Exact dynamics and reconstruction

From selected lengths to the contraction

Algebraic step $\frac{Gx+C}{G+C}=1+\lambda(x-1)$. Input $x$, fixed point $1$, contraction factor $\lambda=\varphi/\Omega_0$. The browser uses binary64 for this illustration. Certified integer equality lives in the floor lab.

1. x = 2.00000000000
2. T^1(x) = 1.33994977134
3. T^2(x) = 1.11556584704
4. T^3(x) = 1.03928658328
5. T^4(x) = 1.01335546500
6. T^5(x) = 1.00454018727
7. T^6(x) = 1.00154343563
8. T^7(x) = 1.00052469059
9. T^8(x) = 1.00017836845

Binary64 λ ≈ 0.339949771344779. Distinct from the rational surrogate q = Q/S. Centered geometric homothety: displacement from 1 is multiplied by λ at each step.

Certified integer floor on D₁₅

Exact BigInt arithmetic in this browser. $Q=339949771344778$, $S=10^{15}$, $q=Q/S$. On $0\le d\le 32767$, $\lfloor\lambda d\rfloor=\bigl\lfloor Qd/S\bigr\rfloor$. $\lambda$ is the real contraction parameter; $q$ is the rational surrogate. They are not the same number.

Parameter interval for agreement of an arbitrary real r with this table: $\frac{4196}{12343}\le r<\frac{7039}{20706}$. Decimal-precision threshold and first-disagreement boundary live on the successor-search page.

Worked reconstruction

Example trajectory $399\to 135\to 45\to 15\to 5\to 1\to 0$. The first step satisfies $Q\cdot 399=S\cdot 135+639958766566422$, so $399=(S\cdot 135+639958766566422)/Q$. Reverse uses only the displayed prefix: the last shown quotient as terminal, and the shown remainders. A malformed remainder fails a named condition and does not reconstruct.

Terminal state plus ordered remainders reconstructs the recorded trajectory. On the canonical domain the first remainder already identifies the seed. Extinction occurs in at most ten transitions; a 23-step record includes padding after extinction. Load 399 restores the six-step example and clears any remainder override. Invalid remainder text is a remainder-format failure; an integer remainder ≥ S is a remainder-bound failure. Neither case reconstructs.