Story Operators: Decomposing the Original $\to$ Sequel Transformation in Embedding Space

# Decomposing the Original → Sequel Transformation in Embedding Space
Source: [https://arxiv.org/html/2606.25379](https://arxiv.org/html/2606.25379)
\(June 2026\)

###### Abstract

I treat a book as a point in a sentence\-embedding space and a literary transformation as an operation on points\. Given an original novel and its sequel, I ask what it takes, geometrically, to turn the first into the second\. Using all\-mpnet\-base\-v2 paragraph embeddings drawn from a precomputed index of the PG19 corpus, I form the displacementd=x¯seq−x¯origd=\\bar\{x\}\_\{\\text\{seq\}\}\-\\bar\{x\}\_\{\\text\{orig\}\}and greedily decompose it along a content basis obtained by PCA over the two books’ own paragraphs\. Each component is an interpretable axis anchored by real passages at its poles\. Across thirteen verified author pairs from Project Gutenberg, the decomposition reveals a small taxonomy of sequels:*formulaic*\(a tiny, low\-rank change: Doyle’s Holmes collections,∥d∥=0\.12\\lVert d\\rVert\{=\}0\.12\),*concentrated*\(one dominant axis: Alcott’sLittle Women→\\toLittle Men, 75% on a single move\), and*compositional*\(many small axes: Twain, Burroughs’s Barsoom, Nesbit\)\. For the canonical case,Tom Sawyer→\\toHuckleberry Finn, the dominant recovered axis is structural—the collapse of sheltering domesticity into a picaresque road—rather than the famous surface themes of vernacular voice or slavery, which ride later, smaller axes; and the transformation routes through adventure\-journey space rather than diluting toward generic realism\. I corroborate the recovered geometry against Twain’s documented authorial intent \(his 1875–76 letters to Howells\), which names the first\-person picaresque move years in advance, and I quantify, with an explicit representation caveat, how much of the realized transformation his stated intentions span\. All computations are reproducible from the released scripts and data\.

## 1Introduction

A sentence\-embedding model maps text to a vector inℝ768\\mathbb\{R\}^\{768\}in which proximity encodes semantic kinship\. Embedding an entire corpus yields a cloud in whichTreasure IslandandKidnappedare neighbors and Darwin sits near the philosophers\. Once books are points, transformations of books become operations on points\. I call a small, named algebra of such operations*Story Operators*\. The canonical algebra has 13 primitives\[[4](https://arxiv.org/html/2606.25379#bib.bib4)\]: 6*point verbs*\(toward,along,scale,reflect,project,blend\), 3*ensemble verbs*\(gather,scatter,carry\), and 4*guards*\(where,clamp,mask,verify\)\. The set is argued to be sufficient under an affine\-closure analysis: every affine endomorphism ofℝd\\mathbb\{R\}^\{d\}decomposes via SVD into rotations, reflections, axis scalings, and translations, all representable as programs over these verbs followed by renormalization toSd−1S^\{d\-1\}\[[4](https://arxiv.org/html/2606.25379#bib.bib4)\]\. This paper studies the operator most directly tied to a documented authorial act: the transformation that carries an*original*novel to its*sequel*\. As I show, the content\-axis decomposition of the original→\\tosequel displacement has the structure of a greedyalongchain—alongbeing the canonical verb for translation along a named semantic direction\.

My contribution is threefold\. \(i\) I give a reproducible method that*decomposes*an original→\\tosequel displacement into a short sequence of interpretable content axes recovered from the two books’ own text, rather than imposing a fixed external dimension set\. \(ii\) I apply it to thirteen verified Gutenberg author pairs and identify a taxonomy—formulaic, concentrated, compositional—supported by comparative metrics\. \(iii\) ForTom Sawyer→\\toHuckleberry FinnI connect the recovered geometry to primary\-source authorial intent and quantify the alignment, with a forthright account of what the measurement can and cannot claim\.

## 2Related Work

Sentence\-BERT\[[1](https://arxiv.org/html/2606.25379#bib.bib1)\]and the mpnet model family\[[2](https://arxiv.org/html/2606.25379#bib.bib2)\]produce the fixed\-length text embeddings I use; I adoptall\-mpnet\-base\-v2\. PG19\[[3](https://arxiv.org/html/2606.25379#bib.bib3)\]provides a large, public, pre\-1919 book corpus with clean provenance\. Prior computational\-narrative work largely measures*similarity*or clusters works; my emphasis is on*decomposing a transformation*into named, composable operators and validating them against text and, where available, authorial record\. The broader Story Operators program situates these operations in a reproducing\-kernel Hilbert space over narrative corpora\[[4](https://arxiv.org/html/2606.25379#bib.bib4)\]\.

## 3Method

### 3\.1Book vectors

I use a precomputed paragraph index of PG19 \(all\-mpnet\-base\-v2, 768\-d, one vector per paragraph; 12,830 distinct books\)\. For a book I discard boilerplate paragraphs \(Project Gutenberg headers/licenses, very short fragments, illustration captions\) and define its vector as theL2L\_\{2\}\-normalized mean of its remaining paragraph embeddings,x¯=𝒩​\(1\|P\|​∑p∈Pep\)\\bar\{x\}=\\mathcal\{N\}\\\!\\big\(\\tfrac\{1\}\{\|P\|\}\\sum\_\{p\\in P\}e\_\{p\}\\big\)\. Paragraph\-level pooling avoids the 384\-token truncation that would otherwise discard most of a novel\.

### 3\.2Content\-axis decomposition

For an \(original, sequel\) pair with book vectorsTTandHH, letd=H−Td=H\-Tandbase=⟨T,H⟩\\mathrm\{base\}=\\langle T,H\\rangle\. I build a*content basis*\{uj\}\\\{u\_\{j\}\\\}by PCA over the union of the two books’ centered paragraph embeddings, so the axes are the directions along which*these two texts actually vary*\. I then greedily composedd: order the axes by\|⟨d,uj⟩\|\|\\langle d,u\_\{j\}\\rangle\|, sign\-align each towardHH, and accumulatewk=wk−1\+⟨d,u\(k\)⟩​u\(k\)w\_\{k\}=w\_\{k\-1\}\+\\langle d,u\_\{\(k\)\}\\rangle u\_\{\(k\)\}fromw0=Tw\_\{0\}=T\. After each step I record the cumulative*angular gap closed*

gk=cos⁡\(wk,H\)−base1−base,g\_\{k\}=\\frac\{\\cos\(w\_\{k\},H\)\-\\mathrm\{base\}\}\{1\-\\mathrm\{base\}\},\(1\)and keep steps whose marginalgk−gk−1g\_\{k\}\-g\_\{k\-1\}exceeds1%1\\%\(the others are dead weight\)\. Each kept step is named from the real paragraphs at the extremes of its axis \(thearg⁡max\\arg\\max/arg⁡min\\arg\\minof the projection ontou\(k\)u\_\{\(k\)\}\), one drawn from the original and one from the sequel\.

In the canonical Story Operators vocabulary\[[4](https://arxiv.org/html/2606.25379#bib.bib4)\], each kept step corresponds to aalongoperation:ALONG​\(x;u,λ\)=𝒩​\(x\+λ​u\)\\mathrm\{ALONG\}\(x;\\,u,\\,\\lambda\)=\\mathcal\{N\}\(x\+\\lambda u\), the canonical verb for translating an embedded object byλ=⟨d,u\(k\)⟩\\lambda=\\langle d,u\_\{\(k\)\}\\ranglealong content axisu\(k\)u\_\{\(k\)\}, where𝒩\\mathcal\{N\}denotesL2L\_\{2\}normalization\. Note the difference from the present method: the canonical verb normalizes after each step, whereas the greedy decomposition here accumulates in the ambient linear space and evaluates angular progress,gkg\_\{k\}, at each step rather than normalizing incrementally\. This gives the decomposition the*conceptual*structure of a greedyalongchain while measuring gap\-closure on the original sphere; the two converge in the limit of small step magnitudes\. The greedy ordering \(largest\|λ\|\|\\lambda\|first\) recovers the dominant semantic move before the residual components\.

### 3\.3Comparative metrics

Per pair I report:cos⁡\(T,H\)\\cos\(T,H\)\(overall similarity\);∥d∥\\lVert d\\rVert\(transformation magnitude\); the*content ceiling*gKg\_\{K\}\(fraction of the angular gap the content basis closes\);neffn\_\{\\text\{eff\}\}\(number of kept steps\); the*dominant share*g1g\_\{1\}; and a*participation ratio*1/∑jp^j21/\\sum\_\{j\}\\hat\{p\}\_\{j\}^\{2\}over the per\-axis energy fractionsp^j∝⟨d,uj⟩2\\hat\{p\}\_\{j\}\\propto\\langle d,u\_\{j\}\\rangle^\{2\}, which is≈1\\approx 1when one axis carries the move and large when the move is diffuse\.

### 3\.4Intent test \(case study\)

For Tom→\\toHuck I additionally embed six articulated “sequel operators” \(e\.g\. vernacular first\-person POV; town episodes→\\toriver journey; confront slavery\) as phrase directionsdir=𝒩​\(eto−efrom\)\\mathrm\{dir\}=\\mathcal\{N\}\(e\_\{\\text\{to\}\}\-e\_\{\\text\{from\}\}\)and measure the fraction ofddlying in their span,∥B​d∥/∥d∥\\lVert B\\,d\\rVert/\\lVert d\\rVertfor an orthonormal basisBBof the directions, against a random\-kk\-subspace baseline\. I flag the representation caveat in §[6](https://arxiv.org/html/2606.25379#S6)\.

## 4Data

I select thirteen original→\\tosequel pairs whose*both*volumes are present in the index with\>80\>80non\-junk paragraphs and which are genuine same\-author continuations \(Table[1](https://arxiv.org/html/2606.25379#S5.T1)\)\. They span children’s fantasy \(Carroll, Baum, Nesbit\), boyhood and pulp adventure \(Twain, Burroughs\), detective \(Doyle\), historical romance \(Dumas\), domestic \(Alcott, Montgomery, Porter\), and utopian \(Bellamy\) modes\. Gutenberg identifiers and per\-book paragraph counts are recorded in the releasedcomparative\.csv\.

## 5Results

### 5\.1A taxonomy of sequels

Table[1](https://arxiv.org/html/2606.25379#S5.T1)and Fig\.[1](https://arxiv.org/html/2606.25379#S5.F1)show that sequels differ along two nearly independent axes:*how far*the sequel moves \(∥d∥\\lVert d\\rVert\) and*how concentrated*the move is \(participation\)\.

- •Formulaic\.Doyle’sAdventures→\\toMemoirs of Sherlock Holmesis the least\-transformed pair \(cos=0\.99\\cos=0\.99,∥d∥=0\.12\\lVert d\\rVert=0\.12, dominant axis only 14%\): the detective template barely changes between story collections\. Dumas’sThree Musketeers→\\toTwenty Years Afteris similar \(∥d∥=0\.18\\lVert d\\rVert=0\.18\)\.
- •Concentrated\.Alcott’sLittle Women→\\toLittle Menputs 75% of the move on a single axis \(participation 1\.3\): the shift from a girls’ domestic coming\-of\-age to a boys’ schoolhouse is nearly one operator\. Bellamy’sLooking Backward→\\toEqualityis comparable \(59%\), the utopian narrative hardening into didactic treatise\.
- •Compositional\.Twain, Burroughs’sPrincess→\\toGods of Mars, and Nesbit spread the move across many small axes \(participation 4\.4–4\.7\): the sequel is a genuine re\-composition, not a single tonal turn\.

Baum’sWizard→\\toMarvelous Land of Ozis the*most*\-transformed pair \(∥d∥=0\.43\\lVert d\\rVert=0\.43, ceiling 90%\), consistent with Baum dropping Dorothy for an entirely new protagonist and cast\.

Table 1:Original→\\tosequel decomposition across thirteen Project Gutenberg pairs, sorted by transformation magnitude\.cos\\cos: original–sequel cosine;∥d∥\\lVert d\\rVert: displacement norm;*ceil*: content ceiling \(gap closed\);*stp*: effective steps;*dom*: dominant\-axis share;*part*: participation ratio\. All values measured\.![Refer to caption](https://arxiv.org/html/2606.25379v1/fig_comparative.png)Figure 1:Comparative profiles\.Left: cumulative gap closed per sequel, segmented by component operator \(color = greedy step order\); short\-and\-dark bars are single\-dominant\-axis sequels, long\-and\-many are compositional\.Right: transformation magnitude∥d∥\\lVert d\\rVertvs\. compositional spread \(participation\), separating formulaic \(lower\-left\) from compositional \(upper\) sequels\.
### 5\.2Case study:Tom Sawyer→\\toHuckleberry Finn

The Twain pair is strongly compositional \(8 effective steps, participation 4\.4, ceiling 85%\)\. The four dominant axes \(74% of the move\), each named from real passages, are: \(1, 32%\)*sheltering domesticity*→\\to*picaresque fraud*\(Aunt Polly praying over Tom→\\tothe con\-man “king”\); \(2, 20%\)*spectated mischief*→\\to*first\-person immersion in danger*; \(3, 12%\)*town set\-pieces*→\\to*the river’s parade of strangers*; \(4, 10%\)*treasure\-as\-reward*→\\to*moral distress*\. The single largest axis is thus*structural*—the protective adult frame collapsing into a road of frauds—rather than the surface themes \(vernacular voice, slavery\) usually named first; those ride later, thinner axes\. Tracing the bent path’s neighbors, the transformation*routes through*adventure\-journey space \(Kidnapped,Treasure Islandappear at the 25–50% marks\), whereas the naive straight interpolation merely dilutes Tom toward generic social realism \(Figs\.[2](https://arxiv.org/html/2606.25379#S5.F2),[3](https://arxiv.org/html/2606.25379#S5.F3)\)\.

![Refer to caption](https://arxiv.org/html/2606.25379v1/fig_reverse_forward.png)Figure 2:Tom→\\toHuck\.Left: the actual decomposition \(cumulative gap by content\-axis step; content ceiling 85%\)\.Right: forward intent test—six articulated authorial moves span 22\.8% of the actual displacement vs\. an 8\.9% random\-6\-dim baseline \(above chance, modest\), with the representation caveat noted in §[6](https://arxiv.org/html/2606.25379#S6)\.![Refer to caption](https://arxiv.org/html/2606.25379v1/fig_components.png)Figure 3:The eight component operators as colored legs looping fromTom Sawyer\(start\) toHuckleberry Finn\(target\) in the plane spanned byH−TH\-Tand the path’s principal bend\. The dashed line is the naive straight blend; the legend lists each operator’s marginal gap\.
### 5\.3Recovered geometry vs\. authorial intent

The recovered dominant axes match Twain’s documented plan\. Nine years before he finishedHuckleberry Finn, Clemens wrote to W\. D\. Howells \(5 July 1875\) that he had stoppedTom Sawyerat boyhood and intended, separately, to “ take a boy of twelve & run him on through life \(in the first person\) … but not Tom Sawyer ”\[[5](https://arxiv.org/html/2606.25379#bib.bib5)\]—naming the first\-person picaresque move \(axes 1–2\) in advance, and invoking the picaresqueGil Blasas the model\. The 9 August 1876 letter dates the sequel’s start \(“It is Huck Finn’s Autobiography … I like it only tolerably well … may possibly pigeon\-hole or burn the MS”\); the publishednoticeandexplanatorynote assert, respectively, the absence of conventional plot \(consistent with the “drifting river\-time” axis\) and the deliberate, researched vernacular\. Twain’s own later verdict on the book—a notebook entry \(c\. 1895\) describing “a sound heart & a deformed conscience \[that\] came into collision & conscience suffers defeat”\[[6](https://arxiv.org/html/2606.25379#bib.bib6)\]—names the moral\-distress axis \(4\)\.

Quantitatively, embedding the six stated intentions as phrase directions, they span22\.8%22\.8\\%of the realized displacement, versus an8\.9%8\.9\\%random\-6\-dimensional baseline \(95th percentile12\.9%12\.9\\%\): above chance but partial\. The dominant structural axis \(shelter→\\tofraud\) is the*least*matched by any stated intention—suggesting that the deepest move of the sequel was less a checklist item than an emergent consequence of following Huck honestly down the river\.

### 5\.4Other notable pairs

Beyond the extremes, several pairs illustrate intermediate regimes\. Carroll’sAlice→\\toThrough the Looking\-Glasshas the lowest content ceiling \(52%\): much of its change lies off the two books’ shared content axes, consistent with a sequel that swaps one self\-contained dream\-logic for another rather than continuously transforming a single world\. Montgomery’sAnne of Green Gables→\\toAnne of Avonleais the most nearly two\-dimensional \(participation 2\.0\): a steady maturation along essentially one pair of axes\. Kipling’s twoJungle Books, Burroughs’s two Tarzan novels, and Porter’s twoPollyannabooks sit in the middle—moderate magnitude, moderate spread—the profile of a continuation that keeps its world but advances its protagonist\. That these regimes recur across unrelated authors suggests the taxonomy reflects modes of sequel\-making rather than idiosyncrasies of single works\.

## 6Discussion and Limitations

Two kinds of difference\.Magnitude and concentration are nearly orthogonal: a sequel can be a large but single\-axis turn \(Baum, Alcott\) or a small but diffuse re\-weighting \(Doyle\)\. The taxonomy is descriptive, but it cleanly separates template continuations from genuine re\-compositions\.

Caveats\.\(i\)*Representation mismatch in the intent test*: intended operators are phrase\-embedding differences whileddis a book\-paragraph\-mean difference; the 22\.8% figure is therefore a noisy*lower bound*on intent\-alignment, not a claim that the remainder was unintended\. A book\-contrast definition of each operator \(centroids of real exemplar books\) would be a stronger test\. \(ii\) The content basis is unsupervised PCA over two books; it captures the texts’ own variance, not a canonical narrative axis set\. \(iii\) Mean\-pooling discards sequence; the operators describe aggregate semantic content, not plot order\. \(iv\) The angular “gap closed” and the Euclidean∥d∥\\lVert d\\rVertmeasure different geometry \(a residual can be long yet nearly orthogonal to the target\)\. \(v\) Thirteen pairs is a small, English, pre\-1919 sample\. None of these undercut the central, fully reproducible result—that an original→\\tosequel transformation decomposes into a few interpretable, text\-anchored operators—but they bound its interpretation\.

Applications\.Decomposed transformations are useful beyond criticism\. The same geometry that names a sequel’s moves can score a*proposed*continuation against its original—does it re\-compose the world or merely dilute it?—place a catalog’s series in the magnitude–concentration plane to diagnose sameness, and supply retrieval\-time structure for curation and recommendation\. Within the canonical 13\-primitive algebra\[[4](https://arxiv.org/html/2606.25379#bib.bib4)\], this paper is consistent with the utility ofalong: the sequel cases confirm that named semantic translations \(content\-axis projections\) compose into interpretable programs and that the dominant move is recoverable from text alone\. I use the full operator algebra in this role within a working publishing program; the present study provides empirical grounding for the core vocabulary on a case where ground\-truth authorial record exists, before relying on it where it does not\.

Reproducibility\.Every number here is produced by released scripts \(sequel\_corpus\_decompose\.py,sequel\_figures\.py,tom\_huck\_\*\) over the public PG19 texts; the comparative table is generated directly from the computedcomparative\.json\.

## 7Conclusion

Treating books as points and transformations as operators turns “what does this sequel do?” into a measurable question\. Across thirteen author pairs the original→\\tosequel move decomposes into a handful of interpretable, text\-anchored axes, and sequels fall into recognizable kinds\. ForTom Sawyer→\\toHuckleberry Finn, the recovered geometry both matches Twain’s stated intent on the first\-person picaresque turn and exposes a deeper structural move—sheltering domesticity collapsing into a picaresque road—that no stated intention fully anticipated\. Story Operators thus offer a vocabulary in which literary transformation is not only describable but decomposable\.

## References

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- \[4\]W\. F\. Zimmerman\.Story Operators\. Story Operators New Media, Ann Arbor, Michigan, 2026\.
- \[5\]S\. L\. Clemens to W\. D\. Howells, 5 July 1875 and 9 August 1876\. InMark Twain–Howells Letters, ed\. H\. N\. Smith and W\. M\. Gibson \(Harvard Univ\. Press, 1960\); also Mark Twain Project critical edition ofAdventures of Huckleberry Finn\.
- \[6\]S\. L\. Clemens, Notebook entry, c\. 1895; see H\. N\. Smith, “A Sound Heart and a Deformed Conscience,” inMark Twain’s Humor\.