IndisputableMonolith.Unification.RecognitionBandwidth
Defines recognition bandwidth from the eight-tick cadence and holographic ledger capacity. Unification and information-bound arguments cite it when converting surface area into a recognition processing rate. The module fixes one full recognition cycle at eight ticks, then sets bandwidth equal to holographic bits divided by recognition cost, with positivity, monotonicity, and linearity lemmas.
claimOne full recognition cycle occupies eight ticks $\tau_0$. For a surface of area $A$, the recognition bandwidth is $B(A)=N_{\mathrm{hol}}(A)/C$, where $N_{\mathrm{hol}}(A)$ is the holographic bit count on that surface and $C$ is the recognition cost per cycle. The module records $B(A)>0$ (for admissible $A$), monotonicity in $A$, linearity in the bit count, and the identity $B=N_{\mathrm{hol}}/C$.
background
Recognition Science treats time in discrete ticks. The fundamental RS time quantum is $\tau_0=1$ tick (Constants). The forcing chain fixes an eight-tick octave: period $2^3$, the minimum closed recognition cycle (T7). This module takes that cadence as the wall-clock cost of one complete $\hat R$ event.
Information capacity enters through the holographic bound (QG-006): ledger information in a region is bounded by boundary area, not volume. The sibling holographicBits packages that area-to-bits map; Planck-area positivity keeps the denominator well-defined.
Cost comes from the RS cost layer and the Boltzmann/ledger-bit development (C-006): temperature and $k_R$ are not free parameters but follow from ledger bit cost. Bandwidth is then bits available per recognition cycle, i.e. holographic capacity divided by the cost of one eight-tick pass.
proof idea
Definition-first module, not a single deep theorem. It introduces the eight-tick cadence constant and proves it is positive and equal to the expected numeric value. Bandwidth is defined as holographic bits over recognition cost; separate lemmas discharge denominator positivity, overall positivity (two forms), monotonicity in area, and linearity in the bit count. The closing identity states bandwidth equals bits over cost by unfolding the definition. Upstream holographic-bound and cost facts are imported rather than re-proved.
why it matters in Recognition Science
Gives Unification a quantitative processing rate: how many recognition events a boundary can support per unit RS time. That rate is the bridge from geometric holography (area, Planck units) to discrete ledger throughput (eight-tick cycles, J-cost accounting).
Downstream, FermionDOFGapBridge imports this module while relating Standard Model degree-of-freedom counts to $D=3$ combinatorial quantities. Even after that bridge was re-scoped to pure arithmetic identities (not a derivation of the SM spectrum), it still needs a coherent bandwidth/cadence vocabulary when DOF counts are compared against recognition capacity.
Framework landmarks touched directly: T7 eight-tick octave, the holographic bound from ledger projection, and cost/ledger-bit normalization used elsewhere for $k_R$ and related constants.
scope and limits
- Does not derive the eight-tick period; it assumes the T7 octave and packages it as cadence.
- Does not prove the holographic bound; that is imported from Quantum.HolographicBound.
- Does not derive Standard Model particle content or fermion spectra.
- Does not fix SI conversion factors beyond RS-native tick and cost units.
- Does not claim dynamical saturation of the bound in curved spacetime.
used by (1)
depends on (5)
declarations in this module (20)
-
def
eightTickCadence -
theorem
eightTickCadence_pos -
theorem
eightTickCadence_eq -
def
bandwidth -
theorem
planckArea_pos -
theorem
bandwidth_denom_pos -
theorem
bandwidth_pos -
theorem
bandwidth_pos' -
theorem
bandwidth_monotone -
theorem
bandwidth_linear -
def
holographicBits -
theorem
bandwidth_eq_bits_over_cost -
theorem
bandwidth_times_cost_eq_rate -
theorem
Clag_eq_phi_neg5 -
theorem
alpha_locked_in_unit -
def
demandedRate -
theorem
demandedRate_pos -
def
IsSaturated -
def
IsSubSaturated -
theorem
saturated_or_sub