Closure
A local state can return while its enclosing relation remains asymmetric
A state can change and return to where it began.
That is a return.
But R2D gives closure a more specific meaning.
A local return becomes a closure when that completed path is compatible with an occurrence at the enclosing boundary.
Thus:
Return is local path completion.
Closure is local return within an enclosing realization relation.
This distinction matters because the local state can close even when the enclosing relations organizing its forward and return paths do not.
That is the basis of asymmetric closure.
A Macrostate Return
At boundary \(B_n\), consider a finite readable path
\[ \mathcal{R}_n: i_0 \rightarrow i_1 \rightarrow \cdots \rightarrow i_m. \]
If
\[ i_m=i_0, \]
the path has returned to its initial macrostate.
Thus:
\[ \boxed{ \mathcal{R}_n: i_0 \rightarrow i_1 \rightarrow \cdots \rightarrow i_0 } \]
is a macrostate return.
A return is defined entirely by the local state path.
No statement about the enclosing relation has yet been made.
Return Is Not Yet Closure
Now consider the enclosing boundary
\[ B_{n+1}. \]
If the completed local return is compatible with at least one enclosing occurrence at \(B_{n+1}\), R2D calls it a closure.
For peer boundary \(\alpha\), write
\[ \boxed{ \mathcal{C}_{n,\alpha}. } \]
So:
\[ \boxed{ \text{return} + \text{compatibility with an enclosing occurrence} = \text{closure}. } \]
The local path is readable at \(B_n\).
Its enclosing significance belongs to \(B_{n+1}\).
This is what makes closure a recursive rather than merely cyclic concept.
The Minimal Closure
The simplest closure has two opposed legs:
\[ \boxed{ \mathcal{C}_{n,\alpha}: i \rightarrow k \rightarrow i. } \]
The first leg is
\[ i\rightarrow k, \]
and the return leg is
\[ k\rightarrow i. \]
The local state closes because the final macrostate equals the initial macrostate.
But this does not require the two legs to occur under identical enclosing conditions.
That is the key point.
Local Reversal Does Not Require Enclosing Reversal
Suppose the forward leg
\[ i\rightarrow k \]
is realized under enclosing condition \(H\).
Suppose the return leg
\[ k\rightarrow i \]
is realized under enclosing condition \(L\).
The local transition has reversed.
But the enclosing compatibility relation need not have reversed.
The extension multiplicities can therefore differ between the two branches:
\[ K_{n+1|n,i}^{H}, \qquad K_{n+1|n,k}^{H}, \]
and
\[ K_{n+1|n,i}^{L}, \qquad K_{n+1|n,k}^{L}. \]
Likewise, the enclosing productive constraints need not be identical on the two legs.
Thus:
\[ \boxed{ \text{local path reversal} \not\Rightarrow \text{reversal of the enclosing relation}. } \]
This is the primitive structure of asymmetric closure.
Possibility Along the Forward Leg
Under enclosing condition \(H\), the directional possibility for
\[ i\rightarrow k \]
is
\[ \boxed{ \Delta P_{n,\alpha,H} = \ln \left( \frac{ K_{n+1|n,k}^{H} }{ K_{n+1|n,i}^{H} } \right). } \]
For the return leg under condition \(L\),
\[ k\rightarrow i, \]
the directional possibility is
\[ \boxed{ \Delta P_{n,\alpha,L} = \ln \left( \frac{ K_{n+1|n,i}^{L} }{ K_{n+1|n,k}^{L} } \right). } \]
If the enclosing compatibility relation were identical on the two legs, then these quantities would be exact opposites:
\[ \Delta P_{n,\alpha,L} = - \Delta P_{n,\alpha,H}. \]
But asymmetric closure does not require that condition.
Therefore,
\[ \boxed{ \Delta P_{n,\alpha,L} \neq - \Delta P_{n,\alpha,H} } \]
is allowed.
Closure Possibility
The directional possibility accumulated around the completed closure is
\[ \boxed{ \Delta P_{n,\alpha,\mathcal C} = \Delta P_{n,\alpha,H} + \Delta P_{n,\alpha,L}. } \]
More generally,
\[ \boxed{ \Delta P_{n,\alpha,\mathcal C} = \sum_{\chi\in\mathcal C_{n,\alpha}} \Delta P_{n,\alpha,\chi}. } \]
The local state can return even when
\[ \boxed{ \Delta P_{n,\alpha,\mathcal C} \neq0. } \]
This means:
The local state can close while enclosing possibility does not.
That distinction is fundamental.
Constraint Can Also Differ Around the Closure
The two branches can also experience different productive constraints.
For the forward leg,
\[ \Delta R_{n,\alpha,H} = C_{n+1,H}^{\mathrm{out}} \lambda_{n,\alpha,i\rightarrow k}. \]
For the return leg,
\[ \Delta R_{n,\alpha,L} = C_{n+1,L}^{\mathrm{out}} \lambda_{n,\alpha,k\rightarrow i}. \]
Because distinguishability is oriented,
\[ \lambda_{n,\alpha,k\rightarrow i} = - \lambda_{n,\alpha,i\rightarrow k}. \]
But the enclosing constraints themselves need not be equal:
\[ C_{n+1,H}^{\mathrm{out}} \neq C_{n+1,L}^{\mathrm{out}}. \]
Therefore the two retention terms need not cancel.
Closure Retention
The productive retention accumulated around the closure is
\[ \boxed{ \Delta R_{n,\alpha,\mathcal C} = \Delta R_{n,\alpha,H} + \Delta R_{n,\alpha,L}. } \]
More generally,
\[ \boxed{ \Delta R_{n,\alpha,\mathcal C} = \sum_{\chi\in\mathcal C_{n,\alpha}} \Delta R_{n,\alpha,\chi}. } \]
Possibility and retention therefore accumulate independently around the same completed local path.
Closure Asymmetry
The net asymmetry remaining around the closure is
\[ \boxed{ \Delta A_{n,\alpha,\mathcal C} = \Delta P_{n,\alpha,\mathcal C} - \Delta R_{n,\alpha,\mathcal C}. } \]
Complete retention satisfies
\[ \boxed{ \Delta A_{n,\alpha,\mathcal C}=0, } \]
or equivalently,
\[ \boxed{ \Delta P_{n,\alpha,\mathcal C} = \Delta R_{n,\alpha,\mathcal C}. } \]
But a completed local return does not require complete retention.
Therefore a closure can have
\[ \boxed{ \Delta A_{n,\alpha,\mathcal C}\neq0. } \]
The Most Important Closure Result
The local path
\[ i\rightarrow k\rightarrow i \]
returns exactly.
But the enclosing relation need not.
That gives the central distinction:
\[ \boxed{ \text{local state closure} \neq \text{closure of the enclosing directional relation}. } \]
A locally closed cycle can therefore remain directionally organized.
This is what makes a turbine-like cycle possible in R2D.
The local machine returns.
The relation driving its opposed branches need not.
Native Multiplicity Closes Exactly
The local multiplicity difference for the forward transition is
\[ \Delta S_{n,\alpha,i\rightarrow k} = \ln \left( \frac{W_{n,k}}{W_{n,i}} \right). \]
For the return,
\[ \Delta S_{n,\alpha,k\rightarrow i} = \ln \left( \frac{W_{n,i}}{W_{n,k}} \right). \]
Adding them gives
\[ \Delta S_{n,\alpha,\mathcal C} = \Delta S_{n,\alpha,i\rightarrow k} + \Delta S_{n,\alpha,k\rightarrow i}. \]
Therefore,
\[ \boxed{ \Delta S_{n,\alpha,\mathcal C}=0. } \]
This is exact.
The native multiplicity ratios are reciprocal because the state path has returned within the same native multiplicity landscape.
Multiplicity Closure Does Not Imply Possibility Closure
The equation
\[ \Delta S_{n,\alpha,\mathcal C}=0 \]
does not imply
\[ \Delta P_{n,\alpha,\mathcal C}=0. \]
These quantities belong to different relations.
Native multiplicity is defined within the local boundary.
Possibility is defined through compatibility with the enclosing boundary.
Thus a closure can satisfy simultaneously
\[ \boxed{ \Delta S_{n,\alpha,\mathcal C}=0 } \]
and
\[ \boxed{ \Delta P_{n,\alpha,\mathcal C}\neq0. } \]
This is one of the cleanest examples of why R2D keeps local and enclosing count relations separate.
Multiplicity Closure Does Not Imply Asymmetry Closure
Likewise,
\[ \Delta S_{n,\alpha,\mathcal C}=0 \]
does not imply
\[ \Delta A_{n,\alpha,\mathcal C}=0. \]
The local multiplicity can return exactly while the enclosing-defined possibility-retention relation remains asymmetric.
Thus:
\[ \boxed{ \Delta S_{\mathcal C}=0 \qquad\text{while}\qquad \Delta A_{\mathcal C}\neq0 } \]
is perfectly consistent.
The first statement concerns possible local structure.
The second concerns enclosing directional organization.
A Turbine Makes the Distinction Intuitive
Imagine a turbine blade returning to the same position after one revolution.
Locally:
\[ \text{position} \rightarrow \text{different position} \rightarrow \text{original position}. \]
The blade has completed a return.
But the pressure difference across the turbine need not have disappeared.
The local mechanical path closes.
The enclosing directional relation remains.
That is the intuitive form of asymmetric closure:
The local structure returns while the larger relation organizing the cycle need not.
R2D abstracts this architecture away from the particular mechanical example.
Muscle Provides a Biological Example
Muscle also contains cyclic local realization.
Molecular switches can repeatedly move among states.
At steady shortening, the ensemble can maintain a statistically stable local cycle while ATP free-energy relations and mechanical load continue to organize the cycle.
The molecular states recur.
The enclosing chemical-mechanical relation does not have to disappear each time one molecular cycle closes.
Again:
\[ \boxed{ \text{local recurrence} \neq \text{loss of enclosing direction}. } \]
This is why closure is especially important for biological steady states.
Occurrence Around a Closure
So far, closure has described the state path.
But the two opposed legs can also be counted as realized occurrences.
Let
\[ \nu_{n,\alpha,H} \]
be the number of completed realizations of the forward oriented leg \(H\),
and
\[ \nu_{n,\alpha,L} \]
the number of completed realizations of the opposed leg \(L\),
within a specified finite realization domain.
These are path occurrence counts.
They are not simply sums of state occurrence counts along the path.
Define
\[ \tau_{n,\alpha,H} = \ln\nu_{n,\alpha,H}, \]
and
\[ \tau_{n,\alpha,L} = \ln\nu_{n,\alpha,L}. \]
This introduces another closure question:
Are the two opposed legs realized equally often?
State-Occurrence Closure Is Exact
The state occurrence coordinate for macrostate \(i\) is
\[ \tau_{n,i} = \ln\nu_{n,i}. \]
For a genuine closed state path, the signed transition differences telescope:
\[ \boxed{ \sum_{\chi\in\mathcal C_{n,\alpha}} \Delta\tau_{n,\alpha,\chi} = 0. } \]
This says that the state-defined occurrence coordinate returns with the closed state path.
It does not mean that the two complete opposed legs occur equally often.
That is a separate count relation.
Reciprocal Occurrence Realization
Define the directional occurrence-count efficiency
\[ \boxed{ \eta_{n,\alpha,\mathcal C} = \frac{ \nu_{n,\alpha,L} }{ \nu_{n,\alpha,H} }. } \]
Choose the orientation so that
\[ 0< \eta_{n,\alpha,\mathcal C} \leq1. \]
If the two opposed legs are realized equally often,
\[ \nu_{n,\alpha,L} = \nu_{n,\alpha,H}, \]
then
\[ \boxed{ \eta_{n,\alpha,\mathcal C}=1. } \]
This is reciprocal occurrence realization.
Nonreciprocal Occurrence Realization
If
\[ \nu_{n,\alpha,L} < \nu_{n,\alpha,H}, \]
then
\[ \boxed{ 0< \eta_{n,\alpha,\mathcal C} <1. } \]
The two opposed legs are realized with unequal directional efficiency.
Define the logarithmic opposed-leg difference
\[ \Delta\tau_{n,\alpha,\mathcal C} = \ln\eta_{n,\alpha,\mathcal C}. \]
Then nonreciprocity gives
\[ \Delta\tau_{n,\alpha,\mathcal C}<0. \]
This does not mean occurrence has been destroyed.
It means the opposed directions are realized with unequal count efficiency.
Closure Asymmetry and Occurrence Nonreciprocity Are Not the Same
This distinction is essential.
A closure can have
\[ \Delta P_{n,\alpha,\mathcal C}\neq0 \]
while still satisfying
\[ \eta_{n,\alpha,\mathcal C}=1. \]
Or it can have
\[ \Delta P_{n,\alpha,\mathcal C}\neq0 \]
together with
\[ 0< \eta_{n,\alpha,\mathcal C} <1. \]
Therefore:
\[ \boxed{ \text{asymmetric enclosing organization} \not\equiv \text{nonreciprocal occurrence realization}. } \]
The first concerns possibility and retention.
The second concerns the realized occurrence counts of the opposed closure legs.
They must remain distinct.
Occurrence-Count Loss
For the chosen orientation, R2D defines
\[ \boxed{ L_{\tau,n,\alpha,\mathcal C} = -\ln\eta_{n,\alpha,\mathcal C}. } \]
Equivalently,
\[ \boxed{ L_{\tau,n,\alpha,\mathcal C} = \ln \left( \frac{ \nu_{n,\alpha,H} }{ \nu_{n,\alpha,L} } \right). } \]
Because
\[ 0< \eta \leq1, \]
we have
\[ \boxed{ L_{\tau,n,\alpha,\mathcal C}\geq0. } \]
Reciprocal realization gives
\[ L_\tau=0. \]
Nonreciprocal realization gives
\[ L_\tau>0. \]
The word loss does not mean count destruction.
It is the logarithmic measure of unequal opposed-leg occurrence efficiency.
The relation between this local nonreciprocity and enclosing background is introduced later in the R2D Second Law.
Four Different Closure Questions
A completed closure therefore contains several mathematically different questions.
Did the local state return?
For a closure,
\[ i\rightarrow k\rightarrow i. \]
Yes.
Did native multiplicity close?
Yes:
\[ \boxed{ \Delta S_{\mathcal C}=0. } \]
Did enclosing asymmetry close?
Not necessarily:
\[ \boxed{ \Delta A_{\mathcal C} \neq0 } \]
is allowed.
Were the opposed legs realized reciprocally?
Not necessarily:
\[ \boxed{ 0<\eta_{\mathcal C}<1 } \]
is allowed.
These statements must not be collapsed into one notion of “closure” or “irreversibility.”
Recurrence Is Yet Another Count
There is still one more question:
How many times does the completed closure recur?
That question is not answered by
\[ \Delta S_{\mathcal C}, \]
by
\[ \Delta A_{\mathcal C}, \]
or by
\[ \eta_{\mathcal C}. \]
It requires a separate whole-closure recurrence count.
Thus R2D separates:
\[ \boxed{ \text{state return} } \]
from
\[ \boxed{ \text{directional asymmetry} } \]
from
\[ \boxed{ \text{opposed-leg occurrence reciprocity} } \]
from
\[ \boxed{ \text{number of closure recurrences}. } \]
Closure Makes Recursive Carry Possible
Once a local realization has completed its return, it can be treated as one completed relation at the local boundary.
When the count boundary changes, that completed local relation need not remain readable through its internal path.
Instead, it can acquire enclosing occurrence meaning.
Thus:
\[ \boxed{ \mathcal C_{n,\alpha} \xmapsto{\kappa_n} \chi_{n+1}^{(r)}. } \]
Closure therefore supplies the completed local realization that recursive carry can reclassify.
The local path is readable below.
Its enclosing meaning is readable above.
Closure Is Not Upward Causation
This does not mean that the completed closure creates the enclosing occurrence.
Causally, R2D retains the opposite orientation:
\[ \boxed{ \chi_{n+1}^{(r)} \Longrightarrow \mathcal C_{n,\alpha}. } \]
The enclosing occurrence defines its compatible local realization.
The completed local closure is then reclassified when the count boundary changes.
Thus:
Closure completes the local realization.
It does not generate the enclosing organization from below.
What Closes and What Need Not Close
The entire structure can now be summarized compactly.
For a completed closure:
Native multiplicity closes
\[ \boxed{ \Delta S_{n,\alpha,\mathcal C}=0. } \]
Enclosing possibility need not close
\[ \boxed{ \Delta P_{n,\alpha,\mathcal C}\neq0 } \]
in general.
Retention need not equal possibility
\[ \boxed{ \Delta R_{n,\alpha,\mathcal C} \neq \Delta P_{n,\alpha,\mathcal C} } \]
in general.
Closure asymmetry can remain
\[ \boxed{ \Delta A_{n,\alpha,\mathcal C} \neq0. } \]
Occurrence realization need not be reciprocal
\[ \boxed{ 0< \eta_{n,\alpha,\mathcal C} <1 } \]
is allowed.
Recurrence count is separate
The number of times the completed closure recurs must be counted independently.
This is the central lesson:
A path can close in one count relation without every relation associated with that path closing.
The Key Distinction
Closure is therefore not simply “coming back to the beginning.”
It is a layered recursive structure.
The local macrostate returns.
Native multiplicity closes exactly.
But the enclosing relation organizing the opposed legs can remain asymmetric.
The realization of those legs can remain nonreciprocal.
And the completed closure can recur at its own count relative to the enclosing boundary.
That separation allows R2D to distinguish cycle, direction, irreversibility, and recurrence before assigning physical units to any of them.
Core Equations
Local closure
\[ \boxed{ \mathcal C_{n,\alpha}: i\rightarrow k\rightarrow i } \]
Native multiplicity closure
\[ \boxed{ \Delta S_{n,\alpha,\mathcal C}=0 } \]
Closure possibility
\[ \boxed{ \Delta P_{n,\alpha,\mathcal C} = \sum_{\chi\in\mathcal C_{n,\alpha}} \Delta P_{n,\alpha,\chi} } \]
Closure retention
\[ \boxed{ \Delta R_{n,\alpha,\mathcal C} = \sum_{\chi\in\mathcal C_{n,\alpha}} \Delta R_{n,\alpha,\chi} } \]
Closure asymmetry
\[ \boxed{ \Delta A_{n,\alpha,\mathcal C} = \Delta P_{n,\alpha,\mathcal C} - \Delta R_{n,\alpha,\mathcal C} } \]
Directional occurrence-count efficiency
\[ \boxed{ \eta_{n,\alpha,\mathcal C} = \frac{ \nu_{n,\alpha,L} }{ \nu_{n,\alpha,H} } } \]
Occurrence-count loss
\[ \boxed{ L_{\tau,n,\alpha,\mathcal C} = -\ln\eta_{n,\alpha,\mathcal C} \geq0 } \]