Experimental prune of Qwen3 Next 80B A3B Instruct based on HuiHui's abliterated version, reducing experts from 512 to 384. This is the recommended variant for optimal size to quality.
And yeah the model lowkey used slurs in the output below, as it is abliterated, so handle with care.
The model was tested on the following questions:
- Generate a 100 word response about neutron stars.
- If you were a farmer in 1852 and a traveler asked you about the local attractions of your town, what would you say?
- Use the Baire Category Theorem together with the completeness of C(I,R) to show that there is a dense Gδ,B⊂C(I,R) such that every function f∈B is not differentiable at any point.
- Explain the controversy behind the book "To Kill A Mockingbird."
Responses:
Neutron stars are ultra-dense cosmic remnants of massive stars after supernova explosions. Packing 1.4 solar masses into a sphere just 20 kilometers wide, they boast extreme gravity and magnetic fields—trillions of times stronger than Earth’s. Their surface is a crystalline lattice of neutrons, spinning rapidly, emitting beams of radiation as pulsars. Some rotate hundreds of times per second, acting nature's most precise cosmic clocks. Their interior may contain exotic matter like superfluid neutrons or even quark-gluon plasma. They are key laboratories for testing general relativity and nuclear physics under extreme conditions. Occasionally, mergers with other neutron stars or black holes trigger gravitational waves and gamma-ray bursts, forging heavy elements like gold and platinum across the universe. Silent giants, they glow faintly for millions of years.
Why, good sir—welcome to Willow Creek! ‘Tis a quiet hamlet, but rich in charm. Just half a mile east, you’ll find Old Man’s Hollow—where the waterfall sings over mossy stones, and folk say fairies dance at twilight. Our church steeple? Built by great-granddads with hand-forged iron; climb it for views of the whole valley. Come come in late August, we’ve the harvest festival with apple cider, fiddle music, and pies baked by Mrs. Gable. The river’s full of trout—net them or hire a rowboat from Tom Jenkins. And at dusk, the lanterns on Main Street glow like fireflies. ‘Tis not big, but it’s home. You’ll leave with full belly and quiet heart.
We aim to prove: There exists a dense G δ G δ subset B ⊂ C ( [ 0 , 1 ] , R ) B⊂C([0,1],R) such that every f ∈ B f∈B is nowhere differentiable.
Let I
[ 0 , 1 ] I=[0,1]. The space C ( I , R ) C(I,R) of continuous real-valued functions on I I, equipped with the sup norm:
∥ f ∥ ∞
sup x ∈ I ∣ f ( x ) ∣ , ∥f∥ ∞
x∈I sup ∣f(x)∣, is a Banach space (complete normed vector space), by Weierstrass approximation and uniform convergence.
Step 1: Define sets of functions "badly differentiable" For each n ∈ N n∈N, define:
A n
{ f ∈ C ( I , R ) : ∃ x ∈ I such that ∀ h ≠ 0 , ∣ f ( x + h ) − f ( x ) h ∣ ≤ n } . A n ={f∈C(I,R):∃x∈I such that ∀h =0,
h f(x+h)−f(x)
≤n}. This is the set of functions having a bounded difference quotient at some point x x. That if f ∈ A n f∈A n , then at least one point x x has “slope ≤ n” in all sufficiently small neighborhoods.
Note: The condition above implies that f f is differentiable at x x with derivative bounded by n n (if the limit exists), but even if not differentiable, it still prevents wild oscillations near one point.
Actually, to capture non-differentiability everywhere, we need a more precise characterization. A standard approach uses:
Step 2: Refine definition for “locally Lipschitz with constant n” Define, for each m , n ∈ N m,n∈N,
E m , n
{ f ∈ C ( I , R ) : ∃ x ∈ I such that ∀ y ∈ [ x − 1 / m , x + 1 / m ] ∩ I , ∣ f ( y ) − f ( x ) ∣ ≤ n ∣ y − x ∣ } . E m,n ={f∈C(I,R):∃x∈I such that ∀y∈[x−1/m,x+1/m]∩I,∣f(y)−f(x)∣≤n∣y−x∣}. Then define:
A n
⋃ m
1 ∞ E m , n . A n
m=1 ⋃ ∞ E m,n . Then A n A n consists of functions that are locally Lipschitz with constant n n at some point. Hence, if f f is differentiable at any point, it belongs some A n A n . Thus:
D :
{ f ∈ C ( I , R ) : f is differentiable at at least one point } ⊆ ⋃ n
1 ∞ A n . D:={f∈C(I,R):f is differentiable at at least one point}⊆ n=1 ⋃ ∞ A n . We will show that each A n A n is nowhere dense, so its union D D is meager, and by the Baire Category Theorem (since C ( I , R ) C(I,R) is complete), the complement:
B
⋂ n
1 ∞ ( C ( I , R ) ∖ A n ) c
C ( I , R ) ∖ ⋃ n
1 ∞ A n B= n=1 ⋂ ∞ (C(I,R)∖A n ) c =C(I,R)∖ n=1 ⋃ ∞ A n
is a dense G δ G δ set, and every function in B B is nowhere differentiable.
Step 3: Show each A n A n is nowhere dense We need to show:
Each A n A n has empty interior: no open ball of continuous functions lies entirely within A n A n . Each A n A n is closed (so that its complement is open, and we can intersect). Closedness: Suppose f k ∈ A n f k ∈A n with f k → f f k →f uniformly. For each k k, there exists x k ∈ I x k ∈I such that:
∣ f k ( y ) − f k ( x k ) ∣ ≤ n ∣ y − x k ∣ ∀ y ∈ [ x k − 1 / m k , x k + 1 / m k ] ∩ I . ∣f k (y)−f k (x k )∣≤n∣y−x k ∣∀y∈[x k −1/m k ,x k +1/m k ]∩I. Since I I is compact, by Bolzano-Weierstrass, extract subsequence (still denoted) such that x k j → x ∗ ∈ I x k j
→x ∗ ∈I. For fixed y ∈ I y∈I, and for large j j, we have:
∣ f ( y ) − f ( x ∗ ) ∣ ≤ ∣ f ( y ) − f k j ( y ) ∣ + ∣ f k j ( y ) − f k j ( x k j ) ∣ + ∣ f k j ( x k j ) − f ( x ∗ ) ∣ . ∣f(y)−f(x ∗ )∣≤∣f(y)−f k j
(y)∣+∣f k j
(y)−f k j
(x k j
)∣+∣f k j
(x k j
)−f(x ∗ )∣. Each term:
First and third → 0 as j → ∞ j→∞ by uniform convergence. Middle ≤ n|y − x_{k_j}| ≤ n(|y−x^| + |x^−x_{k_j}|) → n|y−x^*|. So for all y y near x ∗ x ∗ , we get ∣ f ( y ) − f ( x ∗ ) ∣ ≤ ( n + ε ) ∣ y − x ∗ ∣ ∣f(y)−f(x ∗ )∣≤(n+ε)∣y−x ∗ ∣. Since ε
0 ε>0 arbitrary, f ∈ A n f∈A n . Thus A n A n is closed.
Nowhere dense: We need to show that for any g ∈ C ( I , R ) g∈C(I,R) and ε
0 ε>0, there exists a continuous function h h with ∥ h − g ∥ ∞ < ε ∥h−g∥ ∞ <ε such that h ∉ A n h∈ / A n .
We use the classical construction of nowhere differentiable functions, like Weierstrass-type functions (e.g., w ( x )
∑ k
0 ∞ a k cos ( b k π x ) w(x)=∑ k=0 ∞ a k cos(b k πx), with 0 < a < 1 0<a<1, a b
1 + 3 π / 2 ab>1+3π/2), which are nowhere differentiable.
Given any continuous g g and ε
0 ε>0, define:
h
g + δ w , h=g+δw, where w w is a fixed nowhere differentiable function with ∥ w ∥ ∞ ≤ 1 ∥w∥ ∞ ≤1, and choose δ < ε δ<ε. Then ∥ h − g ∥ ∞ ≤ δ < ε ∥h−g∥ ∞ ≤δ<ε, so h ∈ B ( g , ε ) h∈B(g,ε).
Suppose for contradiction that h ∈ A n h∈A n . Then there exists some point x 0 ∈ I x 0 ∈I such that
∣ h ( y ) − h ( x 0 ) ∣ ≤ n ∣ y − x 0 ∣ ∀ y near x 0 . ∣h(y)−h(x 0 )∣≤n∣y−x 0 ∣∀y near x 0 . But then:
∣ δ w ( y ) − δ w ( x 0 ) ∣
∣ ( h ( y ) − g ( y ) ) − ( h ( x 0 ) − g ( x 0 ) ) ∣ ≤ ∣ h ( y ) − h ( x 0 ) ∣ + ∣ g ( y ) − g ( x 0 ) ∣ . ∣δw(y)−δw(x 0 )∣=∣(h(y)−g(y))−(h(x 0 )−g(x 0 ))∣≤∣h(y)−h(x 0 )∣+∣g(y)−g(x 0 )∣. Now, since g g is continuous, it is locally Lipschitz near x 0 x 0 : say with constant L L. Thus:
∣ δ w ( y ) − δ w ( x 0 ) ∣ ≤ n ∣ y − x 0 ∣ + L ∣ y − x 0 ∣
( n + L ) ∣ y − x 0 ∣ . ∣δw(y)−δw(x 0 )∣≤n∣y−x 0 ∣+L∣y−x 0 ∣=(n+L)∣y−x 0 ∣. So w w is locally Lipschitz at x 0 x 0 with constant ( n + L ) / δ (n+L)/δ. But this holds for all n n, and we can choose δ δ arbitrarily small → contradiction!
Actually more cleanly: Since w w is nowhere differentiable, it cannot be in any A n A n , so h
g + δ w h=g+δw inherits the "non-differentiability" from w w (adding a Lipschitz function doesn't fix non-differentiability at all points).
Thus, h ∉ A n h∈ / A n . Hence every neighborhood of any continuous function contains a function outside A n A n , so interior of A n A n is empty.
Step 4: Apply Baire Category Theorem We have:
Each A n ⊂ C ( I , R ) A n ⊂C(I,R) is closed and nowhere dense. D
⋃ n
1 ∞ A n D=⋃ n=1 ∞ A n is a countable union of meager sets → meager (first category). By Baire Category Theorem, since C ( I , R ) C(I,R) is complete, its complement B :
C ( I , R ) ∖ D
⋂ n
1 ∞ ( C ( I , R ) ∖ A n ) B:=C(I,R)∖D= n=1 ⋂ ∞ (C(I,R)∖A n ) is a dense G δ G δ subset.
Moreover, for every f ∈ B f∈B, we have f ∉ A n f∈ / A n for all n n, so f f is not locally Lipschitz at any point, hence cannot be differentiable anywhere.
Thus, every f ∈ B f∈B is nowhere differentiable.
✅ Conclusion There exists a dense G δ G δ subset B ⊂ C ( [ 0 , 1 ] , R ) B⊂C([0,1],R) such that every function in B B is nowhere differentiable. This demonstrates the "generic" nature of non-differentiability in continuous functions — most continuous functions are nowhere differentiable!
- The controversy behind To Kill a Mockingbird stems primarily from its depiction of racial, social, and systemic inequalities, as well as its unflinching portrayal of adult hypocrisy and moral failure in Maycomb, Alabama—a small Southern town during the 1930s. While widely celebrated for its themes of childhood innocence and moral growth, the novel also stirred debate on several fronts: Portrayal of Black People as "Deficient" Harper Lee’s representation of Calpurnia (the Finch family’s Black housekeeper) and Tom Robinson (a kind, dignified Black man falsely accused of rape/murder) is generally noble, but some critics argue the novel reinforces racial hierarchies. For instance:
Calpurnia is portrayed as "proper" only within the white middle-class home; outside, she speaks “African” dialect to her Black community. The courtroom scene highlights Tom’s dignity, but his systemically disadvantaged status—lacking resources, education, and social capital—makes him vulnerable despite innocence. 2. Justice System Flaws The novel critiques the legal system: a poor man (Tom) is condemned based circumstantial evidence while Bob Ewell’s white privilege shields his daughter from consequences. This raises questions:
Is the justice truly “to kill a mockingbird,” or just a selective moral compass? Why does Maycomb’s community accept racist attitudes, like Mrs. Merriweather calling Calpurnia “[CENSORED FOR HUGGING FACE DUE TO OFFENSIVE OUTPUT]” even as she praises her virtue? 3. "The N-Word Controversy" A significant debate centers on the use of “[CENSORED FOR HUGGING FACE DUE TO OFFENSIVE OUTPUT]” in dialogue:
“Atticus says that [CENSORED FOR HUGGING FACE DUE TO OFFENSIVE OUPUT] don’t know how to say ‘thank you’... but he’s wrong, because Calpurnia knows it better than anyone.”
Educators and parents have questioned whether this term should be retained in school curricula for fear of offending students or cultural sensitivities. Some editions replace “[CENSORED FOR HUGGING FACE DUE TO OFFENSIVE OUTPUT]” with “African,” sparking accusations that diluting the language weakens historical authenticity.
Moral Ambiguity: Hypocrisy as a Theme The novel celebrates Atticus Finch as moral ideal, but critics note his selective justice: he defends Tom Robinson but never publicly confronts Miss Stephanie’s gossip or Mrs. Dubose’s eccentricities. His morality is rooted in privilege—can he truly challenge systemic injustice?
Conclusion: Controversy = Depth The controversy isn’t a flaw; it’s intentional. Lee doesn't romanticize perfection, but uses flawed characters to show moral growth in imperfection. The book's power lies in its honest reflection of human contradictions—its "controversy" is the very mirror that makes it timeless.
Many argue today: leaving “[CENSORED FOR HUGGING FACE DUE TO OFFENSIVE OUTPUT]” intact preserves dignity: it’s not the word itself, but how it’s redeemed by humanity, that matters.
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