How MDL Selects the Polynomial

The Bridge from Path 1

In L06 (GTE Orbits) you saw that the orbit {73, 42, 275} encodes the three charged lepton generations, and that the structure MDL forces is Z₇×Z₃. But how does the actual polynomial formula emerge from this?

The answer is a two-stage process. MDL acts twice in sequence:

This lesson focuses on Selection 2: how the orbit data pins the polynomial and why no other rule is consistent.

The Five-Step Elimination Funnel

MDL doesn't search through candidates one by one. It applies five structural constraints in sequence, each collapsing the space by an enormous factor:

The Direct-Interpolation Lift

After Steps 1–4, we know the rule must be a multilinear polynomial over GF(7). Such a polynomial is determined by its values on the 8 binary input triples (L,C,R) ∈ {0,1}³ — the 8 "binary corner" points. That gives exactly 8 degrees of freedom, each a value in {0,1,2,3,4,5,6}: so 7⁸ = 5,764,801 candidates.

The orbit's parity shadow pins 7 of those 8 corners. The 8th is pinned by the vacuum transparency condition: p(0,0,0) = 0. Together, 8 constraints determine 8 unknowns. Exactly one polynomial survives.

What is the "parity shadow"?

The GTE orbit is a sequence of triples with values in GF(7). When you project an orbit state onto {0,1} by the map v → v mod 2, you get the parity shadow — a binary state. The parity shadow of the orbit {(1,73,823) → (9,42,1023) → (5,275,65535)} generates specific binary neighborhoods as the orbit evolves.

Reading off the (L,C,R) neighborhoods visited and their outputs from the orbit evolution gives us 7 input-output pairs for the polynomial. Each pair is a constraint: p(L,C,R) = output.

The 10 constraints, step by step

The full orbit produces 10 constraints (some redundant). After removing redundancies, exactly 7 are independent. Use step buttons to walk through them:

Verification: p Satisfies All 8 Constraints

The formula p(L,C,R) = C + R − CR − LCR can be checked against all 8 binary inputs by hand. Here are three spot-checks:

The full table is given in L03 (The Polynomial). Every row matches. This agreement is not a design coincidence — it is a consequence of the interpolation theorem.

Why Every Other Rule Fails

The proof of uniqueness is by exhaustive interpolation over GF(7). Any multilinear degree-≤3 polynomial over GF(7) in three variables is determined by 8 values: one for each binary corner (L,C,R) ∈ {0,1}³. The 8 constraints above pin each of those values exactly.

Any other choice of values for any corner would mean:

• Violating a constraint from the orbit — meaning the rule cannot reproduce the known particle structure
• Or violating vacuum transparency — meaning the rule creates excitations from empty space, costing extra description bits

Either way, any competing rule is MDL-penalized relative to the unique survivor. The Lean 4 proof formalizes this by machine-verifying that the Lagrange interpolant is unique over GF(7) and equals C+R−CR−LCR:

Key Takeaways

• MDL acts twice: first to select the Lepton Seed (1,73,823), then a second time via orbit interpolation to select the polynomial itself.
• After Steps 1–4 of the elimination funnel, only multilinear GF(7) polynomials remain — a space of 7⁸ = 5,764,801 candidates, one for each assignment of output values to the 8 binary corners.
• The GTE orbit pins 7 of those 8 values. Vacuum transparency (p(0,0,0) = 0) pins the 8th. Together they yield a unique Lagrange interpolant: p(L,C,R) = C + R − CR − LCR (mod 7).
• This uniqueness is machine-certified in Lean 4:
• The polynomial was not designed to match Rule 110 on binary inputs — it was selected by the orbit data and the vacuum condition. Agreement with Rule 110 is a theorem, not a design choice.