.

diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html
index 548d0cf..8f840b4 100644 (file)
--- a/document/book/TM-2026.html
+++ b/document/book/TM-2026.html
@@ -989,23 +989,24 @@
 
       <img src="TTCA_style_TM_reverse.png" class="rt-diagram" alt="HU reverse machine">
 
+      <p>With data payload examination removed from the controller, long strings of states with a few decision points emerge. There is a loop on the left, and a tail leading to <RT·code>done</RT·code>, on the right. The number of states and arcs is a constant independent of how much data is to be reversed.</p>
+
       <RT·code>
       void TTCA·reverse_string() {
-        // Phase 1: Scan right to EOM and initialize the EOR marker
+        // Initialization: Scan to EOM and setup the EOR boundary
         read_g();
         while(g != EOM){right(); read_g();}
         right();
         write_σ(EOR);
 
-        // Phases 2 through 5: The main shuttle cycle
         while(true){
-
-          // Phase 2: Fetch next unprocessed character and check boundaries
+          // The Fetch Pivot: Locate the next unprocessed character
           left();
           read_g();
           while(g == EOM || g == SP){
             status();
-            if(g == on_leftmost) return;
+            // Termination: Short-circuit for empty string
+            if(g == on_leftmost) return; 
             left();
             read_g();
           }
@@ -1013,19 +1014,10 @@
           read_d();
           status();
 
-          // Phase 6: Final character carry and clean halt
-          if(g == on_leftmost){
-            write_σ(SP);
-            right();
-            read_g();
-            while(g != EOR){right(); read_g();}
-            write_d();
-            right();
-            write_σ(EOR);
-            return;
-          }
+          // Center Break: Q·Check_Last_Char routes to the final chain
+          if(g == on_leftmost) break;
 
-          // Phase 3 & 4: Mark location, carry opaque data, advance EOR
+          // Main Carry Loop: Mark, carry, drop, and return to pivot
           write_σ(SP);
           right();
           read_g();
@@ -1034,14 +1026,40 @@
           right();
           write_σ(EOR);
 
-          // Phase 5: Return left to EOM to begin the next fetch cycle
           left();
           read_g();
           while(g != EOM){left(); read_g();}
         }
+
+        // Final Character Chain: Handle the last payload without a return sweep
+        write_σ(SP);
+        right();
+        read_g();
+        while(g != EOR){right(); read_g();}
+        write_d();
+        right();
+        write_σ(EOR);
+
+        // Termination: Q·Done
+        return;
       }
       </RT·code>
 
+      <p>The total number of steps for reversing an <RT·math>n</RT·math> character string using the TTCA architecture:</p>
+
+      <div style="margin-left: 2em;">
+        <RT·math>
+          \text{steps} = 
+          \begin{cases} 
+            7 & \text{if } n = 0 \\ 
+            4.5n^2 + 11.5n + 5 & \text{if } n \ge 1 
+          \end{cases}
+        </RT·math>
+      </div>
+
+      <p>This machine spends a lot of time shuttling the head between two context area. One with the original string, and one with the result reverse machine. This suggests that a two head version would be faster.</p>
+
+
       <RT·chapter>The Turing Machine architecture/organization</RT·chapter>
 
 <p>