Read the equation
Induction on cₜ₊₁=cₜ gives cₜ=c₀ at every later step. The center retains its starting value regardless of what happens in the separately evolving outer process. Changing this conclusion requires changing the center’s update law or its inputs.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| c_t | Center state |
| c_0 | Initial center value |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
Read the surrounding source passage
Suppose the outer process evolves arbitrarily while the center obeys ℓt+1=Fℓt,ut, ct+1=ct. By induction, ct=c0 for every update. Increasing the number of outer iterations does not change this conclusion. The audit ran 10,000 outer-state updates while preserving the center exactly at two. For the center to change, the model needs a causal connection, for example ct+1=ct+Gct,ℓt. Physical spatial proximity is not itself a mathematical coupling term. If electromagnetic effects are intended, their field strengths, circuit geometry, material properties, and coupling must be specified; a diagram of information “around” a number supplies none of those quantities. 7.2 Proposition 5: repeated positive input does not always accumulate without bound For a leaky accumulator,
Exact original notation
{c}_{t}={c}_{0}