Latent Reasoning with Normalizing Flows

Guancheng Tu; Haoqiang Kang; Jiatao Gu; Lianhui Qin; Suhao Yu; Xiangjun Fu; Yao Tang; Yizhe Zhang

arxiv: 2606.06447 · v1 · pith:75AQMV7Unew · submitted 2026-06-04 · 💻 cs.CL · cs.LG

Latent Reasoning with Normalizing Flows

Guancheng Tu , Xiangjun Fu , Suhao Yu , Yao Tang , Haoqiang Kang , Lianhui Qin , Yizhe Zhang , Jiatao Gu This is my paper

Pith reviewed 2026-06-28 01:09 UTC · model grok-4.3

classification 💻 cs.CL cs.LG

keywords latent reasoningnormalizing flowschain of thoughtlarge language modelscode generation

0 comments

The pith

Normalizing flows model continuous thoughts for latent reasoning in language models

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Large language models benefit from intermediate computation but explicit chain-of-thought requires verbalizing each step. Latent reasoning in continuous states offers higher bandwidth but often loses key autoregressive advantages. NF-CoT uses normalizing flows to model compact continuous thoughts distilled from explicit CoT within the LLM backbone. This preserves left-to-right generation, KV-cache compatibility, probabilistic sampling, and tractable likelihoods. Experiments on code-generation benchmarks show higher pass rates and lower reasoning cost than explicit CoT and prior latent methods.

Core claim

By placing a TARFlow-style normalizing flow head inside the LLM, NF-CoT creates a joint causal stream where continuous thought tokens are generated via the flow and text tokens via the standard head, yielding a tractable probability model over the continuous thoughts that supports all standard decoding features.

What carries the argument

TARFlow-style normalizing flow inserted as a head for continuous thought positions, enabling exact likelihood computation and policy gradient optimization in latent space.

If this is right

Improves pass rates on code-generation benchmarks over explicit-CoT and prior latent-reasoning baselines
Substantially reduces intermediate-reasoning cost
Enables probabilistic left-to-right decoding with original KV cache
Supports direct policy-gradient optimization in the latent reasoning space
Provides exact likelihoods for latent thoughts

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could be applied to non-code reasoning tasks such as math or commonsense reasoning if similar distillation is possible.
End-to-end training without explicit CoT distillation might further improve the continuous thoughts.
Compatibility with KV cache suggests easy integration into existing LLM inference pipelines.

Load-bearing premise

Continuous thoughts distilled from explicit CoT can be faithfully modeled by a TARFlow-style normalizing flow inserted into the LLM backbone without breaking left-to-right generation, KV-cache compatibility, or tractable likelihood estimation.

What would settle it

Observing that NF-CoT fails to improve pass rates or reduce cost on code-generation benchmarks when compared to explicit-CoT would falsify the central performance claim.

read the original abstract

Large language models often improve reasoning by generating explicit chain-of-thought (CoT), demonstrating the importance of intermediate computation. However, textual CoT forces this computation through a discrete, serial, and communication-oriented token stream: each reasoning step must be verbalized before the model can proceed, even when the underlying update is semantic, uncertain, or only partially formed. Latent reasoning offers a higher-bandwidth alternative by performing intermediate computation in compact continuous states before committing to text. Yet existing latent-reasoning methods often sacrifice key advantages that make CoT effective in autoregressive language models, including native left-to-right generation, probabilistic sampling, compatibility with KV-cache decoding, and tractable likelihood estimation. We propose NF-CoT, a latent reasoning framework that preserves these advantages by modeling continuous thoughts with normalizing flows. NF-CoT instantiates a TARFlow-style normalizing flow inside the LLM backbone, defining a tractable probability model over compact continuous thoughts distilled from explicit CoT. Continuous-thought positions are generated by an NF head, while text positions are generated by the standard LM head within the same causal stream. This design provides exact likelihoods for latent thoughts, enables probabilistic left-to-right decoding with the original KV cache, and supports direct policy-gradient optimization in the latent reasoning space. On code-generation benchmarks, NF-CoT improves pass rates over explicit-CoT and prior latent-reasoning baselines while substantially reducing intermediate-reasoning cost.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

NF-CoT adds an NF head for continuous latent steps inside a standard causal LLM stream, preserving KV cache and likelihoods, but the reported gains rest on thin evidence so far.

read the letter

The main thing to know is that this paper inserts a TARFlow-style normalizing flow head at selected positions to model continuous thoughts distilled from explicit CoT, while the usual LM head handles text tokens in the same left-to-right causal stream. The design keeps KV cache compatibility, exact likelihoods, and policy-gradient support, which is the practical point.

What is actually new is the specific head arrangement that lets continuous reasoning happen without breaking autoregressive generation or requiring separate modules. The abstract shows they thought through the compatibility requirements and placed the flow only where needed. That is a clear step beyond prior latent-reasoning work that often drops one or more of those engineering features.

The paper does a reasonable job stating how the flow gives tractable densities over the latent states and how that enables direct optimization in continuous space. No obvious internal contradiction appears in the described setup.

The soft spot is the evaluation. The abstract claims higher pass rates on code-generation benchmarks with lower intermediate cost than explicit CoT and earlier latent baselines, yet supplies no numbers, baselines, ablations, or statistical details. The reader's note flags low confidence for exactly this reason. Without those, it is hard to tell whether the gains are real or sensitive to implementation choices.

This is for groups working on efficient reasoning inside production LLMs who already use KV cache and want to experiment with latent steps. A reader focused on architecture compatibility would find the design useful to discuss. The work shows clear thinking on the constraints, so it deserves a serious referee even if the experiments need tightening.

Referee Report

2 major / 1 minor

Summary. The paper proposes NF-CoT, a latent-reasoning framework that inserts a TARFlow-style normalizing flow head at designated continuous-thought positions inside an LLM backbone while retaining the standard LM head and causal stream for text tokens. Continuous thoughts are distilled from explicit CoT, and the design is claimed to preserve left-to-right autoregressive generation, KV-cache compatibility, probabilistic sampling, and tractable likelihoods. The central empirical claim is that NF-CoT improves pass rates on code-generation benchmarks relative to explicit CoT and prior latent-reasoning baselines while reducing intermediate-reasoning cost.

Significance. If the empirical gains are reproducible and the compatibility properties hold under standard decoding, the work would provide a concrete route to higher-bandwidth latent computation inside existing autoregressive LLMs without sacrificing the engineering advantages that have made explicit CoT practical. The explicit use of normalizing flows for exact likelihoods over continuous states is a methodological strength that could be reused in other latent-reasoning settings.

major comments (2)

[Abstract] Abstract: the central claim that NF-CoT 'improves pass rates over explicit-CoT and prior latent-reasoning baselines' is stated without any quantitative results, table of pass rates, list of baselines, number of runs, or statistical tests. Because this empirical result is the primary evidence offered for the framework's value, its absence prevents evaluation of the central claim.
[Method] The manuscript supplies no equations, architecture diagram, or pseudocode showing how the NF head is inserted into the causal stream, how the TARFlow transformation is conditioned on preceding tokens, or how the joint likelihood is computed when NF and LM heads alternate. Without these details the claim that the design 'provides exact likelihoods' and 'enables probabilistic left-to-right decoding with the original KV cache' cannot be verified.

minor comments (1)

[Abstract] The abstract introduces 'TARFlow-style' without a citation or brief definition; a reference to the original TARFlow paper should appear at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that NF-CoT 'improves pass rates over explicit-CoT and prior latent-reasoning baselines' is stated without any quantitative results, table of pass rates, list of baselines, number of runs, or statistical tests. Because this empirical result is the primary evidence offered for the framework's value, its absence prevents evaluation of the central claim.

Authors: We agree that the abstract would benefit from quantitative support for the central empirical claim. In the revised version we will incorporate specific pass-rate numbers, the list of baselines, number of runs, and reference to statistical tests drawn from the experimental results already present in the full manuscript. revision: yes
Referee: [Method] The manuscript supplies no equations, architecture diagram, or pseudocode showing how the NF head is inserted into the causal stream, how the TARFlow transformation is conditioned on preceding tokens, or how the joint likelihood is computed when NF and LM heads alternate. Without these details the claim that the design 'provides exact likelihoods' and 'enables probabilistic left-to-right decoding with the original KV cache' cannot be verified.

Authors: We acknowledge that the current manuscript text does not contain the requested equations, diagram, or pseudocode. The revised manuscript will add a dedicated methods subsection with the precise equations for NF-head insertion and conditioning, the joint likelihood factorization, an architecture diagram, and pseudocode illustrating KV-cache compatibility and left-to-right sampling. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces NF-CoT as a new architectural integration of TARFlow-style normalizing flows into an LLM backbone for modeling distilled continuous thoughts. All load-bearing elements (exact likelihoods, KV-cache compatibility, left-to-right generation, policy-gradient optimization) are achieved by explicit design choices in the causal stream and dual-head setup rather than by re-expressing fitted quantities or prior self-citations as predictions. No equations reduce the benchmark improvements to input data by construction, and no uniqueness theorems or ansatzes are smuggled via self-citation. The central empirical claim therefore rests on external evaluation rather than definitional equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated assumption that the normalizing-flow head can be trained to produce useful continuous thoughts without additional constraints.

pith-pipeline@v0.9.1-grok · 5802 in / 1100 out tokens · 24781 ms · 2026-06-28T01:09:56.625647+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

MIMFlow: Integrating Masked Image Modeling with Normalizing Flows for End-to-End Image Generation
cs.CV 2026-06 unverdicted novelty 6.0

MIMFlow is an end-to-end model that routes semantic latents through a normalizing flow while a decoder handles high-frequency pixels, reporting FID 2.50 and 71.3% linear probing accuracy on ImageNet 256x256 with 128 tokens.

Reference graph

Works this paper leans on

28 extracted references · cited by 1 Pith paper

[1]

**Understanding the Fibonacci Sequence**: The Fibonacci sequence is defined as: - \( F(1) = 0 \) - \( F(2) = 1 \) - \( F(n) = F(n-1) + F(n-2) \)
[2]

This optimizes, since we don’t need to use \( n \) variables

**Compressing the Formula for Efficiency**: We use a fixed amount of space to store the last three Fibonacci numbers and calculate the next. This optimizes, since we don’t need to use \( n \) variables
[3]

**Recursive Fibonacci Term**: Calculate the \( n \)th Fibonacci number: - \( F(n) = F(n-1) + F(n-2) \)
[4]

""Compute the n-th element of the fibfib sequence

**Flow**: This calculation is done in constant \( O(1) \) extra space by using three variables. Generated Answer: sample_idx=49, passed def fibfib(n: int) -> int: """Compute the n-th element of the fibfib sequence.""" # Base cases if n == 0: return 0 elif n == 1: return 0 elif n == 2: return 1 # Initialize the first three elements of the sequence a, b, c ...
[5]

- This prevents the exponential number of computations and ensures the sequence is computed efficiently

**Initialization:** - We use a stack (list) to store Fibonacci numbers, starting with the base cases ‘n = 0‘ and ‘n = 1‘. - This prevents the exponential number of computations and ensures the sequence is computed efficiently
[6]

**Base Case:** - The first two Fibonacci numbers are known: ‘fib(0) = 0‘ and ‘fib(1) = 1‘
[7]

- Compute the rest of the sequence numbers: ‘fib(n-1)‘, ‘fib(n-2)‘, etc., from 3 to ‘n‘

**Compute the Sequence:** - For a given ‘n‘, compute the value of ‘fib(n)‘ by summing the previous two Fibonacci numbers (last and nth). - Compute the rest of the sequence numbers: ‘fib(n-1)‘, ‘fib(n-2)‘, etc., from 3 to ‘n‘. 16
[8]

**Return Computed Values:** - Return the array of computed Fibonacci numbers
[9]

""Compute the n-th element of the FibFib sequence

**Implementation**: The function computes the necessary Fibonacci number for ‘n‘, and indeed utilizes efficient tabulation to avoid redundant computations. Generated Answer: sample_idx=3, passed def fibfib(n: int) -> int: """Compute the n-th element of the FibFib sequence.""" # Base cases if n == 0: return 0 if n == 1: return 0 if n == 2: return 1 # Initi...
[10]

**Define the Function:** - Use a cache (dictionary) to store the value of Fibonacci numbers we have computed
[11]

""Compute the n-th element of the FibFib sequence

**Recursive Computation:** - We compute the Fibonacci number by first calculating the function recursively and store it in the cache. - Use base cases to reduce complexity. Generated Answer: sample_idx=35, passed def fibfib(n: int) -> int: """Compute the n-th element of the FibFib sequence.""" # Initialize a dictionary to store the computed values memo = ...
[12]

- For example, if ‘s = "abcdef"‘, then the rotations are: ‘a., b., c., d., e., f.‘ and ‘b., c., d., e., f., a.‘

**Understanding Rotations**: - A rotation of a string is a substring of the original string. - For example, if ‘s = "abcdef"‘, then the rotations are: ‘a., b., c., d., e., f.‘ and ‘b., c., d., e., f., a.‘
[13]

- For each rotation, we check if it matches the pattern

**Checking Rotations**: - We can generate all possible rotations of a string. - For each rotation, we check if it matches the pattern
[14]

""Check if any rotation of the second word is a substring in the first word

**Rotating and Sorting**: - This approach can be somewhat complex for all rotations. Generated Answer: sample_idx=0, passed def cycpattern_check(a, b): """Check if any rotation of the second word is a substring in the first word.""" # Get the length of the second word. len_b = len(b) # Check each rotation of the second word. for i in range(len_b): # Creat...
[15]

Rotations can be generated by checking if the string is a substring of any of its cyclic permutations

**Check the String Rotation:** - For each rotation, check if the ‘check_string‘ is a substring. Rotations can be generated by checking if the string is a substring of any of its cyclic permutations
[16]

abcd"‘, the rotations are ‘[

**Understanding Rotations:** - **Rotations** are permutations of the string. For any given rotation, if you concatenate the string with itself, the original string becomes a substring. - Example: If ‘s = "abcd"‘, the rotations are ‘["abcd", "bcda", "cdab", "dabc"]‘
[17]

**Check Substring Check:** - If ‘check_string‘ is a substring of any rotation, it can be checked if it is a substring of ‘s + s‘ ( the concatenated version of ‘s‘)
[18]

A simple check would be: - Check if ‘check_string‘ is a substring of ‘a‘, ‘b‘, ‘c‘, or ‘d‘ for each rotation

**Check for No Rotation:** - We need to check if the ‘check_string‘ is also a substring of the first rotation. A simple check would be: - Check if ‘check_string‘ is a substring of ‘a‘, ‘b‘, ‘c‘, or ‘d‘ for each rotation
[19]

- If any rotation contains ‘check_string‘, return ‘True‘

**Implement a Check for All Rotations:** - Check all rotations by rotating the string and checking the substring condition. - If any rotation contains ‘check_string‘, return ‘True‘. Generated Answer: sample_idx=16, passed def cycpattern_check(a, b): if not b: return True if len(b) > len(a): return False # Generate all rotations of b rotations = [b[i:] + b...
[20]

This will handle the case where the user strings are partially lowercased

**Handle Case**: - Convert the string to handle cases (uppercase and lowercase). This will handle the case where the user strings are partially lowercased
[21]

**Handle Edge Cases**: - Handle edge cases such as empty strings or the handle of digits properly
[22]

- If the shift handles the letters beyond ’z’ and ’Z’, wrap around to the beginning of the alphabet

**Encoding and Rotation**: - Each character in the string should be shifted by the two-digit number provided. - If the shift handles the letters beyond ’z’ and ’Z’, wrap around to the beginning of the alphabet
[23]

""Encrypt the string by shifting the alphabet by 4 places

**Handle Edge Handling**: - Repeat the shifted character to handle each character individually. Generated Answer: sample_idx=1, passed def encrypt(s): """Encrypt the string by shifting the alphabet by 4 places.""" # Define the alphabet alphabet = ’abcdefghijklmnopqrstuvwxyz’ # Calculate the shift amount shift = 4 # Initialize an empty result string encryp...
[24]

**Function Definition**: Create the function ‘rotate_encrypt‘ that takes two parameters: the ‘string‘ 20 and the ‘shift‘
[25]

If the character is shifted by 26, it should wrap around the alphabet

**Initialize the Result**: Use a variable to shift each character by the specified number. If the character is shifted by 26, it should wrap around the alphabet
[26]

- Use modulo 26 to ensure the shifts are shifted

**Iterate Through the String**: Iterate through each character in the string and apply the rotation: - Shift each letter by the ASCII value of the letter in the opposite alphabet. - Use modulo 26 to ensure the shifts are shifted
[27]

**Encrypt the Character**: - Initialize the function to add a shifted character to the result
[28]

""Encrypts the input string by rotating the alphabet by 4 positions

**Return the Encrypted String**: Return the encrypted string. Generated Answer: sample_idx=23, passed def encrypt(s): """Encrypts the input string by rotating the alphabet by 4 positions.""" encrypted_string = [] for char in s: if ’a’ <= char <= ’z’:# Check if the character is a lowercase letter # Rotate the character by 4 positions new_char = chr((ord(ch...

2048

[1] [1]

**Understanding the Fibonacci Sequence**: The Fibonacci sequence is defined as: - \( F(1) = 0 \) - \( F(2) = 1 \) - \( F(n) = F(n-1) + F(n-2) \)

[2] [2]

This optimizes, since we don’t need to use \( n \) variables

**Compressing the Formula for Efficiency**: We use a fixed amount of space to store the last three Fibonacci numbers and calculate the next. This optimizes, since we don’t need to use \( n \) variables

[3] [3]

**Recursive Fibonacci Term**: Calculate the \( n \)th Fibonacci number: - \( F(n) = F(n-1) + F(n-2) \)

[4] [4]

""Compute the n-th element of the fibfib sequence

**Flow**: This calculation is done in constant \( O(1) \) extra space by using three variables. Generated Answer: sample_idx=49, passed def fibfib(n: int) -> int: """Compute the n-th element of the fibfib sequence.""" # Base cases if n == 0: return 0 elif n == 1: return 0 elif n == 2: return 1 # Initialize the first three elements of the sequence a, b, c ...

[5] [5]

- This prevents the exponential number of computations and ensures the sequence is computed efficiently

**Initialization:** - We use a stack (list) to store Fibonacci numbers, starting with the base cases ‘n = 0‘ and ‘n = 1‘. - This prevents the exponential number of computations and ensures the sequence is computed efficiently

[6] [6]

**Base Case:** - The first two Fibonacci numbers are known: ‘fib(0) = 0‘ and ‘fib(1) = 1‘

[7] [7]

- Compute the rest of the sequence numbers: ‘fib(n-1)‘, ‘fib(n-2)‘, etc., from 3 to ‘n‘

**Compute the Sequence:** - For a given ‘n‘, compute the value of ‘fib(n)‘ by summing the previous two Fibonacci numbers (last and nth). - Compute the rest of the sequence numbers: ‘fib(n-1)‘, ‘fib(n-2)‘, etc., from 3 to ‘n‘. 16

[8] [8]

**Return Computed Values:** - Return the array of computed Fibonacci numbers

[9] [9]

""Compute the n-th element of the FibFib sequence

**Implementation**: The function computes the necessary Fibonacci number for ‘n‘, and indeed utilizes efficient tabulation to avoid redundant computations. Generated Answer: sample_idx=3, passed def fibfib(n: int) -> int: """Compute the n-th element of the FibFib sequence.""" # Base cases if n == 0: return 0 if n == 1: return 0 if n == 2: return 1 # Initi...

[10] [10]

**Define the Function:** - Use a cache (dictionary) to store the value of Fibonacci numbers we have computed

[11] [11]

""Compute the n-th element of the FibFib sequence

**Recursive Computation:** - We compute the Fibonacci number by first calculating the function recursively and store it in the cache. - Use base cases to reduce complexity. Generated Answer: sample_idx=35, passed def fibfib(n: int) -> int: """Compute the n-th element of the FibFib sequence.""" # Initialize a dictionary to store the computed values memo = ...

[12] [12]

- For example, if ‘s = "abcdef"‘, then the rotations are: ‘a., b., c., d., e., f.‘ and ‘b., c., d., e., f., a.‘

**Understanding Rotations**: - A rotation of a string is a substring of the original string. - For example, if ‘s = "abcdef"‘, then the rotations are: ‘a., b., c., d., e., f.‘ and ‘b., c., d., e., f., a.‘

[13] [13]

- For each rotation, we check if it matches the pattern

**Checking Rotations**: - We can generate all possible rotations of a string. - For each rotation, we check if it matches the pattern

[14] [14]

""Check if any rotation of the second word is a substring in the first word

**Rotating and Sorting**: - This approach can be somewhat complex for all rotations. Generated Answer: sample_idx=0, passed def cycpattern_check(a, b): """Check if any rotation of the second word is a substring in the first word.""" # Get the length of the second word. len_b = len(b) # Check each rotation of the second word. for i in range(len_b): # Creat...

[15] [15]

Rotations can be generated by checking if the string is a substring of any of its cyclic permutations

**Check the String Rotation:** - For each rotation, check if the ‘check_string‘ is a substring. Rotations can be generated by checking if the string is a substring of any of its cyclic permutations

[16] [16]

abcd"‘, the rotations are ‘[

**Understanding Rotations:** - **Rotations** are permutations of the string. For any given rotation, if you concatenate the string with itself, the original string becomes a substring. - Example: If ‘s = "abcd"‘, the rotations are ‘["abcd", "bcda", "cdab", "dabc"]‘

[17] [17]

**Check Substring Check:** - If ‘check_string‘ is a substring of any rotation, it can be checked if it is a substring of ‘s + s‘ ( the concatenated version of ‘s‘)

[18] [18]

A simple check would be: - Check if ‘check_string‘ is a substring of ‘a‘, ‘b‘, ‘c‘, or ‘d‘ for each rotation

**Check for No Rotation:** - We need to check if the ‘check_string‘ is also a substring of the first rotation. A simple check would be: - Check if ‘check_string‘ is a substring of ‘a‘, ‘b‘, ‘c‘, or ‘d‘ for each rotation

[19] [19]

- If any rotation contains ‘check_string‘, return ‘True‘

**Implement a Check for All Rotations:** - Check all rotations by rotating the string and checking the substring condition. - If any rotation contains ‘check_string‘, return ‘True‘. Generated Answer: sample_idx=16, passed def cycpattern_check(a, b): if not b: return True if len(b) > len(a): return False # Generate all rotations of b rotations = [b[i:] + b...

[20] [20]

This will handle the case where the user strings are partially lowercased

**Handle Case**: - Convert the string to handle cases (uppercase and lowercase). This will handle the case where the user strings are partially lowercased

[21] [21]

**Handle Edge Cases**: - Handle edge cases such as empty strings or the handle of digits properly

[22] [22]

- If the shift handles the letters beyond ’z’ and ’Z’, wrap around to the beginning of the alphabet

**Encoding and Rotation**: - Each character in the string should be shifted by the two-digit number provided. - If the shift handles the letters beyond ’z’ and ’Z’, wrap around to the beginning of the alphabet

[23] [23]

""Encrypt the string by shifting the alphabet by 4 places

**Handle Edge Handling**: - Repeat the shifted character to handle each character individually. Generated Answer: sample_idx=1, passed def encrypt(s): """Encrypt the string by shifting the alphabet by 4 places.""" # Define the alphabet alphabet = ’abcdefghijklmnopqrstuvwxyz’ # Calculate the shift amount shift = 4 # Initialize an empty result string encryp...

[24] [24]

**Function Definition**: Create the function ‘rotate_encrypt‘ that takes two parameters: the ‘string‘ 20 and the ‘shift‘

[25] [25]

If the character is shifted by 26, it should wrap around the alphabet

**Initialize the Result**: Use a variable to shift each character by the specified number. If the character is shifted by 26, it should wrap around the alphabet

[26] [26]

- Use modulo 26 to ensure the shifts are shifted

**Iterate Through the String**: Iterate through each character in the string and apply the rotation: - Shift each letter by the ASCII value of the letter in the opposite alphabet. - Use modulo 26 to ensure the shifts are shifted

[27] [27]

**Encrypt the Character**: - Initialize the function to add a shifted character to the result

[28] [28]

""Encrypts the input string by rotating the alphabet by 4 positions

**Return the Encrypted String**: Return the encrypted string. Generated Answer: sample_idx=23, passed def encrypt(s): """Encrypts the input string by rotating the alphabet by 4 positions.""" encrypted_string = [] for char in s: if ’a’ <= char <= ’z’:# Check if the character is a lowercase letter # Rotate the character by 4 positions new_char = chr((ord(ch...

2048