The primary paper introduces weak, semiweak, and strong simulation for block maps. Its circuit-system block map is effectively strongly universal, while Theorem 4 proves that Conway's Game of Life is semiweakly universal. Question 2 asks whether the Life simulation can be made strong, which requires the selected preimage map to be bijective and hence to contain no extraneous Life preimages.

The focused search dated 2026-07-28 found no later source answering that exact block-map question. The bounded result in this packet tests a necessary consequence of strong universality against every macrotile dimension of area at most 11. It does not produce a universal construction or a counterexample block map, so the canonical problem remains open.

Apply the periodic-fiber count test to \(Y=\{0,1\}^{\mathbb Z^2}\) and its constant map to a singleton. For each positive pair \((m,n)\) with \(mn\leq11\), rotation symmetry lets us assume \(m\leq n\). The replay enumerates all \(2^{mn}\) periodic Life inputs and retains every output with exactly two predecessors, as required by (1) at \((p,q)=(1,1)\). It then counts the predecessors of the same output after doubling the width. Strong simulation would require exactly four. When a torus dimension is 1 or 2, distinct neighbor positions in the infinite periodic configuration can share a residue class. The evaluator retains all eight Moore-neighborhood offsets with their multiplicities.

The base-candidate count and doubled-fiber histogram are given below. A histogram entry \(a:b\) says that \(b\) candidate outputs have \(a\) predecessors after the width is doubled.

For \(1\times1\), the data is \(1\to\{2:1\}\). For \(1\times2\), it is \(1\to\{12:1\}\). The \(1\times3\), \(1\times4\), and \(1\times5\) tori have no base candidates.

The next rows are \(1\times6:6\to\{12:3,16:3\}\), \(1\times7:21\to\{22:7,38:7,44:7\}\), and \(1\times8:20\to\{60:12,264:8\}\).

For \(1\times9\), the data is \(54\to\{10:9,12:9,16:9,42:9,68:9,154:9\}\). For \(1\times10\), it is \(110\to\{22:30,38:20,44:10,54:10,86:10,128:10,188:20\}\). For \(1\times11\), it is \(154\to\{50:11,60:33,68:11,70:11,172:33,232:11,264:11,320:11,344:22\}\).

The \(2\times2\), \(2\times3\), and \(3\times3\) tori have no base candidates. The remaining rows are \(2\times4:12\to\{26:4,36:8\}\) and \(2\times5:35\to\{6:5,26:20,70:10\}\).

No row contains a candidate with doubled fiber size 4. Therefore no strong simulation of this required target can use \(mn\leq11\). This excludes a finite parameter range. It leaves every larger macrotile and the full universality question open.

Repeat the following three rows periodically: \[ P=\begin{pmatrix} 1&1&1&0&0&0\\ 0&1&0&1&1&1\\ 0&0&0&0&1&0 \end{pmatrix}. \] Salo and Törmä call the resulting configuration Köynnös. Every live cell has three live neighbors and every dead cell has four, so \(g(x^P)=x^P\). Their finite forcing certificate proves the stronger statement \[ g^{-1}(x^P)=\{x^P\}. \] Take a singleton alphabet \(D=\{d\}\), a singleton SFT \(Y=\{y\}\), and its only block map. Set \((m,n)=(6,3)\), \(\tau(d)=P\), and map the only Life predecessor to \(y\). This is a strong simulation of that target.

The example closes the easiest singleton specialization. Any obstruction to full strong universality must use a richer target or another invariant. The binary full-shift test in this packet does so.

Let \(D=\{d\}\), let \(Y\) be a two-dimensional SFT, and let \(\phi:Y\to D^{\mathbb Z^2}\) be the constant block map. Suppose Life strongly simulates \(\phi\) with data \((m,n,\tau,h)\). Put \[ z=\tau(d^{\mathbb Z^2}). \] The configuration \(z\) is periodic with periods \((m,0)\) and \((0,n)\). By Definition 5, the domain of the partial map \(h\) is exactly \(g^{-1}(z)\). Strong simulation makes \[ h:(g^{-1}(z),m\mathbb Z\times n\mathbb Z)\longrightarrow Y \] bijective. Definition 5 notes that such an \(h\) is a conjugacy for the blocked shift actions.

Fix positive \(p,q\). Conjugacy preserves points fixed by \((p,0)\) and \((0,q)\). Hence \[ \left|\{x:g(x)=z,\ \sigma^{(pm,0)}x=x,\ \sigma^{(0,qn)}x=x\}\right| =|\operatorname{Fix}_Y(p,q)|. \tag{1} \] For the binary full shift, the right side is \(2^{pq}\). In particular, a candidate macrotile must give fiber size 2 on the \(m\)-by-\(n\) torus and fiber size 4 after doubling either fundamental period. For singleton \(Y\), every count in (1) is 1, and the whole global fiber must contain one configuration.

Equation (1) is a necessary condition. Finite periodic counts alone cannot exclude extra aperiodic preimages, so satisfying the identities would leave a separate global conjugacy problem.

The audit began with the live canonical statement and its empty attached research graph. It then read the complete source of arXiv:2308.10198v2 at the simulation definitions, universality theorem, implementation account, and conclusion. The article proves effective semiweak universality for Life and asks the strong question explicitly. Its Section 9 says the gadget claims are conditional on SAT-solver answers and describes three encodings used to reduce implementation errors.

Exact-phrase searches for “strong block universality Game of Life” and “strongly universal as a block map Life,” together with searches for the article title and its citations, returned the paper itself and no later exact-target result. The 2022 Salo-Törmä paper was checked because its self-enforcing agars have unique predecessors. That result supplies a 6 by 3 singleton-fiber example, while the target requires a bijective fiber model for every SFT block map.

Ríos-Wilson and Theyssier also call a family of finite automata networks obeying the Life local rule strongly universal. Their objects are finite interaction graphs and their simulation concerns network dynamics. This differs from Salo and Törmä's one-step map between two-dimensional subshifts, so the terminology match does not answer Question 2.

The audit ended on 2026-07-28. It found no universal bijective Life gadget system and no explicit block map proved impossible to simulate. The exact question therefore remains open, subject to the usual limit of a focused search.

Continue the constant-map test with the binary full shift. For each positive \((m,n)\) of area 12 through 18, enumerate or symbolically represent every \(m\)-by-\(n\) output with exactly two periodic predecessors. For each survivor, count the lifted fiber on \(2m\)-by-\(n\), \(m\)-by-\(2n\), and \(2m\)-by-\(2n\) tori. The required counts are 4, 4, and 16.

A practical implementation should encode each requested output fiber separately in SAT and enumerate models with blocking clauses, or use row transfer when one dimension is small. Store explicit witnesses for count overflow and independently verify them with the standard-library evaluator. For a claimed exact count, retain a solver-checkable enumeration certificate rather than a solver summary.

Stop after every dimension pair through area 18 has either failed one required count or produced a survivor with certified counts. A survivor should trigger the next periodic tests \((p,q)=(3,1),(1,3),(2,3)\). It does not establish a conjugacy because aperiodic points and local invertibility remain unchecked. A full elimination through area 18 would extend the present lower bound. Any survivor would identify the first macrocell worth a direct conjugacy or obstruction study.

The program evaluates the B3/S23 rule directly on each finite torus. Neighbor coordinates retain multiplicity, which is required when a fundamental period is 1 or 2. A coordinate-set implementation independently agrees with the main evaluator for every state on each listed torus of area at most 12. A 5 by 5 blinker supplies a separate conventional-rule check.

The general fiber survey covers 29 listed tori and 2,844,494 inputs. The stable row data has SHA-256 af7a24b81711a0fd1f49be2aabea8fa0d350e10456938c87aa768779d9a1fca7. The binary-full-shift screen adds the 2 by 1 and 2 by 11 tori, and covers all 16 dimension pairs of area at most 11 up to rotation. Its stable data has SHA-256 cd80fc067571d23ba752537d4dd65f33e0f17c167de684d607e3746be7cbe574. Across both stages, the 31 cached fiber tables enumerate 7,038,802 periodic inputs.

The 3 by 3 torus offers a useful calibration. Its 126 singleton fibers are exactly the weight-four configurations, each fixed by Life. Z3 5.0.0 was used once to find a distinct 6 by 6 periodic predecessor for every lifted output. The replay depends only on the stored integer certificates and the Python standard library. It verifies all 126 witnesses, proving that none of those finite-torus singleton fibers is globally singleton. The canonical certificate encoding is 2,193 bytes with SHA-256 e2f87cdbde0e87c44d2a86c38d63a9029bdf2c45b073bbe23d690fb085180dbc.

The C++ program implements the periodic B3/S23 update with nested coordinate loops and stores complete fiber arrays in std::vector<uint32_t>. It shares no evaluator code with the Python replay. For the 16 dimension pairs of area at most 11 up to rotation, it enumerates 5,966 base inputs. For every base dimension having a two-predecessor output, it enumerates the doubled-width map once, covering another 6,705,172 inputs. The total is 6,711,138 exact Life evaluations.

All 414 base candidates fail the required doubled-width count of four. The output reproduces every histogram in life-sbu-claim-binary-full-shift-area-eleven-exclusion. Three direct executions were byte-identical. The 774-byte output has SHA-256 724665f4e0023a607ff3ebc15b7a5172e964d865bd5a063339fa4f6288951c6a.