phanerozoic/omega-tensor

Omega Tensor: Exact Halting Fractions for 38,416 Instruction Sets

Exhaustive enumeration of all 1,679,616 programs of length 8 for every possible instruction set on a two-register, 8-bit, bounded machine with JNZ-to-zero control flow. Each row is one ISA. Each halting count is exact.

The repository contains two tables:

• omega_tensor.parquet (38,416 rows) — canonical single-initial-state table used in the paper, with the halting fraction at A=0, B=0 and cross-observable columns (cycle length, output entropy, reachability, structural flags).
• init_state_tensor.parquet (38,416 rows, 260 columns) — extends the above to every initial A value in 0..255 at B=0. Halting counts only.

Contents — omega_tensor.parquet

38,416 rows. Each row represents one ordered ISA defined by four opcode assignments (opcodes 1-4) drawn from 14 primitive operations. Opcode 0 is always INC. Opcode 5 is always JNZ.

Column	Type	Description
op1-op4	int	Operation IDs for opcodes 1-4
op1_name-op4_name	str	Human-readable operation names
halting_count	int	Exact count of halting programs (out of 1,679,616)
omega	float	halting_count / 1,679,616
has_swp, has_mov, has_and, has_shr	bool	Structural flags
involution_count	int	Number of involution operations in the ISA
coupling	str	Register coupling type: bidirectional, one-way, or none
avg_cycle	float	Mean cycle length among non-halting programs
max_cycle	int	Maximum cycle length among non-halting programs
entropy_bits	float	Shannon entropy of the output distribution (bits)
reachable_outputs	int	Number of distinct outputs produced by halting programs

Statistics — omega_tensor.parquet

• Range: [0.290710, 0.816916]
• Mean: 0.612418
• Distinct values (6 decimal places): 1,148
• Total programs executed: 64,527,984,256 (38,416 x 1,679,616)
• Avg cycle length range: [23.4, 216.1] (mean 116.2)
• Max cycle length: 255 for all ISAs
• Output entropy range: [0.80, 5.44] bits (mean 2.50)
• Reachable outputs: [9, 211] / 256 (mean 53.7)
• Outputs reachable by ALL ISAs: 9

Extremes

• Minimum (0.2907): INC INC INC INC INC JNZ. Five copies of INC. 71% of programs loop forever.
• Maximum (0.8169): INC AND AND AND AND JNZ. Four copies of AND. Only 18% loop.

Method

Each ISA was evaluated by exhaustive enumeration of all 6^8 = 1,679,616 programs of length 8. Each program was executed from initial state (A=0, B=0, PC=0) for up to 256 steps. Programs that reached PC >= 8 were counted as halting. The computation used a Numba-JIT-compiled Python interpreter at 0.26 seconds per ISA (approximately 2.8 hours total for all 38,416 ISAs).

Every halting count is exact. No sampling, no approximation, no estimation.

Verification

Known ISA values verified against independent FPGA silicon measurements and CPU cross-checks:

• ISA-A (DEC SWP ADD XOR): 978,929 halting (dataset) = 978,929 (FPGA) = 978,929 (CPU)
• ISA-B (NEG MOV SUB AND): 1,235,107 = 1,235,107 = 1,235,107
• ISA-G (SHR MOV SUB AND): 1,248,490 = 1,248,490 = 1,248,490

Contents — init_state_tensor.parquet

38,416 rows, one per ISA (same ISA space as omega_tensor.parquet). 260 columns: four opcode slots plus 256 halting counts, one per initial A value.

Column	Type	Description
op1-op4	int8	Operation IDs for opcodes 1-4
h_000 through h_255	int32	Halting counts at initial state (A=N, B=0) for N=0..255

Join on (op1, op2, op3, op4) to recover names, cycle length, entropy, and structural columns from omega_tensor.parquet. The canonical value omega from that table equals h_000 / 1,679,616.

Statistics — init_state_tensor.parquet

• Total cells: 38,416 x 256 = 9,834,496 exact halting counts
• Total program evaluations: ~16.5 billion
• Omega(init=0) mean: 0.612418 (matches omega_tensor.parquet)
• Omega(init=128) mean: 0.366431, range [0.236044, 0.681003]
• Ratio Omega(0)/Omega(128): mean 1.7429, median 1.6430, range [0.8336, 3.1466]
• 99.92% of ISAs have Omega(0) > Omega(128); 0.08% have the inverse (INC-dominant ISAs)
• 7.91% of ISAs have a perfectly reflection-symmetric bowl (Omega(a) = Omega(256-a))
• No ISA has its Omega minimum at init_a = 128 exactly; 3.23% have argmin in [126, 130]
• Bowl depth Omega(0) - Omega(min) ranges [0.058, 0.513], mean 0.260

Method — init_state_tensor.parquet

Same interpreter and enumeration protocol as the canonical tensor, extended over all 256 values of the initial accumulator A (B held at 0). 9,834,496 (ISA, init_a) cells, each the exact halting count over the full 1,679,616-program space. Computed with the same Numba-JIT interpreter under an 18-worker multiprocessing pool, 110.3 wall hours, zero errors. Every count is exact.

Cross-validation: for every ISA, the h_000 column equals the canonical halting_count from omega_tensor.parquet.

Citation

Based on: The Finite Halting Fraction Across Machine Models