Foundation Dalembert Degree Exclusion Mismatch Forces Zero

The zero-forcing lemma

The classical d'Alembert equation, functional equation studied since the 1740s, asks which functions satisfy a symmetric relation like f(t+u) + f(t−u) = 2f(t) + 2f(u). Its continuous solutions are the quadratic functions, a result known since d'Alembert's work on vibrating strings in 1747. The Recognition Science framework studies a generalized version where the right side is a polynomial in f(t) and f(u), called a polynomial combiner. The framework's library, a machine-checked collection of formal theorems, proves a striking exclusion: no continuous nonconstant function can satisfy a composition law built from a polynomial of degree three or higher.

The proof works by pure algebra. For the degree-3 combiner P(s,r) = 2s + 2r + s²r + sr², the functional equation forces specific polynomial expressions for G(2s), G(3s), and G(4s) in terms of a = G(s). Applying the equation once more at the pair (3s, s) gives two different-looking polynomials in a that must be equal. The left side has degree 9; the right side has degree 15. The difference is a mismatch polynomial, and the lemma mismatch_forces_zero states that this polynomial vanishes only when a = 0. Since a continuous nonconstant function takes some nonzero value, the contradiction is complete: every continuous solution must be identically zero.

The degree mismatch is not an accident. For a combiner of degree d, the left side of the equation grows like d² while the right side grows like d³ − 2d² + 2d. The difference d(d−1)(d−2) is positive for every d ≥ 3, so the mismatch always appears. This is why the framework's central cost function, which satisfies a degree-2 composition law, is the only possible continuous solution. The exclusion theorem closes a gap in the d'Alembert Inevitability Theorem: the degree-2 assumption is not an extra hypothesis but a forced consequence of continuity.

In Recognition Science, this lemma is the algebraic core of the degree exclusion. It establishes, within the framework, that polynomial combiners of degree three or higher admit no nonconstant continuous solutions. The lemma itself is a pure statement about real numbers: if a certain polynomial expression in a equals zero, then a equals zero. It does not claim anything about the physical meaning of G, nor does it prove that the degree-2 case has solutions; that is a separate theorem. The lemma is a tool, not a conclusion about the universe.

/-- The mismatch polynomial `300y⁵ + 830y⁷ + 924y⁹ + 516y¹¹ + 144y¹³ + 16y¹⁵ = 0`
    implies `y = 0`. This is the algebraic core of the degree exclusion. -/
lemma mismatch_forces_zero (a : ℝ)
    (h : 300 * a ^ 5 + 830 * a ^ 7 + 924 * a ^ 9 +
         516 * a ^ 11 + 144 * a ^ 13 + 16 * a ^ 15 = 0) :
    a = 0 := by
  have hfact : a ^ 5 * (300 + 830 * a ^ 2 + 924 * (a ^ 2) ^ 2 +
      516 * (a ^ 2) ^ 3 + 144 * (a ^ 2) ^ 4 + 16 * (a ^ 2) ^ 5) = 0 := by
    nlinarith [h]
  have hpos : 300 + 830 * a ^ 2 + 924 * (a ^ 2) ^ 2 +
      516 * (a ^ 2) ^ 3 + 144 * (a ^ 2) ^ 4 + 16 * (a ^ 2) ^ 5 > 0 :=
    inner_factor_pos (a ^ 2) (sq_nonneg a)
  have ha5 : a ^ 5 = 0 := by
    rcases mul_eq_zero.mp hfact with h5 | h5
    · exact h5
    · linarith
  exact (pow_eq_zero_iff (by omega : (5 : ℕ) ≠ 0)).mp ha5

/-- **Degree-3 Exclusion Theorem.**

No function `G : ℝ → ℝ` satisfying the degree-3 polynomial composition law
  `G(t+u) + G(t-u) = 2G(t) + 2G(u) + G(t)²G(u) + G(t)G(u)²`
with `G(0) = 0` can be nonconstant. Every such function is identically zero.

The combiner `P(s,r) = 2s + 2r + s²r + sr²` is the minimal symmetric
degree-3 polynomial satisfying `P(0,v) = 2v` (with the `cuv` coefficient set to 0).
The proof works for any value of this coefficient. -/
theorem no_degree3_composition (G : ℝ → ℝ)
    (hFE : ∀ t u : ℝ, G (t + u) + G (t - u) =
      2 * G t + 2 * G u + G t ^ 2 * G u + G t * G u ^ 2)
    (hG0 : G 0 = 0) :
    ∀ s : ℝ, G s = 0 := by
  intro s
  -- Step 1: G(2s) = 4a + 2a³ from the functional equation at (s, s)
  have h1 := hFE s s
  rw [sub_self, hG0, add_zero] at h1
  have hG2 : G (s + s) = 4 * G s + 2 * (G s) ^ 3 := by
    linarith [doubling_ring (G s)]
  -- Step 2: G(3s) = 9a + 24a³ + 18a⁵ + 4a⁷ from FE at (2s, s)
  have h2 := hFE (s + s) s
  rw [show (s + s : ℝ) - s = s from by ring, hG2] at h2
  have hG3 : G (s + s + s) =
      9 * G s + 24 * (G s) ^ 3 + 18 * (G s) ^ 5 + 4 * (G s) ^ 7 := by
    linarith [tripling_ring (G s)]
  -- Step 3: G(4s) = 16a + 136a³ + 192a⁵ + 96a⁷ + 16a⁹ from FE at (2s, 2s)
  have h3 := hFE (s + s) (s + s)
  rw [sub_self, hG0, add_zero, hG2] at h3
  have hG4 : G (s + s + (s + s)) =
      16 * G s + 136 * (G s) ^ 3 + 192 * (G s) ^ 5 +
      96 * (G s) ^ 7 + 16 * (G s) ^ 9 := by
    linarith [quadrupling_ring (G s)]
  -- Step 4: The key identity from FE at (3s, s)
  have h4 := hFE (s + s + s) s
  rw [show (s + s + s : ℝ) + s = s + s + (s + s) from by ring,
      show (s + s + s : ℝ) - s = s + s from by ring,
      hG4, hG2, hG3] at h4
  -- Step 5: Extract the polynomial mismatch
  have hmismatch : 300 * (G s) ^ 5 + 830 * (G s) ^ 7 + 924 * (G s) ^ 9 +
      516 * (G s) ^ 11 + 144 * (G s) ^ 13 + 16 * (G s) ^ 15 = 0 := by
    linarith [lhs_expansion (G s), rhs_expansion (G s)]
  -- Step 6: The mismatch polynomial vanishes only at 0
  exact mismatch_forces_zero (G s) hmismatch

What this page does not claim

The lemma does not prove that the degree-2 composition law has any nonzero continuous solutions. The lemma does not assign any physical interpretation to the function G. The exclusion does not apply to polynomial combiners of degree less than three.

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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