The 19-Bit Accounting
The polynomial needs exactly 19 bits to specify completely. An exhaustive lookup table for the same rule would take 963 bits — 50× more. MDL says pick the shortest. The polynomial wins.
What "19 Bits to Specify" Means
MDL compares the description lengths of competing theories: which one requires
fewer bits to specify completely? "19 bits to specify the polynomial" means that if
you wanted to communicate the complete rule
p(L,C,R) = C+R−CR−LCR over GF(7) to someone who knew only the framework
but not the specific rule, you could do it in 19 binary digits.
Compare this to an alternative: just write down all 7³ = 343 possible input triples (L,C,R) ∈ {0,…,6}³ and the corresponding output for each. That lookup table approach requires 963 bits — 50× more.
Among all rules that behave identically on the orbit data, the polynomial has the shortest description. MDL says choose that one. 19 < 963 — the polynomial wins.
The 19-Bit Breakdown
The 19 bits break into five components. Click each block to see what it specifies:
binary floor
spec
form
selector
pattern
The breakdown in detail
Each Component Explained
8 bits — the Rule 110 binary floor
Rule 110 is specified by its 8-row truth table: one output bit for each of the 8 binary input triples (L,C,R) ∈ {0,1}³. Each output is either 0 or 1, so each row costs 1 bit. Total: 8 × 1 = 8 bits.
This is the binary floor established in Step 1 of the funnel: any valid GTE rule must match Rule 110 on all 8 binary inputs. Since the outputs are 0 or 1, each costs exactly 1 bit.
Eight bits is the minimum needed to specify any binary CA rule. Rule 110, taken alone, costs just 8 bits — and certifies exactly one thing: Turing universality. It proves the substrate can compute anything. But Rule 110 alone cannot certify particle generations, charge quantization, color charge, chirality, or baryon number — because those structures require the non-binary, 7-state algebraic field GF(7) to even express. The additional 11 bits (the GF(7) field specification) are what allow the certificate to encompass the full Standard Model structure. More compact does not mean more powerful: a 19-bit GF(7) polynomial does strictly more physics than any 8-bit binary rule.
3 bits — modulus specification
After pinning the 8 binary corners via Rule 110, we need to specify that the rule operates mod 7 — not mod 5 or mod 11. The modulus is the prime 7.
How many bits does it take to specify "the prime is 7"? The smallest prime is 2, and 7 is the 4th prime. To pick from all primes, we need ⌈log₂7⌉ = 3 bits (since 2³ = 8 ≥ 7).
2 bits covers values 0–3 (4 options)
3 bits covers values 0–7 (8 options) → includes 7 ✓
Specifying "prime = 7" costs 3 bits
Running total: 8 + 3 = 11 bits so far.
5 bits — algebraic form selector
After specifying that the rule operates over GF(7), we need to say which algebraic form it takes. Among all multilinear polynomials of degree ≤ 3 in three variables over GF(7), how many are there?
A multilinear polynomial in L, C, R of degree ≤ 3 has the form:
a₀ + a₁L + a₂C + a₃R + a₄LC + a₅LR + a₆CR + a₇LCR
That's 8 coefficients, each from GF(7). But the constraint that the rule must be the unique multilinear interpolant pinned by the orbit data eliminates all but one choice — the only remaining freedom is which of the ~32 possible algebraic "shapes" (patterns of nonzero terms) is present. This selection costs 5 bits (2⁵ = 32 options).
Running total: 8 + 3 + 5 = 16 bits so far.
1 bit — chirality selector
The polynomial could run in two chiral orientations. The GTE framework distinguishes "right-moving" and "left-moving" excitations — this corresponds to the chirality symmetry ℤ₂ ⊂ GF(7)*.
One bit selects which chirality: 0 = standard orientation
(as in the PSC-admissible orbit), 1 = mirror orientation.
The standard orientation is selected by the orbit structure.
0 → p(L,C,R) = C + R − CR − LCR
1 → p(L,C,R) = C + L − CL − LCR (mirror)
Running total: 8 + 3 + 5 + 1 = 17 bits so far.
2 bits — modular reduction signs
The last 2 bits specify the sign pattern on the product terms.
The polynomial has the form C + R ∓ CR ∓ LCR.
Each of the two product terms can carry a + or − sign independently.
The orbit interpolation forces the signs to be (−CR, −LCR).
This uses 2 bits: 1 bit for the sign of CR, 1 bit for the sign of LCR.
Bit 1: 1 → sign of CR is negative (−CR)
Bit 2: 1 → sign of LCR is negative (−LCR)
Result: p = C + R −CR −LCR ✓
Final total: 8 + 3 + 5 + 1 + 2 = 19 bits.
Summary: the complete 19-bit specification
The 19-bit coding theorem is machine-certified in Lean 4: — zero sorry. The polynomial specification is informationally closed: no bits can be removed without ambiguity.
Why the Lookup Table Costs 963 Bits
A general GF(7) CA rule with radius 1 is a function from {0,…,6}³ to {0,…,6}. There are 7³ = 343 possible input triples. For each triple, the output is one of 7 values, which takes ⌈log₂7⌉ = 3 bits to specify.
343 input triples × log₂(7) bits each
= 343 × 2.807…
≈ 963 bits
Using the information-theoretic bit count log₂(7) ≈ 2.807 bits per output symbol (rather than the ceiling ⌈log₂7⌉ = 3) gives the minimal prefix-free code for 343 independent 7-valued outputs. This is the Shannon entropy per entry when outputs are uniform over {0,…,6}.
The polynomial expression eliminates all of that redundancy because the 4-term algebraic structure implicitly determines all 343 outputs — even the 335 non-binary ones — from just 19 bits.
Why This Settles the MDL Argument
The MDL principle says: among all theories consistent with the data, choose the one with the shortest description. The "data" here is the orbit structure and the vacuum transparency condition.
Both the polynomial and the full lookup table are consistent with that data — they agree on all 8 binary corners. But the polynomial describes the rule in 19 bits while the lookup table requires 963 bits.
MDL's verdict is unambiguous: the polynomial is the shorter description. It must be selected. This is not a subjective judgment about "elegance" — it is a quantitative comparison of bit counts.
A theory with a 963-bit description secretly contains 944 bits of unexplained structure. Where do those extra bits come from? There is no answer — they would have to be free parameters, set by hand. The polynomial needs no free parameters: all 19 bits are derived from the framework constraints. The description is informationally closed.
Key Takeaways
-
The polynomial
p(L,C,R) = C+R−CR−LCRrequires exactly 19 bits to specify completely: 8 (Rule 110 floor) + 3 (modulus) + 5 (algebraic form) + 1 (chirality) + 2 (sign pattern) = 19. - An exhaustive lookup table for the same GF(7) CA rule would require approximately 963 bits — about 50× more. MDL selects the polynomial because it is the shorter description.
- The 19-bit description is informationally closed: no bits can be removed without ambiguity, and no bits need to be added. The rule is fully specified.
- The 19-bit coding theorem is machine-certified:
- This is why the polynomial has no free parameters: all 19 bits are derivable from the framework constraints (PSC, MDL, the orbit, vacuum transparency). Nothing is tuned or chosen by hand.
See Also
mdl_ca_rule_coding_closed— view on GitHub ↗