IndisputableMonolith.Papers.GCIC.GCICDerivation
The GCICDerivation module assembles lemmas that derive discrete gauge identification and phase alignment from J-cost minimization on graphs, using the Recognition Science forcing chain. Researchers modeling holographic information or discrete symmetries in physics cite its chain from constant fields to vanishing couplings. The module proceeds by importing rigidity and potential results then chaining implications to obtain gcic_from_forcing_chain and related statements.
claimOn a connected graph with positive field $x$, the ratio energy $C_G[x] = 0$ if and only if $x$ is constant; the reduced phase potential satisfies $J̃_b(δ) = cosh(λ d_ℤ(δ)) - 1$ with $λ = ln b$, and 8-tick neutrality plus φ-self-similarity forces the discrete gauge $r ~ r + n · ln φ$.
background
The module sits inside the Recognition Science framework where the J-function is defined by $J(x) = (x + x^{-1})/2 - 1$. The Cost module supplies the total J-cost as the sum of per-edge terms $J(x_v/x_w)$. GraphRigidity proves that this sum vanishes on any finite connected graph precisely when the positive field is constant. DiscreteGauge encodes the gap-A result that 8-tick neutrality together with φ-self-similarity forces the discrete scaling identification $r ~ r + n · ln φ$. ReducedPhasePotential introduces the reduced mismatch potential $J̃_b(δ) = cosh(λ d_ℤ(δ)) - 1$ induced by the scaling quotient.
proof idea
The module imports the four supporting modules and assembles a linear chain of implications. It first invokes the rigidity theorem that zero total J-cost forces constant field, then applies the discrete-gauge identification to introduce the scaling quotient, and finally uses the reduced-phase-potential definition to show that phase alignment minimizes $J̃$ and that couplings vanish exactly when fields are aligned. All steps are one-line wrappers or direct algebraic reductions from the imported statements.
why it matters in Recognition Science
This module supplies the GCIC foundation required by the downstream BrainHolography derivation, which obtains surface-area scaling of accessible information from the same ledger. It closes the GCIC paper's acknowledged gap on discrete identification by linking the forcing chain (T0-T8) through graph rigidity and the reduced potential. The module thereby converts the abstract 8-tick octave into concrete statements about phase stiffness and collective cost.
scope and limits
- Does not derive the full T0-T8 forcing chain.
- Does not treat infinite or continuous graphs.
- Does not compute explicit numerical values for α or G.
- Does not address mass-ladder assignments or Berry thresholds.
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