ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck pass

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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.

Direct check of the flat vacuum data with a cosmological parameter against the Friedmann-system equations

Example

Fix real constants $c > 0$ , $G_{N} > 0$ , $A > 0$ , $t_{0} \in R$ and $Λ \in R$ . Then:

If $Λ > 0$ , put $H_{Λ} : = c \sqrt{Λ / 3}$ and define $a : R \to (0, \infty)$ by $a (t) : = A \exp (H_{Λ} (t - t_{0}))$ , together with $ε : = 0$ and $p : = 0$ on $I : = R$ . Then the Friedmann-system datum $(c, G_{N}, Λ, I, 0, a, ε, p)$ satisfies (F), (A), (C) and (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum on all of $I$ .
If $Λ < 0$ , then for every nonempty open interval $I \subseteq R$ and every smooth $a : I \to (0, \infty)$ the datum $(c, G_{N}, Λ, I, 0, a, 0, 0)$ fails the Friedmann equation (F) at every point of $I$ ; in particular no such flat vacuum datum satisfies the Friedmann-system equations.

Facts & Assumptions

Given: The constants $c, G_{N}, A, t_{0}, Λ$ with $c > 0$ , $G_{N} > 0$ , $A > 0$ , the functions $a, ε, p$ of the example, and the equations (F), (A), (C), (S) of The Friedmann, acceleration and continuity equations of an FLRW model datum.

[L1]

The real exponential is strictly positive on $R$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ) and smooth with $\exp^{'} = \exp$ ; consequently for real constants $α, β$ the function $t \mapsto β \exp (α (t - t_{0}))$ is smooth with derivative $α β \exp (α (t - t_{0}))$ and second derivative $α^{2} β \exp (α (t - t_{0}))$ (The real exponential function and the number $e$ by a power series, The exponential function is smooth and $(\exp)^{'} = \exp$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L2]

For $Λ > 0$ the real power $\sqrt{Λ / 3} = (Λ / 3)^{1 / 2}$ is a strictly positive real number with $(Λ / 3)^{1 / 2} \cdot (Λ / 3)^{1 / 2} = (Λ / 3)^{1} = Λ / 3$ , and real powers satisfy $(x^{r})^{s} = x^{r s}$ and $(x y)^{r} = x^{r} y^{r}$ for $x, y > 0$ (Real powers for positive bases, with the zero-base positive-exponent convention, The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents). Hence $H_{Λ} = c (Λ / 3)^{1 / 2}$ satisfies $H_{Λ} > 0$ and $H_{Λ}^{2} = c^{2} Λ / 3$ . For $Λ < 0$ the number $Λ c^{2} / 3$ is negative.

[L3]

The equations of The Friedmann, acceleration and continuity equations of an FLRW model datum with $k = 0$ and $ε = p = 0$ read

$H^{2} = \frac{Λ c^{2}}{3}, \frac{\ddot{a}}{a} = \frac{Λ c^{2}}{3}, \dot{ε} + 3 H (ε + p) = 0, - \frac{2 a \ddot{a} + {\dot{a}}^{2}}{c^{2}} + Λ a^{2} = 0,$

where the four displays are (F), (A), (C) and (S) at every point of $I$ ; the third holds identically for $ε = p = 0$ .

[L4]

For every real number $x$ one has $x^{2} \geq 0$ : if $x = 0$ this is clear, and if $x \neq 0$ then $x^{2} > 0$ (Squares of nonzero elements are positive). Since $c > 0$ gives $c^{2} > 0$ , for $Λ < 0$ the product $Λ c^{2} / 3$ is negative (Order on the reals).

Verification

1.1L1givenalgebra

(The exponential datum is admissible.) The function $a (t) = A \exp (H_{Λ} (t - t_{0}))$ is smooth on $R$ and strictly positive because $A > 0$ and $\exp > 0$ ; by [L1] its logarithmic derivative is $H (t) = \dot{a} / a = H_{Λ}$ and $\ddot{a} / a = H_{Λ}^{2}$ . The zero functions $ε, p$ are smooth, and $(c, G_{N}, Λ, I, 0, a, ε, p)$ with $I = R$ is a Friedmann-system datum.

1.2L2L3L4algebra

( $Λ < 0$ obstruction.) Suppose $Λ < 0$ , let $I \subseteq R$ be a nonempty open interval and let $a : I \to (0, \infty)$ be smooth, and suppose the datum $(c, G_{N}, Λ, I, 0, a, 0, 0)$ satisfies (F). By [L3] the Friedmann equation reads $H (t)^{2} = Λ c^{2} / 3$ at every $t \in I$ ; by [L4] the right-hand side is negative while $H (t)^{2} \geq 0$ for the real number $H (t)$ , a contradiction. Hence no such datum satisfies (F), proving claim 2.

2.1L2L3step 1.1algebra

(Friedmann equation for $Λ > 0$ .) By [L3] and step 1.1, (F) reads $H^{2} = Λ c^{2} / 3$ ; the left-hand side is $H_{Λ}^{2}$ and $H_{Λ}^{2} = c^{2} Λ / 3 = Λ c^{2} / 3$ by [L2]. Hence (F) holds on $I$ .

2.2L2L3step 1.1algebra

(Acceleration equation for $Λ > 0$ .) By [L3] and step 1.1, (A) reads $\ddot{a} / a = Λ c^{2} / 3$ , and $\ddot{a} / a = H_{Λ}^{2} = Λ c^{2} / 3$ by [L2]. Hence (A) holds on $I$ .

2.3L2L3step 1.1algebra

(Continuity and spatial-component equations for $Λ > 0$ .) For $ε = p = 0$ every term of (C) vanishes, so (C) holds identically. For (S), step 1.1 gives $2 a \ddot{a} + {\dot{a}}^{2} = 2 a \cdot H_{Λ}^{2} a + (H_{Λ} a)^{2} = 3 H_{Λ}^{2} a^{2}$ , hence $- (2 a \ddot{a} + {\dot{a}}^{2}) / c^{2} + Λ a^{2} = - 3 H_{Λ}^{2} a^{2} / c^{2} + Λ a^{2} = - 3 (c^{2} Λ / 3) a^{2} / c^{2} + Λ a^{2} = 0$ , which is (S) with zero pressure. This completes claim 1.

3.1step 2.1step 2.2step 2.3step 1.2∎

(Conclusion.) Steps 2.1, 2.2 and 2.3 verify (F), (A), (C) and (S) for the exponential datum of claim 1 on all of $R$ , and step 1.2 proves the failure of (F) for every flat vacuum datum with $Λ < 0$ , which is claim 2. The verification used only differentiation of the exponential and the sign of a square; no field equation, no reference to the A page and no physical premise was used.

Remarks

What is checked. The example verifies the two sign regimes directly in the defining equations of The Friedmann, acceleration and continuity equations of an FLRW model datum: for $Λ > 0$ the exponential datum satisfies all four scalar equations, and for $Λ < 0$ the Friedmann equation alone is already unsatisfiable because a square cannot equal a negative number. The physical reading of the datum as a vacuum solution of the field law belongs to the A page and is not asserted here.
What is not claimed. No uniqueness of the solutions for $Λ > 0$ , no statement about the flat $Λ = 0$ case, no numerical value of $Λ$ or $H_{Λ}$ , and no physical claim about an expanding universe is made.

Direct check of the flat vacuum data with a cosmological parameter against the Friedmann-system equations

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources