Gravity Analysis Regge Ttsymbol Specification Audit Tendsto Const Mul Punctured

The scaling guard

In analysis, a common maneuver is to rescale a variable and ask whether a limit still behaves. The declaration tendsto_const_mul_punctured is a precise, machine-checked statement of one such maneuver: for any nonzero real number c, the map t ↦ c·t sends every punctured neighborhood of 0 back into a punctured neighborhood of 0. In plain words, multiplying by a nonzero constant cannot push a point that is close to, but not equal to, zero onto zero itself, nor can it drag a point from far away into that punctured vicinity. The result is a theorem in the framework's machine-checked library of formal theorems, proved without any additional axioms beyond the standard logical ones.

Why does this matter? The theorem is the technical hinge in a larger argument about a proposed constant value, −1/4, for a certain continuum symbol in a gravity analysis. The symbol is defined through a limit as an amplitude t approaches zero, but the amplitude is not the only thing that can be rescaled. The polarization matrix E, which describes the wave's transverse-traceless character, can also be multiplied by a constant c. The framework proves that this rescaling of E is equivalent to rescaling the amplitude t by the same factor, and that the symbol's value then scales by c². Without a guard like tendsto_const_mul_punctured, the limit defining the symbol could be manipulated by rescaling, and the fixed value −1/4 would be meaningless: the same physical family would report both H and c²·H.

The theorem itself does not establish that the symbol's value is −1/4. It only certifies that the limit operation is well-behaved under rescaling, which is a prerequisite for any fixed-value claim to be well-posed. The framework then adds a normalization condition, fixing the Frobenius norm of the polarization to 1, so that the only rescalings that stay within the allowed class are c² = 1, under which the reported value is invariant. The declaration tendsto_const_mul_punctured is thus a silent but load-bearing part of that well-posedness argument: it ensures the limit that defines the symbol is stable under the very rescalings that would otherwise make a fixed constant contradictory.

What a reader can now see is the difference between a statement being meaningful and a statement being true. The framework's library shows that the −1/4 target is scaling-coherent: it is a statement that can be true or false without falling apart under reparametrization. Whether the value actually equals −1/4 remains an open target, explicitly not proved and not evidenced by this theorem. The declaration is a guardrail, not a destination.

/-- Multiplication by a nonzero constant maps the punctured neighborhood
filter of `0` to itself. -/
theorem tendsto_const_mul_punctured {c : ℝ} (hc : c ≠ 0) :
    Filter.Tendsto (fun t : ℝ => c * t)
      (nhdsWithin 0 {(0 : ℝ)}ᶜ) (nhdsWithin 0 {(0 : ℝ)}ᶜ) := by
  have h1 : Filter.Tendsto (fun t : ℝ => c * t) (nhds 0) (nhds 0) := by
    simpa using (continuous_const.mul continuous_id).tendsto (0 : ℝ)
  refine tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
    (h1.mono_left nhdsWithin_le_nhds) ?_
  filter_upwards [self_mem_nhdsWithin] with t ht
  have ht' : t ≠ 0 := ht
  exact mul_ne_zero hc ht'

/-- Multiplication by a nonzero constant maps the punctured neighborhood
filter of `0` to itself. -/
theorem tendsto_const_mul_punctured {c : ℝ} (hc : c ≠ 0) :
    Filter.Tendsto (fun t : ℝ => c * t)
      (nhdsWithin 0 {(0 : ℝ)}ᶜ) (nhdsWithin 0 {(0 : ℝ)}ᶜ) := by
  have h1 : Filter.Tendsto (fun t : ℝ => c * t) (nhds 0) (nhds 0) := by
    simpa using (continuous_const.mul continuous_id).tendsto (0 : ℝ)
  refine tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
    (h1.mono_left nhdsWithin_le_nhds) ?_
  filter_upwards [self_mem_nhdsWithin] with t ht
  have ht' : t ≠ 0 := ht
  exact mul_ne_zero hc ht'

/-- **GATE A0 VERDICT (THEOREM): the `-1/4` target statement is
well-posed under the `polEdgeCoeff` scaling convention.**

The three clauses, all kernel-checked, assemble the well-posedness
argument:

1. (pin) every `E` in the target's hypothesis class `IsTTPolarization`
   has Frobenius square-sum exactly `1`;
2. (quadratic scaling) the Bloch symbol value transforms as `H ↦ c²·H`
   under `E ↦ c·E`, so WITHOUT a normalization the fixed-value claim
   would be contradictory (the same family would report `H` and `c²·H`);
3. (pin bites) the only rescalings that stay inside the hypothesis class
   are `c² = 1`, and for those `c²·H = H` — the reported value is
   invariant on the quantified class.

Hence `ReggeTTContinuumIsotropyTarget`, which quantifies over
`IsTTPolarization` (pin included), assigns a scaling-coherent meaning to
the fixed constant `reggeTTContinuumCoefficient = -(1/4)`.  Nothing here
proves (or evidences) that the value IS `-1/4`; that target remains OPEN
with status flag `false`. -/
theorem reggeTT_target_scaling_wellPosed {c : ℝ} (hc : c ≠ 0)
    (m : Fin 3 → ℤ) (E : Fin 3 → Fin 3 → ℝ) (H : ℝ)
    (hE : IsTTPolarization m E) :
    frobeniusSq E = 1 ∧
      (TTBlochSymbolIs N E m H ↔ TTBlochSymbolIs N (c • E) m (c ^ 2 * H)) ∧
      (IsTTPolarization m (c • E) ↔ c ^ 2 = 1) :=
  ⟨isTTPolarization_frobenius_pinned m E hE,
    TTBlochSymbolIs_smul N hc E m H,
    isTTPolarization_smul_iff m E c hE⟩

What this page does not claim

The value −1/4 is not proved or evidenced by this theorem. The theorem does not assert that the limit defining the symbol exists for any particular polarization. The theorem does not address whether the −1/4 target is physically correct, only that it is well-posed under the stated scaling convention.

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/Analysis/ReggeTTSymbolSpecificationAudit.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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