Encyclopedia Gravity Gravity Seven Gaps Stationarity Bridge Closure Quadratic Source Family Source Do

ARTICLE 3 claims 3 theorems

Gravity Seven Gaps Stationarity Bridge Closure Quadratic Source Family Source Do

A machine-checked theorem shows a specific family of source strengths stays small enough for a derived bridge to hold, and the proof is not vacuous.

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The quadratic source family

In the Recognition Science framework, a recognition event is a discrete record of a comparison, and its cost is the price the ledger pays for that record. The framework's library, a machine-checked collection of formal theorems, studies how these costs behave. One question is whether a bridge structure, a derived relation between two recognition ratios, can be built from a stationarity principle instead of being assumed. The declaration quadraticSourceFamily_source_dominated is a theorem that helps answer yes for a particular family of source strengths.

The theorem concerns a family where the deficit, a measure of geometric imbalance, scales with the square of the mesh size h. Specifically, the deficit is (n/kappa) * h^2, where n is the number of channels and kappa is a coupling constant. The theorem states that if the mesh is fine (0 < h) and not too coarse (h ≤ 1), then the absolute value of kappa times the deficit is bounded by n * h. In plain terms, the source strength, the product of the coupling and the deficit, is dominated by the mesh scale. This domination condition is exactly what the bridge constructor requires as a hypothesis.

This is not a trivial point. The framework's library also contains a kill record, a theorem showing that a linear deficit family, where the deficit scales with h instead of h^2, fails the admissibility predicate for any constants. The quadratic family is the correctly-typed analogue that satisfies the condition. The theorem quadraticSourceFamily_source_dominated is one of several that together show this family is admissible and non-vacuous, meaning it produces a genuine, non-zero deficit and positive log ratios. It is a piece of the proof that the bridge is inhabited by derivation, not by assumption.

What the theorem does not claim is just as important. It does not claim that the bridge is derived from the bare recognition ledger alone. The deficit-source coupling inside the action is a disclosed constitutive MODEL premise, inherited from the core stationarity module. The theorem also does not claim to build a mesh-indexed tower of bridge structures; it applies at each fixed mesh, with the family theorem supplying uniform constants. The derivation from a bare ledger is permanently false, a status flag that stays false.

THEOREM quadraticSourceFamily_source_dominated · IndisputableMonolith/Gravity/SevenGaps/StationarityBridgeClosure.lean
quadraticSourceFamily_source_dominated · IndisputableMonolith/Gravity/SevenGaps/StationarityBridgeClosure.lean:436 
/-- **THEOREM (family members feed the constructor).** For every mesh
0 < h <= 1 the quadratic family's source obeys the constructor's
domination side condition |kappa * delta(h)| <= n*h (since
|kappa * delta(h)| = n h^2 <= n h). So each family member is directly a
`recognitionRatioBridge_ofStationarity` input: the uniform-family
admissibility (T2) and the bridge-by-derivation construction (T3) cover
the same objects. -/
theorem quadraticSourceFamily_source_dominated (n : ℕ) (hn : 1 ≤ n)
    (kappa h : ℝ) (hκ : kappa ≠ 0) (hh0 : 0 < h) (hh1 : h ≤ 1) :
    |kappa * (quadraticSourceFamily n kappa).deficit h| ≤ (n : ℝ) * h := by
  have hn0 : (0 : ℝ) < n := by exact_mod_cast hn
  show |kappa * ((n : ℝ) / kappa * h ^ 2)| ≤ (n : ℝ) * h
  have heq : kappa * ((n : ℝ) / kappa * h ^ 2) = (n : ℝ) * h ^ 2 := by
    field_simp
  rw [heq, abs_of_nonneg (by positivity : (0 : ℝ) ≤ (n : ℝ) * h ^ 2)]
  have hsq : h ^ 2 ≤ h := by nlinarith [hh0, hh1]
  exact mul_le_mul_of_nonneg_left hsq hn0.le
THEOREM linear_deficit_family_not_isAdmissible · IndisputableMonolith/Gravity/SevenGaps/StationarityBridgeClosure.lean
linear_deficit_family_not_isAdmissible · IndisputableMonolith/Gravity/SevenGaps/StationarityBridgeClosure.lean:318 
/-- **THEOREM (kill record for the panel's literal linear family).** For
every n >= 1, kappa ≠ 0, h0 > 0 and EVERY pair of constants (C_K, C_R),
the linear-deficit sourced family delta(h) = (n/kappa) * h (the
mesh-indexed form of the panel's delta_m = 1/(m+1), h_m = |kappa|/(n(m+1)),
which has kappa*delta/n = h) is NOT admissible: the curvature conjunct
|delta(h)| <= C_K h^2 of `RecognitionRatioFamily.IsAdmissible` fails at
small h because a linear deficit cannot be dominated by h^2 uniformly.
This is why T2 is closed with the quadratic-deficit family below; the
deviation from the panel's literal spec is forced by the predicate's
curvature conjunct, and this theorem is the receipt. -/
theorem linear_deficit_family_not_isAdmissible (n : ℕ) (hn : 1 ≤ n)
    (h₀ kappa C_K C_R : ℝ) (hh₀ : 0 < h₀) (hκ : kappa ≠ 0) :
    ¬ (sourcedRatioFamily n kappa
        (fun h => (n : ℝ) / kappa * h)).IsAdmissible h₀ kappa C_K C_R := by
  intro hadm
  have hn0 : (0 : ℝ) < n := by exact_mod_cast hn
  have hk : 0 < |kappa| := abs_pos.mpr hκ
  set M := max C_K 0 with hMdef
  have hM0 : 0 ≤ M := le_max_right _ _
  have hden : 0 < |kappa| * (M + 1) := by positivity
  set h := min (h₀ / 2) ((n : ℝ) / (|kappa| * (M + 1))) with hdef
  have hhpos : 0 < h := lt_min (by linarith) (div_pos hn0 hden)
  have hhlt : h < h₀ := lt_of_le_of_lt (min_le_left _ _) (by linarith)
  obtain ⟨hcurv, _⟩ := hadm h ⟨hhpos, hhlt⟩
  have hcurv' : (n : ℝ) / |kappa| * h ≤ C_K * h ^ 2 := by
    have hc : |(n : ℝ) / kappa * h| ≤ C_K * h ^ 2 := hcurv
    rw [abs_mul, abs_div, abs_of_pos hn0, abs_of_pos hhpos] at hc
    exact hc
  have hCM : C_K * h ^ 2 ≤ M * h ^ 2 :=
    mul_le_mul_of_nonneg_right (le_max_left _ _) (sq_nonneg h)
  have e1 : (n : ℝ) * h ≤ |kappa| * M * h ^ 2 := by
    calc (n : ℝ) * h = |kappa| * ((n : ℝ) / |kappa| * h) := by
          field_simp
      _ ≤ |kappa| * (M * h ^ 2) :=
          mul_le_mul_of_nonneg_left (le_trans hcurv' hCM) hk.le
      _ = |kappa| * M * h ^ 2 := by ring
  have e2 : h * (|kappa| * (M + 1)) ≤ (n : ℝ) := by
    have hmin : h ≤ (n : ℝ) / (|kappa| * (M + 1)) := by
      rw [hdef]
      exact min_le_right _ _
    exact (le_div_iff₀ hden).mp hmin
  have e3 : h * (|kappa| * (M + 1)) * (M * h) ≤ (n : ℝ) * (M * h) :=
    mul_le_mul_of_nonneg_right e2 (mul_nonneg hM0 hhpos.le)
  have e4 : (n : ℝ) * h * (M + 1) ≤ |kappa| * M * h ^ 2 * (M + 1) :=
    mul_le_mul_of_nonneg_right e1 (by linarith)
  have hnh : 0 < (n : ℝ) * h := mul_pos hn0 hhpos
  nlinarith [e3, e4, hnh]
THEOREM quadraticSourceFamily_isAdmissible · IndisputableMonolith/Gravity/SevenGaps/StationarityBridgeClosure.lean
quadraticSourceFamily_isAdmissible · IndisputableMonolith/Gravity/SevenGaps/StationarityBridgeClosure.lean:376 
/-- **THEOREM (T2, uniform admissibility of the quadratic family).** For
kappa ≠ 0 the quadratic-deficit family is admissible on (0, h0) with the
UNIFORM constants C_K = n/|kappa| and C_R = n * h0^3 / 6: both constants
sit OUTSIDE the mesh scale, per the admissibility predicate, so the
derivation covers a genuine h -> 0 family, not a single point. Proof:
`sourced_ratio_isAdmissible` (reused, not re-derived) with the curvature
bound an exact equality |delta(h)| = (n/|kappa|) h^2, followed by the
constant simplification |kappa|^3 (n/|kappa|)^3 h0^3/(6 n^2)
= n h0^3/6. (As in `sourced_ratio_isAdmissible`: for h0 <= 0 the predicate
is vacuously true; the statement carries content exactly when 0 < h0.) -/
theorem quadraticSourceFamily_isAdmissible (n : ℕ) (hn : 1 ≤ n)
    (h₀ kappa : ℝ) (hκ : kappa ≠ 0) :
    (quadraticSourceFamily n kappa).IsAdmissible h₀ kappa
      ((n : ℝ) / |kappa|) ((n : ℝ) * h₀ ^ 3 / 6) := by
  have hn0 : (0 : ℝ) < n := by exact_mod_cast hn
  have hne : (n : ℝ) ≠ 0 := ne_of_gt hn0
  have hκ' : |kappa| ≠ 0 := abs_ne_zero.mpr hκ
  have hδ : ∀ h ∈ Set.Ioo (0 : ℝ) h₀,
      |(n : ℝ) / kappa * h ^ 2| ≤ (n : ℝ) / |kappa| * h ^ 2 := by
    intro h _
    rw [abs_mul, abs_div, abs_of_pos hn0, abs_of_nonneg (sq_nonneg h)]
  have hbase := sourced_ratio_isAdmissible n hn h₀ kappa
    ((n : ℝ) / |kappa|) (fun h => (n : ℝ) / kappa * h ^ 2) hδ
  have hconst : |kappa| ^ 3 * ((n : ℝ) / |kappa|) ^ 3 * h₀ ^ 3
      / (6 * (n : ℝ) ^ 2) = (n : ℝ) * h₀ ^ 3 / 6 := by
    field_simp
  rw [← hconst]
  exact hbase

What this page does not claim

This theorem does not derive the bridge from the bare recognition ledger alone. This theorem does not establish the physical recognition-to-linking bridge for three spatial dimensions. This theorem does not construct a mesh-indexed tower of bridge structures.

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

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Derived articles

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