LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends

Statement

Fix $a \geq 100$ , let $R = \sqrt{a} + 3$ , $L = \sqrt{a - 2 R}$ , let $x (s)$ and $h$ be the coded sequence and coding map of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences, The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution and Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ , and let $F (x, y) = (y, y^{2} - a - x)$ . For $u \in {- 1, + 1}$ write $\overset{ˉ}{u} \in X_{\pm}$ for the constant code ${\overset{ˉ}{u}}_{n} : = u$ , and put

$r_{u} : = 1 + u \sqrt{a + 1},$

so that $r_{+} > 0 > r_{-}$ , and write $W^{s}, W^{u}$ for the stable and unstable sets of Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family. Then:

$x_{n} (\overset{ˉ}{u}) = r_{u}$ for every $n$ , and $r_{u}^{2} = a + 2 r_{u}$ ; consequently $h (\overset{ˉ}{u}) = (r_{u}, r_{u})$ is a fixed point of $F$ ;
if $s \in X_{\pm}$ satisfies $s_{n} = u$ for all $∣ n ∣ \geq N$ for some $N \geq 0$ , then

$F^{n} (h (s)) ⟶ (r_{u}, r_{u}) as n \to + \infty and as n \to - \infty;$

if moreover $s$ is not the constant code $\overset{ˉ}{u}$ , then $h (s) \neq (r_{u}, r_{u})$ , $F^{n} (h (s)) \neq (r_{u}, r_{u})$ for every $n \in Z$ , and $h (s) \in W^{s} (r_{u}, r_{u}) \cap W^{u} (r_{u}, r_{u})$ ; hence $h (s)$ is a homoclinic point of the fixed point $(r_{u}, r_{u})$ ;
distinct codes $s \neq t$ of the kind in clause 2 give distinct points $h (s) \neq h (t)$ ; in particular the finite-word deviations from the two constant codes produce infinitely many distinct homoclinic points.

Facts & Assumptions

Given: The parameter $a \geq 100$ , the constants $R, L$ , the coded sequence $x (s)$ and map $h$ , the phase map $F$ , the constant codes $\overset{ˉ}{u}$ , the numbers $r_{u}$ , and a code $s$ with $s_{n} = u$ for $∣ n ∣ \geq N$ .

[F1]

Coding data (The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ ): $x (s) \in B_{R}$ is the unique fixed point of $T_{s}$ , $sign x_{n} (s) = s_{n}$ , $∣ x_{n} (s) ∣ \geq L$ , $x_{n + 1} (s) + x_{n - 1} (s) = x_{n} (s)^{2} - a$ ; locality: $(T_{s}^{m} 0)_{n}$ depends only on $s_{j}$ with $∣ j - n ∣ \leq m - 1$ ; uniform error: $∥ T_{s}^{m} 0 - x (s) ∥_{\infty} \leq 2 R L^{- m}$ ; $h$ is injective; $F (x_{n - 1} (s), x_{n} (s)) = (x_{n} (s), x_{n + 1} (s))$ .

[F2]

Conjugacy and invariant set (The quadratic map $F (x, y) = (y, y^{2} - a - x)$ has a compact coded invariant set conjugate to the two-sign shift, with chaos under ${A C}_{ω}$ ): $F (h (s)) = h (S_{\pm} s)$ for all $s$ , and the coded set is invariant.

[F3]

Square roots and the equation $r^{2} = a + 2 r$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Absolute value in an ordered field): $a + 2 r_{u} > 0$ , $\sqrt{a + 2 r_{u}} = ∣ r_{u} ∣$ and $r_{u} = u \sqrt{a + 2 r_{u}}$ , because $r_{u}^{2} = a + 2 r_{u}$ and $sign r_{u} = u$ .

[F4]

Stable and unstable sets (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family): $W^{s} (\overset{ˉ}{z}) = {z : F^{n} (z) defined for all n \geq 0, F^{n} (z) \to \overset{ˉ}{z}}$ , $W^{u} (\overset{ˉ}{z}) = {z : F^{- n} (z) \to \overset{ˉ}{z}}$ , and a homoclinic point of a fixed point $\overset{ˉ}{z}$ is a point of $W^{s} (\overset{ˉ}{z}) \cap W^{u} (\overset{ˉ}{z})$ different from $\overset{ˉ}{z}$ ; convergence is metric convergence (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[F5]

The sequence $L^{- m}$ tends to $0$ because $L > 1$ (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ , For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[F6]

Absolute value and the base arithmetic (Basic properties of the absolute value, Absolute value in an ordered field): $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ , $∣ u + w ∣ \leq ∣ u ∣ + ∣ w ∣$ , and $10^{2} = 100$ .

[F7]

Squaring is strictly increasing on the nonnegative reals (Squaring is monotone on the nonnegatives): if $0 \leq x < y$ then $x^{2} < y^{2}$ , and if $0 \leq x \leq y$ then $x^{2} \leq y^{2}$ .

Proof

technique · direct; check the constant codes against the recurrence, use the locality of the finite windows for the two-sided convergence, and invoke injectivity of the coding for the distinctness of finite-word homoclinic points

1.1givenF3F6F7algebra

The constants $r_{u}$ : since $a \geq 100$ , [F7] applied to $10^{2} \leq a = (\sqrt{a})^{2}$ gives $\sqrt{a} \geq 10$ . Also $(\sqrt{a} + 1)^{2} = a + 2 \sqrt{a} + 1 > a + 1$ , so [F7] gives $\sqrt{a + 1} < \sqrt{a} + 1$ ; and $1^{2} < a + 1$ gives $\sqrt{a + 1} > 1$ . Thus $a + 2 - 2 \sqrt{a + 1} > a - 2 \sqrt{a} = \sqrt{a} (\sqrt{a} - 2) > 0$ , so $a + 2 r_{u} = a + 2 + 2 u \sqrt{a + 1} > 0$ for both signs. The number $r_{u} = 1 + u \sqrt{a + 1}$ satisfies $r_{u}^{2} = 1 + 2 u \sqrt{a + 1} + (a + 1) = a + 2 + 2 u \sqrt{a + 1} = a + 2 r_{u}$ , and $sign r_{u} = u$ by $\sqrt{a + 1} > 1$ . Hence by [F3] $r_{u} = u \sqrt{a + 2 r_{u}}$ .

2.1step 1.1F1F2F6F7algebra

Constant codes: let $\tilde{x}$ be the constant sequence ${\tilde{x}}_{n} : = r_{u}$ . By step 1.1, $(T_{\overset{ˉ}{u}} \tilde{x})_{n} = u \sqrt{a + r_{u} + r_{u}} = u \sqrt{a + 2 r_{u}} = r_{u} = {\tilde{x}}_{n}$ for all $n$ ; so $\tilde{x}$ is a fixed point of $T_{\overset{ˉ}{u}}$ in $B_{R}$ (indeed $∣ r_{u} ∣ \leq 1 + \sqrt{a + 1} < \sqrt{a} + 2 < R$ by [F7] and $a \geq 100$ ). By uniqueness ([F1]) $x (\overset{ˉ}{u}) = \tilde{x}$ , hence $h (\overset{ˉ}{u}) = (r_{u}, r_{u})$ ; and $F (r_{u}, r_{u}) = (r_{u}, r_{u}^{2} - a - r_{u}) = (r_{u}, r_{u})$ because $r_{u}^{2} - a = 2 r_{u}$ .

3.1step 2.1F1F2F5algebra

Convergence in the forward direction: let $m \geq 0$ and let $n \geq N + m - 1$ ; then every $j$ with $∣ j - n ∣ \leq m - 1$ satisfies $j \geq n - (m - 1) \geq N$ , hence $s_{j} = u = {\overset{ˉ}{u}}_{j}$ . By [F1], $(T_{s}^{m} 0)_{n} = (T_{\overset{ˉ}{u}}^{m} 0)_{n}$ . Therefore $∣ x_{n} (s) - x_{n} (\overset{ˉ}{u}) ∣ \leq ∣ x_{n} (s) - (T_{s}^{m} 0)_{n} ∣ + ∣ (T_{\overset{ˉ}{u}}^{m} 0)_{n} - x_{n} (\overset{ˉ}{u}) ∣ \leq 4 R L^{- m}$ for $n \geq N + m - 1$ , using the uniform error bound of [F1] twice and $x (\overset{ˉ}{u}) = \tilde{x}$ of step 2.1. Given $ε > 0$ , choose $m$ with $4 R L^{- m} < ε$ , possible by [F5]; then $∣ x_{n} (s) - r_{u} ∣ < ε$ for all $n \geq N + m - 1$ . Hence $x_{n} (s) \to r_{u}$ as $n \to + \infty$ , and therefore $F^{n} (h (s)) = (x_{n - 1} (s), x_{n} (s)) \to (r_{u}, r_{u})$ by [F1].

3.2step 2.1F1F2algebra

Nonconstancy: suppose $s \neq \overset{ˉ}{u}$ ; then there is $k$ with $s_{k} \neq u$ , and by [F1] $sign x_{k} (s) = s_{k} \neq u = sign r_{u}$ , so $x_{k} (s) \neq r_{u}$ and $h (s) = (x_{- 1} (s), x_{0} (s))$ ; if $h (s) = (r_{u}, r_{u})$ then $x_{- 1} (s) = x_{0} (s) = r_{u}$ , and then by the recurrence $x_{n} (s) = r_{u}$ for all $n$ , so $s_{n} = sign x_{n} (s) = u$ for all $n$ , a contradiction. Hence $h (s) \neq (r_{u}, r_{u})$ . Since $F^{n} (h (s)) = h (S_{\pm}^{n} s)$ by [F2] and each $S_{\pm}^{n} s$ is again not constant, the same argument gives $F^{n} (h (s)) \neq (r_{u}, r_{u})$ for all $n \in Z$ .

4.1step 3.1F1algebra

Convergence in the backward direction: if $n \leq - N - (m - 1)$ and $∣ j - n ∣ \leq m - 1$ , then $j \leq n + (m - 1) \leq - N$ , so $∣ j ∣ \geq N$ and $s_{j} = u = {\overset{ˉ}{u}}_{j}$ . Locality [F1] therefore gives $(T_{s}^{m} 0)_{n} = (T_{\overset{ˉ}{u}}^{m} 0)_{n}$ , and the same two uniform error bounds as in step 3.1 yield $∣ x_{n} (s) - r_{u} ∣ \leq 4 R L^{- m}$ . Hence $x_{n} (s) \to r_{u}$ as $n \to - \infty$ . Since $F$ is a bijection ([F1]) and $F^{n} (h (s)) = (x_{n - 1} (s), x_{n} (s))$ , this says $F^{n} (h (s)) \to (r_{u}, r_{u})$ as $n \to - \infty$ .

5.1step 3.1step 4.1step 3.2F1F4

Homoclinic points: by steps 3.1 and 4.1, the point $h (s)$ lies in $W^{s} (r_{u}, r_{u}) \cap W^{u} (r_{u}, r_{u})$ (the forward and backward orbits are defined for all times because $F$ is a bijection and $F^{n} (h (s)) = (x_{n - 1} (s), x_{n} (s))$ for all $n$ ), and by step 3.2 it differs from the fixed point. By the definition [F4] it is a homoclinic point of $(r_{u}, r_{u})$ .

6.1step 5.1F1∎

Distinctness: if $s \neq t$ then $h (s) \neq h (t)$ because $h$ is injective by [F1]; the nonconstant finite-word codes are infinite in number (for example the codes differing from $\overset{ˉ}{u}$ at exactly the coordinates $0, 1, \dots, j$ for $j \geq 0$ are pairwise distinct), so there are infinitely many distinct homoclinic points.

Remarks

The convergence rates are uniform in the code: for a code that agrees with $\overset{ˉ}{u}$ for $∣ n ∣ \geq N$ , the deviation satisfies $∣ x_{n} (s) - r_{u} ∣ \leq 4 R L^{- m}$ whenever $∣ n ∣ \geq N + m - 1$ .
The two constant codes give fixed points, not homoclinic points; homoclinic points arise precisely from nonconstant codes with constant tails, which are the finite-word deviations promised by the design.

Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources