. . . . "analysis.py evolves all 256 elementary rules from 64 six-bit programs and four input pairs for 16 steps and reads the cell at position -1. It runs one radius-4 multiplexer, whose new center is the program bit indexed by the two input bits, and a Turing machine that walks from the input bits to that same program bit and writes it on the output cell. A function is realized when one program makes all four inputs match it. Criteria were frozen in commit 093ca8002a77b774b469b965ff599d123fdaefbb before this run." . . "Measured: 0 boolean functions of two bits realized by one ceiling system and not the other (n = 16, exact). Interpretation: The claim (NKS p. 675) says that as rules grow more capable they pass a threshold of universality, after which the set of computations they can perform is the same, so a more sophisticated rule and a different kind of system gain nothing further about what they can compute. Universality over all partial recursive functions is not a finite enumeration. The computation uses a finite universe that can actually be saturated: all 16 boolean functions of two bits. The radius-4 rule and the Turing machine are different constructions aimed at that ceiling; the elementary rules, including rule 0, are the progressive family underneath. Capability is the set of functions, not the encoding, so the comparison across kinds is extensional. A computation shows these sets are equal as mathematics, not that organisms compute." . "Claim test of wolfram2002-ch11-13 (pr-130): supports, confidence 0.84" . . . "Only configurations evolved by analysis.py from the pre-registered encodings. Elementary rules use Wolfram numbers 0..255. The radius-4 rule is the boolean multiplexer stated in the pass criterion, not a Wolfram code number. No external dataset." . "claim-test/0.1.0" . "computational" . "Rule 0 realizes all 16 functions under the elementary encoding, or some elementary rule realizes more than 16 functions." . "The computation shows that these programs realize the same set of two-bit boolean functions once they can realize all 16, and that rule 0 realizes fewer. It does not show that organisms compute, and it does not prove Turing universality." . "The largest elementary capability under this encoding is 16 functions, at rule 22. Some elementary rule reaches the same set of 16." . "The universe of computations is the 16 functions of two bits. Past that ceiling there is no further function of two bits to gain. A richer universe, such as all partial recursive functions, was not enumerated." . "The radius-4 cellular automaton and the lookup Turing machine each realize all 16 boolean functions of two bits and those two sets are equal, rule 0 realizes at most 4 of them, and no elementary rule realizes more than those 16." . "Pre-registered. REFUTES if |Cap(rule 0)| >= 16 (the null rule already performs every computation in U, so this encoding does not measure a threshold) or if any elementary rule has |Cap| > 16. INCONCLUSIVE with reason 'underpowered' if |Cap(radius-4)| < 16 or |Cap(TM)| < 16 (a system built to sit at the ceiling did not reach it, so equality past the ceiling was not tested). MIXED if both ceiling systems equal U and |Cap(rule 0)| <= 4, but some elementary rule has |Cap| = 16 whose set of functions is a proper subset of U (impossible if cardinality is 16, retained so a counting bug cannot be scored as support) or if the two ceiling sets are both size 16 and unequal. Two size-16 subsets of a 16-element set are equal, so a reported inequality at size 16 is a counting error and is MIXED rather than a refutation of the book. Confidence: mixed 0.70, refutes 0.80, inconclusive 0.50." . "0.84"^^ . . "supports" . "Once a system can realize every boolean function of two bits, a structurally richer system that can also realize every such function realizes exactly the same set, while a system with no information flow realizes a strictly smaller set." . "Compare the sets of realized truth tables. Supports when both ceiling systems equal the full 16 and rule 0 is at most 4. Inconclusive (underpowered) when a ceiling system misses the full set. Refutes when rule 0 is already at the ceiling or a cardinality exceeds 16. Confidence: supports 0.84, mixed 0.70, refutes 0.80, inconclusive 0.50." . "Pre-registered. The universe U is the 16 functions from {0,1}^2 to {0,1}, truth tables ordered by inputs (b0,b1) = (0,0), (0,1), (1,0), (1,1). Three systems. (1) Elementary cellular automata, Wolfram numbers 0..255, fixed-0 boundaries. Encoding: output cell at position -1 starts at 0; input bits at positions 0 and 1; a program of 6 bits at positions 2..7 (64 programs); all other cells 0; evolve t = 1..16 steps on the tape from -17 to 23 inclusive. A function f is realized if some program and some t make cell -1 equal f(b0,b1) for all four inputs. (2) One radius-4 binary cellular automaton, one step. Neighborhood positions relative to the center, from -4 to +4. Input b0 is at +1, b1 at +2. Program bits p00, p01, p10, p11 sit at -4, -3, -2, -1 and the 16 programs are exactly the 16 truth tables in the input order above. The local rule sets the new center to the program bit indexed by b0 + 2*b1, ignoring every other neighbor. Initial center is 0. Read the center after 1 step. (3) A 2-symbol Turing machine on the tape [output=0, b0, b1, p00, p01, p10, p11]. From a head start on b0 it stores b0, moves right, stores b1, moves to p00, steps right b0+2*b1 cells, reads that bit, returns to the output cell, writes it, and halts. The same 16 truth-table programs are used. Let Cap(S) be the set of functions realized. SUPPORTS if Cap(radius-4) = Cap(TM) = U, |Cap(rule 0)| <= 4, every elementary rule has |Cap| <= 16, and every elementary rule with |Cap| = 16 has Cap equal to U. Confidence if supports: 0.84." . "false"^^ . . "the systems whose sets of two-bit boolean functions are compared" . "" . . "Dataset: Boolean-function capabilities of elementary rules 0..255, one radius-4 multiplexer, and one lookup Turing machine, generated by analysis.py" . . . "delphisci claim tester (Grok 4.7; cursor/wolfram-claim-tests-7ca2)" . . . "Grok 4.7" . . . . . . . . . "2026-11-08T16:24:31.939Z"^^ . . . . . "Claim test of wolfram2002-ch11-13 (pr-130): supports, confidence 0.84" . . "RSA" . 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