Gravity Discriminator Matrix Cell Cdt Leading Log Distinct

The CDT cell

Black hole entropy carries a quantum correction. The famous area law, where entropy is proportional to the horizon area, is the leading term; the next term, the leading-log correction, is a coefficient that multiplies the logarithm of that area. In the Recognition Science framework, this coefficient, written c_RS, is not a free parameter. The framework derives it from the same forced cost function that produces other constants, and the derivation yields a definite sign: c_RS is negative.

The theorem cell_CDT_LeadingLog_distinct states exactly that inequality, c_RS < 0. Its meaning is comparative. Causal Dynamical Triangulations, or CDT, a rival discrete approach to quantum gravity, predicts no such quantum-gravity signal in this sector; its leading-log coefficient is zero by construction. The framework's prediction of a strictly negative value therefore separates the two approaches: if the coefficient is measured to be negative, CDT's silent prediction is falsified, and if it is measured to be zero, the framework's derivation is wrong. The theorem does not say how negative the coefficient is, only that it is not zero and not positive.

The claim is a structural theorem in the framework's machine-checked library of formal theorems, carrying no empirical input and no fitted parameters. It belongs to a larger discriminator matrix that compares the framework against four rival approaches across three observational sectors. For CDT and Bohmian approaches, which predict no signal, any positive or negative deviation suffices to distinguish; the theorem supplies the negative sign. For Loop Quantum Gravity and string theory, which predict specific nonzero values, the matrix instead provides explicit numerical margins, such as a gap of more than one quarter between the framework's coefficient and LQG's value of negative one half.

The theorem's scope is deliberately narrow. It establishes a sign, not a magnitude, and it says nothing about whether the coefficient can be measured with current instruments. The framework's own plan keeps the dataset-tied falsifier register separate, requiring specific experimental sensitivity numbers from observatories like LIGO, Virgo, and LISA before a measurement can be declared decisive. What the theorem does change is the logical setting: a single sign, if observed, would eliminate an entire class of discrete quantum-gravity theories at once.

/-- (CDT, LeadingLog): CDT does not produce a φ-rational leading-log
coefficient (no recognition ledger). RS predicts `c_RS < 0` distinct
from any CDT zero-prediction. -/
theorem cell_CDT_LeadingLog_distinct : c_RS < 0 :=
  c_RS_neg

/-- (CDT, LeadingLog): CDT does not produce a φ-rational leading-log
coefficient (no recognition ledger). RS predicts `c_RS < 0` distinct
from any CDT zero-prediction. -/
theorem cell_CDT_LeadingLog_distinct : c_RS < 0 :=
  c_RS_neg

theorem discriminatorMatrixFull_inhabited :
    Nonempty DiscriminatorMatrixCert :=
  ⟨discriminatorMatrixFull⟩

What this page does not claim

The theorem does not predict a numerical magnitude for c_RS, only its negative sign. The theorem does not claim that current instruments can measure the leading-log coefficient. The theorem does not address the other sectors of the discriminator matrix, such as echo damping or rung phase.

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)

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