explanation
How the source derives seven recursive depths
The missing bridge is explicit: the source reuses a continuity address as recursive turning depth.
The current source-specific derivation, including its ten-part metric and explicit address-to-depth premise; supersedes blanket “not derived” wording.
First locate the continuity shares
Under equal-start additive inheritance, the larger share proceeds through 1/2, 2/3, 3/5, 5/8 and onward toward 1/φ. Its complementary share remains present. Together their equal-start envelope lies between one third and two thirds.
The source divides the normalized contraction-to-expansion interval into ten equal, one-based addresses. Both the expansion-facing golden share, approximately 0.618, and the upper boundary 2/3 fall in its seventh address.
a(q) = min(10, 1 + floor(10q))
a(1/φ) = a(2/3) = 7
Then make the recursive connection
The September 1 author correction states D:=a(1/φ)=a(2/3). That is the bridge from continuity address to recursive depth. With it, the source obtains seven turning depths within an active dimension. The returned first depth is marked as changed because it receives the consequence of the preceding passage.
Seven is therefore not an estimate of how many Fibonacci steps happen before convergence. Convergence continues, and its stopping time depends on a chosen tolerance. Nor is seven the same count as the twelve I/O/R addresses within a depth.
L₁,k → L₂,k → … → L₇,k → L′₁,k
Keep the stronger question exact
The ten-part metric and its depth-address rule are architectural premises of this derivation. A stronger result would derive a first qualifying depth from an independently specified recursive update and closure condition, without supplying that address rule. That remains a precise research task; it does not erase the source’s existing conditional derivation.
Questions to carry forward
Are seven explanation layers and seven engine depths interchangeable?
They serve different purposes. Explanation layers organize understanding; recursive depths belong to the source’s operating architecture. Any correspondence needs an explicit mapping.
Where this belongs in the journey
- Explore geometry →geometry
- Explore matrix readings →matrix
- Follow the connected course →core
- Continue with seven depth return →The teaching journey