A question for the path space
Does admissibility make
history measurable?
Keep the target fixed. Change the histories allowed to construct it. Measure what the constraints cost.
Instrument
dH(x | F)
Target structure x admissibility regime
01
Active experiment
Constrained construction
01Choose xSelect a fixed target case.
02Set FChoose which histories remain admissible.
03Read dHCompare the shortest free and surviving paths.
Historical load
dH(x | F)=
+3
Admissibility adds 3 assembly steps.
Added formation burden, not downstream generative capacity.
free5
admissible8
Shortest constructions
Same target, different histories
Free optimum
Admissible optimum
Filtered state
How to readEach dot is a construction state; Sn marks an intermediate. Lines are transitions, not physical time. The red cross is the first state removed by F.
Invalid under FConstitutive shortcut
Valid under FStable bridge
Target identity
The object does not change
The path space does.
=same x
Trace / first divergence
Where history becomes more expensive
START
S2
Constitutive shortcutclosed(S4) = false
Stable bridgeshortest surviving continuation
X
The free optimum is removed at S4. The shortest surviving construction reaches the same target in 8 rather than 5 steps.
02
Interpretation
What does the number mean?
Standard assembly asks for the shortest construction of x. Historical load asks how much longer that construction becomes when inadmissible intermediates are removed.
Unit of comparison: one declared construction transition. The result compares minima inside the finite candidate set shown by this prototype; it is not a universal physical timescale.
object xshortest constructiona0
object x + filter Ffiltered path spacedH
03
The microstructure is unchanged.
The essential distinction
The microstructure is unchanged.
Reachability is not.
Historical load is not an intrinsic label attached to a structure. It is a result of pairing that structure with a declared admissibility regime.
structure x+filter FdH(x | F)
dHphase response
Constraint thresholds can produce discrete transitions in the reachable path space.
A valid answer
dH = 0
04
The claim can fail.
For some objects and admissibility regimes, constraints add no historical burden. In that case, the strongest historical-dependence claim is unsupported by the example.
05
Finite examples
Results, not declarations.
All values use the Physical preset. Select a row to load its disclosed toy path space into the instrument.