Gravity Seven Gaps Exact Shell Gauge Uv Z Rsuvshell Zero Phase Re Pos

The zero-phase sum

The declaration zRSUVShell_zeroPhase_re_pos concerns a mathematical object called Z_RS_uv, which in the Recognition Science framework is a path sum: a way of adding up contributions from many possible discrete configurations, here called complexes, each made of vertices, edges, and tetrahedra. The sum is built from shells, where each shell collects all complexes with the same complexity, measured as the largest of three counts: vertices, edges, or tetrahedra. The theorem states that when every configuration is assigned the same phase (zero phase, meaning no oscillating weight), and when a Gaussian regulator of the form exp(-ρ·n²) is applied with any positive strength ρ, the real part of the total sum is strictly greater than zero.

This is a theorem in the machine-checked library of formal theorems, proved with no gaps and no extra axioms. It is not a statement about physics directly. The regulator is a mathematical device inserted by hand, not derived from any principle. The action or phase that would normally appear in a path sum is here just a parameter, an arbitrary function on equivalence classes of complexes; no physical action is derived. The theorem is a statement about the convergence and positivity of a particular formal series, given those choices.

The proof rests on several earlier results. Each shell is a finite set, and its size is bounded by (n+1)^(12·(n+1)). Each configuration class carries a measure equal to 1 divided by the size of its automorphism group, a standard symmetry-factor convention. The regulated shell term satisfies a modulus bound, and the shell series is summable for every positive ρ. The cutoff partial sums converge to Z_RS_uv. At zero phase, the regulated sum is real and strictly positive. This positivity is called non-vacuity: it shows the construction is not empty or degenerate.

What the theorem does not claim is as important as what it does. It does not claim that the regulator can be removed, that is, that the limit as ρ approaches 0 from above exists; that remains an open problem, recorded as a named target. It does not claim that the complexity cutoff represents a physical continuum limit; the cutoff is not mesh refinement, and the continuum limit stays unproved. It does not claim that any physical action or coupling constant is derived. The theorem is a formal result about a specific regularized sum, not a statement about the physical world.

In the Recognition Science framework, this theorem is a step toward organizing the configuration space of discrete gravity into exact shells. The practical consequence is that one can work with a well-defined, convergent, positive object at finite regulator strength, and study its properties without worrying about divergences from double-counting. The next step, removing the regulator, remains open.

/-- **THEOREM (S2e, non-vacuity).**  At zero phase the regulated path sum
has strictly positive real part for every `ρ > 0`: the regulated theory
is not the zero functional.  (All terms are nonnegative real and the
`n = 0` term is positive; positivity passes to the `tsum`.) -/
theorem Z_RS_uv_zeroPhase_re_pos (ρ : ℝ) (hρ : 0 < ρ) :
    0 < (Z_RS_uv ρ zeroPhase).re := by
  have hsC : Summable
      (fun n : ℕ => ((Real.exp (-ρ * (n : ℝ) ^ 2) * shellMass n : ℝ) : ℂ)) :=
    (summable_zRSUVShell ρ hρ zeroPhase).congr
      (fun n => zRSUVShell_zeroPhase_eq ρ n)
  have hsR : Summable
      (fun n : ℕ => Real.exp (-ρ * (n : ℝ) ^ 2) * shellMass n) :=
    Complex.summable_ofReal.mp hsC
  have hZ : Z_RS_uv ρ zeroPhase =
      ((∑' n : ℕ, Real.exp (-ρ * (n : ℝ) ^ 2) * shellMass n : ℝ) : ℂ) := by
    unfold Z_RS_uv
    rw [tsum_congr (fun n => zRSUVShell_zeroPhase_eq ρ n),
      ← Complex.ofReal_tsum]
  rw [hZ, Complex.ofReal_re]
  exact hsR.tsum_pos
    (fun n => (mul_pos (Real.exp_pos _) (shellMass_pos n)).le) 0
    (mul_pos (Real.exp_pos _) (shellMass_pos 0))

/-- **THEOREM (shell entropy bound, S1c).**  The number of combinatorially
distinct exact complexes of complexity `n` is at most `(n+1)^(12·(n+1))`.
Counting audit: `≤ (n+1)³` signature choices (`shellSig_card_le`), times
`≤ (n+1)^(6·(n+1))` labeled configurations per signature
(`exactComplex_card_le`, quotient card ≤ labeled card via the surjection
`Quotient.mk`), and `3 + 6·(n+1) ≤ 12·(n+1)`. -/
theorem exactPathClass_card_le (n : ℕ) :
    Nat.card (ExactPathClass n) ≤ (n + 1) ^ (12 * (n + 1)) := by
  have hfiber : ∀ s : ShellSig n,
      Nat.card (Quotient (exactSetoid (sigV s) (sigE s) (sigT s))) ≤
        (n + 1) ^ (6 * (n + 1)) := by
    intro s
    have hsurj : Nat.card (Quotient (exactSetoid (sigV s) (sigE s) (sigT s))) ≤
        Nat.card (ExactComplex (sigV s) (sigE s) (sigT s)) :=
      Nat.card_le_card_of_surjective
        (Quotient.mk (exactSetoid (sigV s) (sigE s) (sigT s)))
        (fun q => Quotient.exists_rep q)
    have hlab : Nat.card (ExactComplex (sigV s) (sigE s) (sigT s)) ≤
        (n + 1) ^ (6 * (n + 1)) := by
      rw [Nat.card_eq_fintype_card]
      exact exactComplex_card_le n _ _ _ (le_of_lt s.1.1.isLt)
        (le_of_lt s.1.2.1.isLt) (le_of_lt s.1.2.2.isLt)
    exact le_trans hsurj hlab
  rw [Nat.card_sigma]
  calc ∑ s : ShellSig n,
        Nat.card (Quotient (exactSetoid (sigV s) (sigE s) (sigT s)))
      ≤ ∑ _s : ShellSig n, (n + 1) ^ (6 * (n + 1)) :=
        Finset.sum_le_sum fun s _ => hfiber s
    _ = Fintype.card (ShellSig n) * (n + 1) ^ (6 * (n + 1)) := by
        rw [Finset.sum_const, Finset.card_univ, smul_eq_mul]
    _ ≤ (n + 1) ^ 3 * (n + 1) ^ (6 * (n + 1)) :=
        Nat.mul_le_mul_right _ (shellSig_card_le n)
    _ = (n + 1) ^ (3 + 6 * (n + 1)) := by rw [← pow_add]
    _ ≤ (n + 1) ^ (12 * (n + 1)) :=
        Nat.pow_le_pow_right (Nat.succ_pos n) (by omega)

/-- **THEOREM (S1d, unbounded support).**  EVERY shell is inhabited: the
`n`-isolated-vertices complex has complexity exactly `n`, so no shell is
eventually empty. -/
theorem exactPathClass_unbounded_support (n : ℕ) :
    0 < Nat.card (ExactPathClass n) :=
  Nat.card_pos

What this page does not claim

The regulator removal limit exists. The complexity cutoff represents a physical continuum limit. Any physical action or coupling constant is derived.

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/ExactShellGaugeUV.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.

Derived articles

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