16-10-2026, 10:19 PM
This is a follow-up analysis to my You are not allowed to view links. Register or Login to view. All of the code for this work can be found in the You are not allowed to view links. Register or Login to view..
Problem / Goal
The goal of this analysis was to answer "how uncertain / unstable are points on the proposed chronology". In particular, some follow-up analysis I have been conducting shows high sensitivity around the inclusion of f58, suggesting the placement of the missing H quire could change the ordering.
The core problem is, when tracing a path, the ends of the path have the least certainty. This is because points in the middle of the path have preceding and following points, which also have preceding / following points. These multiple overlapping observations provide much more certainty about the ordering.
However, points on the end do not have this. By definition, the first and final points would only have either a preceding or following connection. Depending on the direction and amount of movement, this can create uncertainty around which group is the first or last.
f58 and the missing H bifolio add this same uncertainty in the middle of the manuscript. We are missing part of the path, which creates additional uncertainty, specifically in the Herbal B ordering.
Method
I tested the ordering stability using both 207 individual pages and 50 pooled bifolia, holding the transition-distance/MDS representation fixed.
I utilized the ensemble procedure described in You are not allowed to view links. Register or Login to view. which involves fitting curves to 100 random approximately 90% subsets, using eight starting routes and two regularization settings. A whole-bifolium sampling control brought the total to 4,800 fits.I then compared recurring positions, order reversals and competing routes’ scores within each sample.
I ran You are not allowed to view links. Register or Login to view. to attempt to identify any discontinuous regions along the gradient.
To establish toe potential impact of the missing pages, and the creation of the the H-position MDS charts below, I used the You are not allowed to view links. Register or Login to view., applied derivative-based continuation at λ=.01, checking assignment changes and different movement paths.
A You are not allowed to view links. Register or Login to view. of exactly how each methodology was applied can be found on the voynich-clusters web page. All code is You are not allowed to view links. Register or Login to view.
Results
Instability Across the Gradient
The chart below shows the variance in the ordering of the individual pages based on different ensemble simulations described in ElPiGraph's paper. This shows where across the 4,800 different fit variants each page and bifolia fell. This allows us to determine how sensitive the positioning is to different starting conditions, assumptions, and modeling parameters. Darker regions indicate more variants placed the page in that area.
Observations from the chart:
Checking for discontinuity
scEGRET is a method described by Zheng et al that applies directly to our problem. It is a toolkit designed to follow paths and identify discontinuous regions, allowing us to more firmly establish if we are missing any sections of the overall curve. The code is freely available on the authors github.
The general idea is this, along a well sampled progression, successive groups should generally change gradually. An unusually large change indicates the trajectory crosses a gap, for example, because intermediate data is missing.
scEGRET identifies no fully discontinuous regions across the gradient. Note, this does not mean certain areas along the curve are not missing pages. It simply means scEGRET was unable to detect one.
F58 does show a modest local signal. The scEGRET scores measuring relative change jump from .184 to .271, around a 1.5x increase. Calibration places this within the 95% confidence region. Removing f58 along with adjacent astro material entirely does produce discontinuity flags, but it depends on certain model settings and is therefore not conclusive.
So there appears to be no obvious gaps in the gradient, f58's region appears suspicious but is not enough to claim the region is discontinuous.
Impact of Missing Quire H on Herbal B
The chart below shows the impact the missing Quire H bifolia could have on the ordering of Herbal B's 3a/3b sections.
You can see the current f58 placement is almost directly on this red/blue frontier. This means without those missing pages, it is probably not possible to say for certain which Herbal B group comes first, because any slight changes from the current f58 point could alter the ordering.
Conclusion
The overall gradient is clear, but how confident we can be around the relative ordering of certain sections along that gradient varies. The largest uncertainty is:
Problem / Goal
The goal of this analysis was to answer "how uncertain / unstable are points on the proposed chronology". In particular, some follow-up analysis I have been conducting shows high sensitivity around the inclusion of f58, suggesting the placement of the missing H quire could change the ordering.
The core problem is, when tracing a path, the ends of the path have the least certainty. This is because points in the middle of the path have preceding and following points, which also have preceding / following points. These multiple overlapping observations provide much more certainty about the ordering.
However, points on the end do not have this. By definition, the first and final points would only have either a preceding or following connection. Depending on the direction and amount of movement, this can create uncertainty around which group is the first or last.
f58 and the missing H bifolio add this same uncertainty in the middle of the manuscript. We are missing part of the path, which creates additional uncertainty, specifically in the Herbal B ordering.
Method
I tested the ordering stability using both 207 individual pages and 50 pooled bifolia, holding the transition-distance/MDS representation fixed.
I utilized the ensemble procedure described in You are not allowed to view links. Register or Login to view. which involves fitting curves to 100 random approximately 90% subsets, using eight starting routes and two regularization settings. A whole-bifolium sampling control brought the total to 4,800 fits.I then compared recurring positions, order reversals and competing routes’ scores within each sample.
I ran You are not allowed to view links. Register or Login to view. to attempt to identify any discontinuous regions along the gradient.
To establish toe potential impact of the missing pages, and the creation of the the H-position MDS charts below, I used the You are not allowed to view links. Register or Login to view., applied derivative-based continuation at λ=.01, checking assignment changes and different movement paths.
A You are not allowed to view links. Register or Login to view. of exactly how each methodology was applied can be found on the voynich-clusters web page. All code is You are not allowed to view links. Register or Login to view.
Results
Instability Across the Gradient
The chart below shows the variance in the ordering of the individual pages based on different ensemble simulations described in ElPiGraph's paper. This shows where across the 4,800 different fit variants each page and bifolia fell. This allows us to determine how sensitive the positioning is to different starting conditions, assumptions, and modeling parameters. Darker regions indicate more variants placed the page in that area.
Observations from the chart:
- the ordering with respect to 1a and 1b is ambiguous, however 1a pages generally exhibit a slight initial ordering preference over 1b
- 1.5a and 2a's placements are generally stable.
- f58, astro and cosmo are generally stable as a group. The only exception is Rosettes, which is the single isolated band not with the others in 5a, on the page level chart. It clearly was not made at the same time as the other 5a group, probably instead being produced alongside the Herbal B sections, but possibly even being made as late as stars 3c.
- Herbal 3a and 3b's ordering is not stable, and will be discussed below in the next section.
- At the bifolium level Stars 3c is generally stable, with f85r1 and r2 (the two unassigned bars ) also having the strongest preference for the same timing.
- Bio / Stars 4c are more ambiguous, with fits generally preferring the ordering Stars 4c-> Bio
Checking for discontinuity
scEGRET is a method described by Zheng et al that applies directly to our problem. It is a toolkit designed to follow paths and identify discontinuous regions, allowing us to more firmly establish if we are missing any sections of the overall curve. The code is freely available on the authors github.
The general idea is this, along a well sampled progression, successive groups should generally change gradually. An unusually large change indicates the trajectory crosses a gap, for example, because intermediate data is missing.
scEGRET identifies no fully discontinuous regions across the gradient. Note, this does not mean certain areas along the curve are not missing pages. It simply means scEGRET was unable to detect one.
F58 does show a modest local signal. The scEGRET scores measuring relative change jump from .184 to .271, around a 1.5x increase. Calibration places this within the 95% confidence region. Removing f58 along with adjacent astro material entirely does produce discontinuity flags, but it depends on certain model settings and is therefore not conclusive.
So there appears to be no obvious gaps in the gradient, f58's region appears suspicious but is not enough to claim the region is discontinuous.
Impact of Missing Quire H on Herbal B
The chart below shows the impact the missing Quire H bifolia could have on the ordering of Herbal B's 3a/3b sections.
- This is using the same MDS dimensions as the plots in the previous chronology theory,
- The star shows the current placement of f58, the only remaining portion of quire H.
- The colors indicate which ordering of 3a and 3b is preferred, based on where the missing pages of quire H would shift the current Quire H centroid on the MDS dimensions
- Red regions mean if the missing pages shift the centroid there, the preferred ordering is 3a->3b. If the missing bifolia instead shift the centroid into the blue regions, then the preferred ordering is 3b->3a.
- The black line divides the frontier between these regions
You can see the current f58 placement is almost directly on this red/blue frontier. This means without those missing pages, it is probably not possible to say for certain which Herbal B group comes first, because any slight changes from the current f58 point could alter the ordering.
Conclusion
The overall gradient is clear, but how confident we can be around the relative ordering of certain sections along that gradient varies. The largest uncertainty is:
- The ordering of Early Herbal A sections 1a and 1b, although 1a is generally preferred as the starting point
- The ordering of 3a and 3b, basically indeterminant without the missing Quire H
- The ordering of 4c and 4a / Stars-T and Bio is indeterminant, although bio is generally preferred as the endpoint.
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