Gravity Discriminator Matrix Cell Bohmian Echo Damping Positive

The Bohmian echo cell

In quantum gravity research, a discriminator is a measurable quantity that would give different values under different theories. The Recognition Science framework has built a 4 by 3 matrix of such discriminators, comparing its own predictions against four rival approaches: loop quantum gravity, string theory, causal dynamical triangulations, and Bohmian or stochastic-collapse models. Each cell of the matrix records a theorem-grade inequality, a formal statement proved in the framework's machine-checked library of formal theorems.

The specific declaration cell_Bohmian_EchoDamping_positive concerns the sector the framework calls echo damping. This is a predicted amplitude ratio, the factor by which a signal's strength is reduced as it passes through successive discrete steps of the framework's recognition ledger, a discrete record of events. The theorem states that this ratio is strictly greater than zero. That is the entire content of the cell: a positive number, nothing more.

The meaning of that positivity is comparative. The framework's entry for the Bohmian rival in the echo-damping sector is that these theories predict no quantum-gravity signal at all in this sector. Therefore, any positive value, no matter how small, would distinguish the framework's prediction from the rival's null prediction. The same positive-existence inequality appears in the cell for causal dynamical triangulations. Against loop quantum gravity and string theory, the matrix provides stronger claims with explicit numerical margins, such as an echo-damping ratio greater than one half.

What the declaration does not claim is equally precise. It does not assert that the echo-damping ratio has any particular value, only that it is positive. It does not claim that any experiment has measured this ratio. The matrix organizes theorem-grade discriminators; it does not supply the observational sensitivity numbers that a real detector would need, which remain a separate matter tied to instruments like LIGO, Virgo, and LISA. The declaration is a structural result about what the framework's mathematics entails, not a report of data.

/-- (Bohmian, EchoDamping): Bohmian / DP substrates do not produce this
φ-rung algebra.  This is not a claim of a closed observable echo mechanism. -/
theorem cell_Bohmian_EchoDamping_positive : 0 < echoDampingRatio :=
  echoDampingRatio_pos

/-- Rival predictions for each sector. `Option ℝ`: `some r` if the rival
has a specific value; `none` if the rival predicts no signal in this
sector (RS discriminates by positive existence). -/
def rivalPrediction : Rival → Sector → Option ℝ
  | .LQG,     .LeadingLog   => some (-1 / 2)   -- LQG area-quantum: c = -1/2
  | .LQG,     .EchoDamping  => some (1 / 2)    -- Uniform-discreteness echoes
  | .LQG,     .RungPhase    => some (1 / 2)    -- Half-quantum phase delay
  | .String,  .LeadingLog   => some (-3 / 2)   -- Strominger-Vafa: c = -3/2
  | .String,  .EchoDamping  => some (1 / 2)    -- String fuzzball uniform damping
  | .String,  .RungPhase    => none             -- Depends on string scale; not a clean prediction
  | .CDT,     .LeadingLog   => none             -- CDT does not give phi-rational leading-log
  | .CDT,     .EchoDamping  => none             -- No matching φ-rung amplitude algebra
  | .CDT,     .RungPhase    => none             -- No matching φ-rung phase algebra
  | .Bohmian, .LeadingLog   => none             -- Bohmian/DP do not produce phi-rational signals
  | .Bohmian, .EchoDamping  => none             -- No matching φ-rung amplitude algebra
  | .Bohmian, .RungPhase    => none             -- No matching φ-rung phase algebra

What this page does not claim

The declaration does not assign a specific numerical value to the echo-damping ratio. The declaration does not report any experimental measurement of the ratio. The declaration does not provide the sensitivity requirements for a real detector.

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/DiscriminatorMatrix.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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