YES. NOR pins it to NO. XNOR accepts both, and therefore certifies the face without committing to a bit. 3 What kind of test this is Four properties are already inside the construct. None of them is an extra checker bolted on afterward. Existence. The probe passes only when the set of fixed points is non-empty. Emptiness is the negative result. Uniqueness of the witness, not always of the bit. Across candidate faces, at most the true face passes. Inside one probe the surviving bit may be unique (implication, NOR) or not (XNOR). The face is what must be unique if you want a reveal. Polarity. Implication and NOR do not merely say “this face is possible.” They also constrain the answer bit. A later reader can check that the returned bit really is a fixed point, which is a local audit. Fail-closed behavior. If you ask every face and none completes, you reveal nothing. There is no fallback guess. A wrong face cannot be dressed up as a completed answer, because completion is defined as the existence of the fixed point. What the construct does not test is also worth stating. It does not test whether (P) was computed honestly by someone else. If the machine that evaluates (F) already knows the die and is willing to lie about whether a fixed point exists, the test collapses. The veracity is internal to the equation, not a defense against a corrupt evaluator. It is also not Gödel’s incompleteness theorem. Gödel’s result concerns sentences of arithmetic that a sufficiently strong formal system can neither prove nor refute. These probes are Boolean equations with a parameter. The analogy is the shape of the failure: self-reference makes a question unanswerable, and that unanswerability is informative. 4 The cheap path, restated without metaphor Suppose the hidden face is some (n) in ({1, \ldots, N}), and for each candidate you run one probe. Every wrong candidate produces the contradiction (A = \neg A) and stops. The true candidate produces a fixed point and is allowed to continue. You never need a second stage that compares costs. The branches that cannot be completed are already gone, so the completed branch is the only route left. That is the sense in which completeness is the cheap path. The cost is one fixed-point check per candidate, and failed checks do not spawn further work. 5 Synthetic constructs you can build The three probes are generators. Larger questions are formulas or circuits built out of the same fixed-point demand. A majority probe. Compute the three formulas and accept a candidate only if at least two of them have a fixed point. For a wrong face all three fail, so majority fails. For the true face all three succeed, so majority succeeds. The poster’s “are at least two of these three statements true?” is this construct asked as one question. Its veracity test is stricter in presentation, looser in the bit it returns: you learn the face, not which of the three internal answers was chosen. A pinned probe. Use implication if you need the surviving answer to be YES, NOR if you need it to be NO. Pinning matters when the answer bit is itself a command. YES can mean “forward the packet”; NO can mean “close the branch.” XNOR cannot drive that command, because both bits are legal. A conjunction of faces. To test whether the hidden value lies in a set (S), set (P) true when the face is in (S), and use any of the three formulas. The probe completes exactly on that set. This is a filter, not an identifier: it says “in (S)” or “incomplete,” not which member of (S) it was. A binary search shape. Ask the implication probe for “the face is greater than (m).” The probe completes only if that comparison is true, and then only with YES. Incomplete means “not greater.” Logarithmically many probes identify the face. Each probe still carries its own fixed-point test; the search tree adds no new notion of truth. A veto. Run NOR on a forbidden face. Completion with NO means “this forbidden face is the actual one,” which is the alarm. Incompletion means “not this face.” The veto does not reveal the true face among the others. A code, not just a face. Let the hidden object be a bit string. For each bit position, run a pinned probe. The pattern of completions is the string. A single mismatch makes that bit’s probe incomplete, so a corrupted guess cannot be returned as a completed codeword. This is an error-detecting shape, not an error-correcting one: you detect the bad position by its silence, and you do not repair it from inside the same equation. A meta-probe. Let (Q) be the statement “at least one candidate face completes.” Form (A = F(A, Q)) with implication. This completes only if some face really completes, and then only with YES. It is a single question whose fixed point certifies that the family of probes is not empty. It does not name the face; the inner probes do that. All of these inherit the same limit. They are synthetic only in the arrangement of (P) and of the connectives. They do not create information that was not already in the hidden fact and in the fixed-point rule. 6 A worked check Let the die show 4, and let the formula be implication. For candidate 4, (P) is true, and the only solution of (A = (\neg A \lor P)) is (A = \text{true}). For candidate 5, (P) is false, and both candidate answers fail the equation. The same split appears for NOR, with the surviving answer NO, and for XNOR, with both answers surviving. Majority of the three agrees on candidate 4 and rejects the rest. That is the entire reveal. 7 How to read a result If a probe returns a bit, check the bit before trusting it: substitute it back into (F) and confirm equality. Implication must return YES on success. NOR must return NO. XNOR may return either, so the bit is not evidence; the existence of a bit is the evidence. If a probe returns incomplete, treat that as a proved rejection of that (P) under the formula, not as a missing measurement. If a whole family returns incomplete, the fail-closed rule says to reveal nothing rather than pick the least-bad candidate. Adding a guess at that point would throw away the only veracity test the construct has. The idea a newcomer should keep is small. Self-reference turns a Boolean formula into a lock. The lock opens only for the hidden fact that was written into it, and the shape of the opening — YES, NO, or either — is chosen by the connective. Everything called a Gödel oracle in the posters is that lock, repeated across candidates, with silence doing the pruning.