Gravity Seven Gaps Zq Continuum Blocker Zero Phase Fails Both Removal Routes

The two failed routes

In a path sum, you add up contributions from many possible paths. When the list is infinite, you often impose a cutoff: keep only paths up to some complexity B, compute the finite sum, then let B grow toward infinity. The question is whether the finite sums settle on a single value. This is the same question as whether a sequence of partial sums converges, and the standard test is the Cauchy criterion: the sums must get arbitrarily close to each other once B is large enough.

The Recognition Science framework's machine-checked library of formal theorems studies this question for a specific family of path sums in its gravity program. The sums are built from "shells," each shell collecting paths of a fixed complexity class. A phase can be attached to each path, a complex number of unit magnitude that rotates the contribution. The zero phase attaches no rotation at all: every path contributes its plain weight. The framework proves a sharp negative result for this zero phase: the sequence of finite sums does not converge, and it fails by a wide margin. The failure is not borderline. For any cutoff N, one can find a later block of shells whose combined contribution has magnitude greater than 1, no matter how far out one goes. The tail never dies down.

The framework then asks whether a different removal route could save the zero phase. One route removes the cutoff by a regulator, a smoothing device that tames the tail. The other route keeps the capped API but requires a compatibility bridge between two ways of writing the sums. The theorem zeroPhase_fails_both_removal_routes proves that both routes fail for the zero phase: neither the exact-shell cutoff limit nor the regulator removal exists. The proof is a conjunction of two earlier theorems, each showing one route is blocked. A companion theorem shows that even if the two APIs were made compatible, the zero phase still could not have a limit.

What this establishes is a precise obstruction, not a general statement about all phases. The zero phase is the simplest choice, and it does not work. The framework's library proves that a substrate-derived phase, one that comes from the physical structure rather than being set to zero, is what the cancellation criterion demands. The zero phase result is the discriminating witness that shows why a nontrivial phase is necessary. It does not prove that any particular nontrivial phase works, nor does it say anything about the continuum limit, mesh refinement, or the physical ledger of recognition events. Those remain open targets.

/-- **C1 EPSILON-ONE WITNESS.** Beyond every proposed tail threshold there
is a contiguous one-shell block whose zero-phase norm is greater than one.
The bridge is direct: `shellMass_lower` concerns the same cap-free exact
quotient `ExactPathClass` and the same `classMu` used by
`exactShellAmplitude`; it is neither a pre-quotient nor a cap-dependent mass.
-/
theorem zeroPhase_epsilon_one_failure (N : ℕ) :
    ∃ m n : ℕ, N ≤ m ∧ m ≤ n ∧
      1 < ‖∑ k ∈ Finset.Ico m n, exactShellAmplitude zeroPhase k‖ := by
  let k : ℕ := max 2 N
  have hNk : N ≤ k := le_max_right 2 N
  have hsingle :
      ∑ j ∈ Finset.Ico k (k + 1), exactShellAmplitude zeroPhase j =
        exactShellAmplitude zeroPhase k := by
    rw [Finset.sum_Ico_eq_sub (exactShellAmplitude zeroPhase)
      (Nat.le_succ k), Finset.sum_range_succ, add_sub_cancel_left]
  have hk2 : 2 ≤ k := le_max_left 2 N
  have hpowN : k ≤ k ^ (3 * k) :=
    Nat.le_self_pow (by omega) k
  have hpowR : ((k : ℕ) : ℝ) ≤ ((k : ℕ) : ℝ) ^ (3 * k) := by
    calc
      ((k : ℕ) : ℝ) ≤ ((k ^ (3 * k) : ℕ) : ℝ) := by
        exact_mod_cast hpowN
      _ = ((k : ℕ) : ℝ) ^ (3 * k) := Nat.cast_pow _ _
  have hkR : (1 : ℝ) < (k : ℕ) := by
    exact_mod_cast (show 1 < k by omega)
  have hmass : (1 : ℝ) < shellMass k := by
    have hlower := RegulatorRemovalNoGo.shellMass_lower k
    linarith
  refine ⟨k, k + 1, hNk, Nat.le_succ k, ?_⟩
  rw [hsingle, exactShellAmplitude_zeroPhase, Complex.norm_real,
    Real.norm_eq_abs, abs_of_pos (shellMass_pos k)]
  exact hmass

/-- Zero phase fails both available removal routes: the unregulated
complexity cutoff and the positive Gaussian regulator-removal limit. Neither
statement is a mesh-refinement or physical-continuum claim. -/
theorem zeroPhase_fails_both_removal_routes :
    (¬ HasExactComplexityCutoffLimit zeroPhase) ∧
      (¬ HasZRSRegulatorRemoval zeroPhase) :=
  ⟨not_hasExactComplexityCutoffLimit_zeroPhase,
    RegulatorRemovalNoGo.not_hasZRSRegulatorRemoval_zeroPhase⟩

/-- A zero-phase capped family cannot both agree with the exact-shell
decomposition and have a complexity-cutoff limit. This is a concrete
discriminant for any proposed bridge implementation. -/
theorem zeroPhase_compatibility_and_limit_impossible :
    ¬ (CapShellCompatibility zeroCapPhaseFamily zeroPhase ∧
      HasPhasedZqComplexityLimit zeroCapPhaseFamily) := by
  rintro ⟨hcompat, hlimit⟩
  exact zeroPhase_not_exactShellTailCancellation
    ((hasPhasedZqLimit_iff_exactShellTail_of_compatibility
      zeroCapPhaseFamily zeroPhase hcompat).mp hlimit)

What this page does not claim

This result does not prove that any particular nontrivial phase works, only that the zero phase fails. This result carries no claim about continuum geometry, observations, a convergence rate, or a derived measure. This result does not address the full-theory ledger of recognition events.

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expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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