Lilienfeld 1925 Complete the Conducting Path

# Complete the Conducting Path: One Bad Segment Can Still Ruin Every Good One

Step 5 treated the conducting path as a single uniform resistor, set by one gap length; Step 7 and Step 8 then showed that the film actually filling that gap is neither uniformly thick nor uniformly converted. Completing the conducting path means confronting what that variation actually does to the path as a whole, and the answer is not forgiving. A conductor spanning the terminal gap is not one resistance — it is many small segments in series, one after another along the path, and a series path's overall behavior is dominated by its single worst segment, not by an average over all of them.

## 1. Series Segments Add; One Open Segment Breaks the Whole Path

Whatever variation Step 7's shadowing and Step 8's reaction completeness left behind, the path's total resistance is simply the sum of every segment's own resistance along its length:

$$ R_{\text{path}} \;=\; \sum_i R_{\text{segment},i} $$

A path made of segments that are each individually reasonable, but none perfect, still sums to a reasonable total resistance — exactly what Step 5's simple $R_{\text{film}}$ estimate assumed all along. But if even one segment along that path never actually cleared Step 7's percolation threshold, or never reached a conducting composition under Step 8's reaction, that segment's resistance is effectively infinite, and an infinite term in a sum makes the sum infinite too — regardless of how good every other segment happens to be. This is the real content behind this step's instruction to "retain a continuous" film: continuity is not a nice-to-have property that slightly improves performance, it is the one property a series path cannot do without at all.

## 2. Real Diagram: Nine Good Segments, One Gap, Zero Conduction

A Series Path Is Only as Good as Its Worst Link Every other segment conducting perfectly changes nothing here open eight good segments conduct fine; the path as a whole still conducts nothing

## 3. Across a Whole Batch, Continuity Itself Has a Yield

Treating each segment's own chance of actually clearing Step 7's and Step 8's requirements as an independent probability, the probability that the entire path ends up continuous — not any single segment's probability, but the whole chain's — is the product of every segment's own survival probability:

$$ P_{\text{path}} \;=\; \prod_i P_{\text{segment},i} $$

Because this is a product of numbers each less than one, $P_{\text{path}}$ falls steadily as the number of segments along the path grows, even if each individual segment is quite reliable on its own. A longer gap, or a rougher step-coverage profile that effectively divides the path into more independently-risky segments, lowers the overall yield of continuous devices from a batch in a way that no single segment's own quality can fully compensate for — which is the quantitative reason this step exists as its own checkpoint, rather than being folded silently into Step 8's sulfurization and assumed to work out.

## Real Diagram: Yield Falls as the Chain Gets Longer

Continuous-Path Yield Versus Segment Count Each segment's own high reliability still compounds downward number of segments along the path P_path, continuous-path yield short gap, few segments long gap, rough coverage

## Complete the Conducting Path's Place in the Process Lineage

Complete the Conducting Path follows Step 8, Sulfurize the Deposited Copper, whose segment-by-segment conversion this step now has to confirm actually adds up to a working whole; it precedes Step 10, Attach Electrical Connections, which assumes a continuous path already exists and has nothing further to offer if it doesn't. It is the ninth and final step of this concept's film-formation sequence, and the point where Step 5's single-resistance model, Step 7's shadowed-thickness concern, and Step 8's variable-composition concern all converge onto the one property that actually determines whether any current flows between the terminals at all.

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