Editing Openai/6926a9e8-c270-8004-89f7-c49d6ee67c80 (section)

=== 🔍 Detailed Comparison: How Each Uses Partial Information ===

==== 1. Use of the partial response’s conceptual direction ====

The partial response says:

: “We construct a sequence (like geometric progression) to get a lower bound for λ.”

===== Solution A =====
* Tries to analyze extremal configurations
* Thinks about making neighbors small to push a term toward 1
* Considers periodic patterns → It follows the partial response’s general direction (maximizing S through structure).

===== Solution B =====
* Does not follow the partial reasoning at all
* Begins re-deriving the problem from scratch
* Wanders into unstructured discussions (Cauchy-Schwarz, equal values) → It essentially ignores the partial response.

==== 2. Use of the geometric-sequence construction ====
𝑎
𝑖
=
𝑥
𝑖
−
1
a
i
	

=x
i−1

The partial solution ends with:

: “Let 
: 𝑎
: 𝑖
: =
: 𝑥
: 𝑖
: −
: 1
: a
: i
: 	
: 
: =x
: i−1
:  for small 
: 𝑥
: x. Now continue.”

===== Solution A =====

❌ Does not pick up the geometric sequence
BUT
✔ It continues the spirit — exploring small vs large neighborhoods and asymmetry.

===== Solution B =====

❌ Completely ignores the geometric sequence
❌ Does not continue from the ending point
→ This violates the instruction: “Continue from where the partial response ends.”

==== 3. Use of homogeneity / ratios ====
𝑎
𝑖
+
𝑗
/
𝑎
𝑖
a
i+j
	

/a
i
	

Partial response implicitly sets up for ratio analysis (because geometric sequences make ratios constant).

===== Solution A =====

✔ Rewrites each term in terms of ratios
✔ Uses this to reason about extremal constructions
→ This is consistent with the intended direction.

===== Solution B =====

✔ Mentions the rewrite
❌ Does not use ratios in any meaningful argument
→ A shallow restatement only.

==== 4. Use of cyclic dependency structure ====

Important because every aja_jaj affects k+1k+1k+1 terms.

===== Solution A =====

✔ Mentions cyclic dependence
✔ Uses it to argue constraints on making many terms ≈ 1
→ Correct use.

===== Solution B =====

✔ Mentions cyclic indexing
❌ Does not incorporate it into reasoning.

==== 5. Progress toward bounding or maximizing S ====

===== Solution A =====
* Attempts patterns to maximize S
* Thinks about how many “large terms” can be made ≈ 1
* Although not finished, this is exactly what the partial response wants.

===== Solution B =====
* Repeatedly resets its reasoning
* No coherent path toward maximizing S
* Effectively restarts the problem rather than continuing it.