Editing Openai/6948e03d-a2c8-8004-b437-592576c8ff41 (section)

=== Assistant: To quantify compression fidelity in the context of your chain/comprehension experiment, you’d want a metric that measur… ===
To quantify compression fidelity in the context of your chain/comprehension experiment, you’d want a metric that measures how much of the original logical structure is retained when the reader switches from full tracking to a reconstructive or compressed mode. Here’s how I would formalize it:

===== 1. Define the “full information” =====
* Each chain has N logical units (clauses, dependencies, or ideas).
* Let the full chain be represented as a sequence S=[s1,s2,…,sN]S = [s_1, s_2, …, s_N]S=[s1,s2,…,sN].

===== 2. Capture what the reader retains =====
* After reading, the reader produces a reconstructed version R=[r1,r2,…,rM]R = [r_1, r_2, …, r_M]R=[r1,r2,…,rM].
* MMM may be ≤ N, and some elements may be misordered or misinterpreted.

===== 3. Compute fidelity =====
* Exact match fraction (simple version):

F=# of correctly retained unitsNF = \frac{\text{\# of correctly retained units}}{N}F=N# of correctly retained units
* Weighted match (more nuanced): - Give partial credit for partially retained dependencies or correctly inferred relationships. - For nested or hierarchical clauses, consider tree similarity metrics (e.g., tree edit distance).
* Compression ratio (optional):

C=MNC = \frac{M}{N} C=NM
* This measures how much the chain was compressed in output; fidelity should ideally normalize for this, giving a sense of information retained per unit of compression.

===== 4. Adjust for subjective scoring =====
* If using self-report, ask the reader to rate their confidence or perceived completeness for each unit.
* Can combine subjective and objective measures:

Compression Fidelity=F×confidence factor\text{Compression Fidelity} = F \times \text{confidence factor}Compression Fidelity=F×confidence factor

===== 5. Aggregate over chains =====
* For a phase-transition analysis, compute fidelity for each chain.
* Plot fidelity vs. chain size to see the critical point where reconstruction begins and fidelity drops.

Essentially, compression fidelity measures the proportion of original information that survives cognitive compression, weighted by correctness or confidence, normalized by chain length.

If you want, I can draft a concrete formula and example calculation for one of your 16 chains, showing how to turn your self-reported results into a compression fidelity metric.