IndisputableMonolith.Foundation.HamiltonianEmergence
The HamiltonianEmergence module derives quadratic energy terms from the J-cost function by restricting to near-equilibrium bond configurations. It defines SmallDeviationState for multipliers of the form 1 + ε_i with |ε_i| small and shows that totalJcost then approximates a quadratic form suitable for Hamiltonian dynamics. Researchers deriving effective dynamics from recognition axioms cite this step for the emergence of classical and quantum energy functionals. The module structure consists of targeted definitions and approximations built on Jx
claimLet bond multipliers satisfy $x_i = 1 + ε_i$ with $|ε_i| < δ$ for small $δ$. Then the total J-cost satisfies $J_{total} ≈ (1/2) ∑ ε_i^2$ under the Recognition Composition Law, yielding a quadratic energy functional on the deviation space.
background
Recognition Science obtains all physics from the J functional obeying J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The upstream JcostCore module supplies the definition of totalJcost as the sum of J(x_b) over bonds b together with the associated defectDist measure. This module specializes those objects to the small-deviation regime where each multiplier lies close to unity, permitting a Taylor expansion of J around 1 that produces the quadratic term. The local setting is the foundation layer that precedes discrete evolution and Hamiltonian operator construction.
proof idea
This is a definition module, no proofs. It introduces SmallDeviationState, totalJcost, quadratic_emergence, quadraticEnergy, DeviationHilbert, embed, DiscreteEvolution, diagonalHamiltonian and H_HamiltonianIsGenerator to organize the quadratic approximation and its embedding into a Hilbert space.
why it matters in Recognition Science
The module supplies the quadratic energy approximation required for Hamiltonian emergence in the Recognition framework. It feeds the discrete-evolution and generator theorems that appear downstream, closing the step from J-cost to dynamics while remaining inside the phi-ladder and eight-tick octave structure. The construction directly supports the later identification of the Hamiltonian as the generator of time evolution.
scope and limits
- Does not treat large deviations from equilibrium.
- Does not derive explicit numerical constants such as alpha or G.
- Does not address statistics beyond the quadratic bosonic sector.
- Does not prove global existence or stability of the resulting dynamics.
depends on (1)
declarations in this module (13)
-
structure
SmallDeviationState -
def
totalJcost -
theorem
quadratic_emergence -
def
quadraticEnergy -
theorem
per_bond_remainder_bounded -
theorem
totalJcost_approx_quadratic -
abbrev
DeviationHilbert -
def
embed -
theorem
embed_norm_sq -
structure
DiscreteEvolution -
def
diagonalHamiltonian -
def
H_HamiltonianIsGenerator -
theorem
emergence_scalar_proved