Gravity Seven Gaps Zq Continuum Blocker Has Phased Zq Complexity Limit Iff Cauch

The Cauchy criterion

In mathematics, a sequence of numbers has a limit if and only if it is a Cauchy sequence: for any chosen tolerance, all terms far enough out lie within that tolerance of each other. This is not a special property of real numbers; it is the definition of completeness for the complex numbers. The theorem hasPhasedZqComplexityLimit_iff_cauchy applies that classical fact to a specific family of sequences that arise in a model of gravity.

The sequences come from a ledger, a discrete record of events, that sums amplitudes over path classes up to a complexity cutoff. The framework models a family of such finite sums, one for each cutoff value, and asks whether they settle down to a single complex number as the cutoff grows. The theorem proves that this happens exactly when the sequence meets the Cauchy criterion: for any small tolerance, all sufficiently late sums differ by less than that tolerance. This is a direct application of the completeness of the complex numbers, and it is machine-checked in the framework's library of formal theorems.

The theorem is discriminating. For the zero phase, where every path gets the same weight, the library proves an explicit failure: no matter how far out you start, you can find two later sums that differ by more than one. The zero-phase sequence is not Cauchy, so it has no limit. This shows the criterion is not vacuous; it separates phases that cancel cleanly from those that do not.

One bridge is still absent. The capped sequence uses one kind of path class, while the exact shell decomposition uses another. The statement that connects convergence of the capped sequence to the exact-shell tail-cancellation criterion requires a compatibility statement, named CapShellCompatibility, that the two finite sums agree at every cap. Until that bridge is proved, the theorem applies to the exact-shell sums, not yet to the capped Zq sequence.

What the theorem does not claim is as important as what it proves. It says nothing about mesh refinement, continuum geometry, observations, a convergence rate, a derived measure, or the full-theory ledger. It is a statement about removing a complexity cutoff in one specific family of finite sums, and nothing more.

/-- **HEADLINE IFF.** The existing phased quotient sums have a
complexity-cutoff limit exactly when their cross-cap differences are Cauchy.
This theorem assumes no convergence and no physical continuum interpretation.
-/
theorem hasPhasedZqComplexityLimit_iff_cauchy (P : CapPhaseFamily) :
    HasPhasedZqComplexityLimit P ↔ PhasedZqCauchyCriterion P := by
  constructor
  · rintro ⟨L, hL⟩
    have hC := (Metric.cauchySeq_iff.mp hL.cauchySeq)
    intro ε hε
    obtain ⟨N, hN⟩ := hC ε hε
    refine ⟨N, fun m hm n hn => ?_⟩
    simpa only [dist_eq_norm] using hN m hm n hn
  · intro h
    have hC : CauchySeq (phasedZqSequence P) := by
      rw [Metric.cauchySeq_iff]
      exact h
    exact cauchySeq_tendsto_of_complete hC

/-- **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

What this page does not claim

The theorem does not claim that the capped Zq sequence converges, only the exact-shell sums under a missing bridge. It does not claim anything about mesh refinement, continuum geometry, or observations. It does not claim a convergence rate or a derived measure.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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