IndisputableMonolith.Papers.GCIC.GCICDerivation
GCICDerivation assembles the derivation of GCIC results from Recognition Science primitives on graphs. It shows J-cost minimization forces constant positive fields and incorporates discrete gauge from 8-tick neutrality plus phi-self-similarity. Researchers on holographic models or discrete scaling in RS cite the module for closing the GCIC paper gap. The structure composes lemmas from GraphRigidity and ReducedPhasePotential into statements such as gcic_from_forcing_chain.
claimOn a finite connected graph $G$ with positive field $x$, the ratio energy $C_G[x] = 0$ if and only if $x$ is constant. The reduced phase potential is defined by $J̃_b(δ) = cosh(λ · d_ℤ(δ)) - 1$ where $λ = ln b$ and $d_ℤ$ is distance to the nearest integer. Discrete gauge identification $r ~ r + n · ln φ$ follows from 8-tick neutrality and φ-self-similarity.
background
The module sits inside the GCIC Response paper treatment of Gap A on dynamical discrete identification. It imports Cost for the J functional, GraphRigidity for ratio-energy vanishing, ReducedPhasePotential for the phase-mismatch induced by discrete scaling quotients, and DiscreteGauge for the 8-tick neutrality mechanism. GraphRigidity states: 'Machine-verified proof that on any finite connected graph, the ratio energy C_G[x] = Σ J(x_v/x_w) vanishes if and only if x is a constant positive field.' ReducedPhasePotential supplies J̃_b(δ) = cosh(lam · d_ℤ(δ)) − 1.
proof idea
This is a derivation module whose argument assembles imported results rather than introducing new core proofs. It applies the rigidity lemma to obtain constant-field conclusions, composes the reduced potential with phase-alignment statements, and invokes the discrete-gauge result to reach gcic_from_forcing_chain and the listed sibling lemmas.
why it matters in Recognition Science
The module supplies the mathematical backbone that feeds BrainHolography, which derives that every local ledger region encodes global state and that accessible information scales with surface area. It closes GCIC Gap A by linking the forcing chain (T7 eight-tick octave, T6 phi self-similarity) to the discrete gauge and constant-field statements required for the holographic derivation.
scope and limits
- Does not prove GCIC results outside the imported Cost, rigidity, and phase-potential components.
- Does not treat infinite graphs or continuous-field limits.
- Does not verify numerical values such as the alpha band.
- Does not address dynamical evolution beyond the static minimization statements.
used by (1)
depends on (4)
declarations in this module (15)
-
theorem
J_stationary_implies_constant_field -
theorem
constant_field_constant_phase -
theorem
gcic_from_forcing_chain -
theorem
phase_alignment_minimizes_Jtilde -
theorem
coupling_vanishes_iff_aligned -
theorem
theta_coupling_stiffness -
theorem
neg_Jtilde_coupling_nonpos -
theorem
neg_Jtilde_coupling_zero_iff -
theorem
neg_Jtilde_coupling_periodic -
theorem
phi_coupling_stiffness -
theorem
aligned_collective_cost_zero -
theorem
perturbed_cost_positive -
theorem
list_sum_pos -
theorem
aligned_beats_perturbed -
theorem
gcic_derivation_cert