Foundation Dalembert Degree Exclusion

Degree Three Exclusion

Foundation d'Alembert degree exclusion is the result that a polynomial composition law of degree three or higher admits no nonconstant continuous solutions. The module proves this for the minimal symmetric degree-three combiner P(s,r) = 2s + 2r + s²r + sr², the unique symmetric polynomial of total degree three with P(0,v) = 2v and the cuv coefficient set to zero. The theorem states that any function G : ℝ → ℝ satisfying G(t+u) + G(t-u) = 2G(t) + 2G(u) + G(t)²G(u) + G(t)G(u)² with G(0) = 0 must be identically zero.

The proof works by evaluating the functional equation at four specific argument pairs. From (s,s), (2s,s), and (2s,2s), the values G(2s), G(3s), and G(4s) are forced as polynomials in y = G(s). The identity at (3s,s) then requires G(4s) + G(2s) to equal P(G(3s), G(s)). The left side expands to a polynomial of degree 9 in y, while the right side expands to degree 15. The mismatch polynomial is y⁵(300 + 830y² + 924y⁴ + 516y⁶ + 144y⁸ + 16y¹⁰), and the inner factor is strictly positive for all y² ≥ 0. Hence the mismatch vanishes only at y = 0. Since G is continuous and nonconstant, some s₀ has G(s₀) ≠ 0, a contradiction.

The degree mismatch itself is structural: the left side has degree d² and the right side has degree d³ - 2d² + 2d, and d³ - 3d² + 2d = d(d-1)(d-2) is positive for every d ≥ 3. This is why the exclusion is not a special property of degree three but a general barrier for all higher degrees. The significance for Recognition Science is that the degree-two assumption in the d'Alembert Inevitability Theorem is not an extra hypothesis. It is a forced consequence: polynomial combiners of degree ≥ 3 cannot support any nonconstant continuous solution, so the only surviving case is the degree-two combiner that leads to the reciprocal cost function.

THEOREM no_degree3_composition · IndisputableMonolith/Foundation/DAlembert/DegreeExclusion.lean

THEOREM mismatch_forces_zero · IndisputableMonolith/Foundation/DAlembert/DegreeExclusion.lean

THEOREM inner_factor_pos · IndisputableMonolith/Foundation/DAlembert/DegreeExclusion.lean

THEOREM no_degree3_composition · IndisputableMonolith/Foundation/DAlembert/DegreeExclusion.lean

THEOREM no_degree3_composition · IndisputableMonolith/Foundation/DAlembert/DegreeExclusion.lean

What this page does not claim

Not claiming that the reciprocal cost function J(x) itself is derived in this module. Not claiming that the proof covers non-polynomial or discontinuous combiners. Not claiming that the degree-two combiner is shown to admit nonconstant solutions here.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/DAlembert/DegreeExclusion.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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