Thesis defect corrections, the step drift diagnosed, and the build matched to the published thesis

[salindne]

The notation pass this PR was scoped for is cancelled on measurement. What lands instead is a set of thesis defect corrections, the step-numbering drift the author reported diagnosed to its cause, and the reconstructed build brought into agreement with the published thesis on pages and on algorithm numbers.

Five attempted corrections in this PR turned out to be wrong and are retracted with their evidence. That is the most useful part of it, so it gets its own section.

Why the notation pass is cancelled

The plan was to move coefficient-level formulas from two-digit indexing (u_{11}) to primes (u^{\prime\prime}_1), matching the generated tables. Measured in chapter5.tex:

	count
u_1 as a polynomial, divisor 1 (/u_1, \pmod{u_1})	123
u_1 as a coefficient of degree 1	1
u_{11} two-digit, divisor 1 degree 1	49

Mapping u_{11} \to u_1 would collide the coefficient with 123 polynomial uses. The two-digit form is not redundant with the prime notation; it is what disambiguates coefficients from polynomials, and removing it would create the ambiguity the PR existed to remove. The only escape is to rename the polynomials too, which would split the prose from the pseudocode that the 2026-08-09 decision keeps on divisor-level indexing.

Worth recording because it is the opposite of where the risk was expected: 21 of 28 align* blocks in chapter 5 carry coefficient notation only and would have been mechanical. The blocks were viable and the prose is what makes the pass unsound.

The original problem resolves the other way round. The generated tables should carry two-digit notation, and ideally from the generator rather than by hand. Folded into PR31.

The step drift the author reported is \EndIf

The algpseudocode environment, whose engine is algorithmicx, numbers \EndIf lines. The prose never counted them as steps, so every reference drifts by the number of \EndIf lines above it, and the drift grows down each algorithm.

One preamble line fixes it, \algtext*{EndIf}. Precisely what it does, because the mechanism is the argument: \algtext* binds the entity's text to \ALG@x@notext, and algorithmicx.sty:193-197 then emits \item[] rather than \item. The list's default label is {\ALG@step}, which is what advances ALG@line, so an empty label bypasses it. The line loses its text, loses its number, and consumes no number. That last part is why later references move.

Seven anchors across the two genus-3 split algorithms confirm it:

algorithm	step named in prose	prose says	\EndIf numbered	suppressed
3DBL	monic quotient q', remainder r'	10--11	12--13	10--11
3DBL	compute the six ratios	14	17	14
3DBL	M_2' = (r(2v_1 + h) + qk)/u_1	15	18	15
3DBL	z, then v_n	17--18	20--21	17--18
3ADD	compute the six ratios	14	17	14
3ADD	M_2'	16	19	16
3ADD	u_n	17	20	17

Every one lands with suppression, and without it each is off by exactly the count of \EndIf lines above it. The first row is off by two rather than three, which is what rules out a constant offset from some other cause. The author reported the Degree 3 Addition description as "3 lines off" before any diagnosis was attempted, and three is that algorithm's \EndIf count.

The first diagnosis was wrong and was reverted inside this PR. It blamed the \State Go to ... and See description escape lines and edited 20 \Statex lines. It fitted alg:g3explSPLIT3ADD only because that algorithm happens to have three \EndIf and three escape lines. alg:explRAM2ADD discriminates: two \EndIf, zero escape lines, and it drifts by two.

The corrections that stand

where	defect	how it is decided
chapter6.tex:1091	u_2 = x^3 + u_{12}x^2 + \ldots; the leading coefficient belongs to u_1, declared in the same sentence. Now u_{22}	present in the published PDF's own rendering. Confined to that one declaration: u_{12} at :1160, :1162, :1170, :1239 genuinely means divisor 1 and is untouched
chapter5.tex:403-407 into :416-417, and the twin block at chapter4.tex:1227-1233	bare u_1/u_0 where the result's u_{n_1}/u_{n_0} are meant. 11 names per block	two arguments covering different names — see below. Chapter 4 was disagreeing with itself: :1238-1239 already wrote h_2u_{n_1}
chapter4.tex:1224-1225	"u_{n_1} = s_0'' - t_2 and therefore t_2 = s_0'' - u_{n_1}", both signs inverted	arb_ramifiedG2_DBL.mag:205 computes upp1 := spp0 + t2; the same page says "Notice that t_2 = u_{n_1} - s_0''" ten lines later; chapter5.tex:394/:644 already had it right
chapter6.tex:1260-1261	two malformed r-subscripts. On u_{n_1} a phantom subscript swallowed three summands; on u_{n_0} it swallowed two and detached the factor (t_7 + r_{r_{01}}), which then multiplied the phantom r. Plus r_{q_01} twice and a missing +	matched term for term against arb_splitG3_ADD.mag:10308-10310
chapter6.tex:1235 and :1257	two paragraphs that name a step in prose and then label their own block with a different one	:1235 announces "In Step15 ... terms of k" and labelled the block Step14; the block gives k_3/k_2, and Step14's formula is explicitly deferred to the doubling section two sentences earlier. :1257 announces Step17 and labelled the block Step~15. A scan finds no third instance in either chapter
chapter5.tex:1838	unbalanced parenthesis, -w_4( never closed	by reading. Not gate-visible: an unbalanced ( is legal in math mode, so the build is silent
chapter5.tex:647	dangling text, "the degree 2 coefficient k_2 = of k"	by reading
chapter5.tex:1589-1591	the two n_n adjustments cited as Steps 17/18 of alg:g2nudupl	retargeted to alg:g2balnudupl Steps 19 and 20, confirmed in the rendering

The u_{n_1}/u_{n_0} block has two arguments and they cover different names, which the errata now states rather than flattening:

• Five names by algebra. Each block states a formula and then its factored form, so the two lines must be equal. Expanding (u_{n_1} - s_0'')(u_{11} - u_{n_1}) forces u_{n_1}s_0'', u_{n_1}u_{11}, u_{n_0}s_0'' and u_{n_0}u_{n_1}. Verified symbolically: the corrected lines are equal and the published ones are not.
• Six h_2u_i names by the code. They sit outside the factorised group and appear identically on both lines, so comparing the lines constrains them not at all. They rest on arb_ramifiedG2_DBL.mag:209-210, vpp1 := ... + h2*upp1, which multiplies h_2 by the result.

The five retractions

1. \vh_1 is h_1 + v_{11}, not v_{21}. The argument for v_{21} was that alg:explRAM2ADD is an addition whose XGCD takes h + v_1 + v_2, supported by the code idiom vh1 := h1 + vp1. That idiom is from Deg12ADD, a different function, where the primed group is the degree-2 divisor. The function this passage describes is Deg2ADD, whose unprimed group is first and whose lines read

t4   := h1 + v1;
upp0 := spp0*(t3 - m3) + m1 + w3*(h2*(spp0 - up1) + t4 + v1) + w4*(u1 + up1 - f4);

so the block's \vh_1 + v_{11} is t4 + v1, that is h_1 + 2v_{11}. The passage says so two lines above the block, stating what it computes as u_n' = (s(-su_1 - 2v_1 - h) + k)/u_2, whose 2v_1 + h has degree-1 coefficient h_1 + 2v_{11}. v_2 reaches this step only through s and k. A code idiom is evidence only from the function under discussion, and the reading this overrode was the one a reader would naturally guess.

2. The generated-table repriming is reverted. Eight OUT rows in latexTables/split_ADD.tex were double-primed on the premise that single primes name the second input and so cannot name the result. In those four blocks there is no second input: the IN rows read D' = [], and the implementing functions Deg01ADDDWN, Deg01ADDUP, Deg02ADDDWN and Deg02ADDUP take no primed parameter at all, so up*/vp* legitimately name the result and the source returns them. Prime depth in these tables is a mechanical transcription of the Magma identifier and tracks arity, not the file.

The edit therefore made four self-consistent tables inconsistent: their bodies define only u^{\prime}_*, so the double-primed OUT rows named quantities appearing nowhere in the block and nowhere in the source. It also hand-edited generator output, which a regeneration would silently undo. The real fault is the label: latexConverter.py:479,561 append $D + D' = D'' = to every addition block regardless of arity. The claim that split_ADD.tex was the sole outlier is withdrawn too. latexTables/ is out of this PR's diff entirely.

3. alg:g3nucomp's S' is corrected in the opposite direction. The algorithm bound S and then tested and divided by an undefined S', and the first repair renamed the binding to S'. That is the wrong half: :561 reads u_n = (s(\vt - v_1) + kS)/u_2, and that factor must be the second gcd because k is formed at :545 from the undivided u_1.

The implementations use the overwrite convention the published text already had. reduced_basis_arithmetic.mag:147-153 reads S,a1,b1 := XGCD(u1,u2), then S,a2,b2 := XGCD(S,v2 + t1), then divides by S. The genus-3 ramified additions scale k by that quantity, k := k*S1 at nine live sites across arb, nch2 and ch2, and omit the multiplier entirely where the second gcd is 1. So :549 and :550 now read S, which is also the only repair well defined on every path: under S' naming, S = 1 at the outer test leaves S' unbound.

This is not a pure notation fix and should not be described as one. Two related defects are recorded and deliberately not corrected, both needing an editorial decision: alg:g3balnucomp assigns and tests S' but then uses \deg(S) and kS, and alg:g2balnucomp assigns and tests S' while dividing by S. alg:g2nucomp is the only one of the four internally consistent as printed, so alg:g3balnucomp is not a clean model to copy.

4. The prose Step~0 references were correct and are restored. Three algorithms carry \setcounter{ALG@line}{-1} immediately after \begin{algorithmic}[1], in the frozen published source as well as here, and the built PDF prints exactly three lines numbered 0:, every one the negative reduced normalization of v_1. So the published Step~0 was deliberate. An intermediate commit read it as an off-by-one and shifted four citations by +1; reverted, and confirmed against Algorithm 50's rendering, which prints 0 for the normalization, 1 to 3 for the d test and its escape, 4 for the z step the next sentence singles out, and 13 to 15 for the u_n computation.

The trap is that a source-level count of \State lines cannot see the counter reset, so Step~1 looked right for a line the document prints as 0. The arbiter for a step reference is the PDF, never the .tex.

5. The \Statex escape-line fix is reverted, as described above.

Two recorded descriptions were also wrong. D' = [] in four IN rows is not ERRATA.md E4's drained filter(); all four are degree-zero additions whose second divisor genuinely is the identity, so E4's claim that the committed .tex is unaffected stands. And the coordinate ordering was recorded as split ascending against ramified descending; measured, every file mixes orderings internally. Folded into PR31.

The build now matches the published thesis on pages and on algorithm numbers

Three reconstruction defects were found by measuring against the published PDF.

1. \frontmatter and \mainmatter were missing, so the Abstract, Preface and Acknowledgments were numbered as Chapters 1 to 3, shifting every chapter number.
2. \settocdepth{subsubsection} was missing, so the table of contents printed 33 of the 144 entries the .toc holds, two pages against the published four.
3. subroutine had its own per-chapter counter and its own list. The eleven subroutines numbered 5.1 to 6.5, were absent from the List of Algorithms, and every algorithm above 34 was numbered too low: the document ended at Algorithm 76 against the published 87.

	published, cover discounted	rebuilt
total pages	266	266
front matter	i to xiii	i to xiii
Chapter 1 / 7 / Bibliography begin on	14 / 260 / 263	14 / 260 / 263
Chapter 1 to Chapter 7 span	246	246
algorithm and subroutine numbering	1 to 87	1 to 87
subroutine numbers	35, 38, 40, 44, 48, 51, 60, 61, 62, 70, 71	identical
extracted words	106,780	106,586

The published PDF is 267 pages because its page 1 is a PRISM repository cover sheet added by the institutional repository. The body was exact all along at 246 pages; the entire page gap was front matter.

Item 3 matters beyond typography, and it is why the citation check was redone. A step reference verified against the rebuild alone could have been verified against the wrong algorithm: alg:g3explSPLIT3ADD rendered as Algorithm 60 here against 69 in print. The eighteen citations now read Algorithms 28, 37, 46, 50 and 69, matching the published document, and all eighteen land.

Item 3 also retired a wrong explanation of its own. The List of Algorithms had been recorded as one page short because of "entry indentation and wrapping"; it was short because eleven entries were missing. One residual remains and is genuinely cosmetic: the table of contents runs five pages against four and the list of algorithms three against four. They happen to cancel, which is why the front matter is thirteen roman pages in both, and that cancellation should not be read as evidence of anything.

Thesis/thesis.aux and Thesis/thesis.toc are committed artifacts of the 2020 build and are what the include order was read off. A bare pdflatex thesis overwrites them, so both READMEs now build out of tree. The root README still shipped the destructive command.

E-T14 records a defect the rebuild reproduces rather than introduces: the three front-matter list references are one page low in the published thesis, because each contents line is written before the page break that starts its list. Explicit at frontmatter.tex:65. Note this is not a convergence artifact and an earlier draft of Thesis/README.md said it was: the published PDF's own table of contents prints viii too, so the committed .toc agrees with the converged build to the page.

What is not claimed

No global "every step reference resolves" statement. Eighteen citations were checked against the rendered PDF across five algorithms, and all eighteen land. Nothing outside them was checked. Three earlier attempts to automate this were each wrong, in four distinct ways: counting only \State lines, missing plural Steps~N--M forms, mis-pairing a step number with an unrelated nearby \ref, and an alternation that listed If but not EndIf so both hypotheses were identical by construction.

No Magma gate applies here. No .mag file is touched, so opcount, whitebox, driver, blockcheck and the Magma suite have nothing to say about this PR. Where a correction is matched against code, that is a reading of the source, not an execution. Nothing in the Magma constrains LaTeX numbering, so the \EndIf item can have no code cross-check at all.

The page total is not evidence for the corrections. It is evidence about the reconstruction. The numbering alignment is what supports E-T11.

Word counts state their extraction method (pdftotext then wc -w) because a different tokenizer moves the total by tens of words, and this table has previously carried two different numbers for the same frozen document for exactly that reason.

ThesisPublished/ is untouched.

Left for the author

• A convention for S against S' across all four NUCOMP algorithms. Three different mixtures are in print.
• chapter6.tex's "the computation of z ... in Step~17", which names a step that is 18 in the explicit algorithm and 34 in alg:g3balnucomp, citing neither explicitly.
• latexTables/split_ADD.tex:253, [0,1,y_1,y_0,_1nn^{\prime}_1] in the balancing-weight slot. It compiles, since ,_1 subscripts the comma, so no build catches it.
• The hardcoded OUT label in latexConverter.py, and the negation sign the generator drops at :191/:206.

Recorded

Thesis/ERRATA.md gains E-T14, rewrites E-T11, E-T12 and E-T13, and brings its Status block and defect table current — the table had still said none of its entries was corrected. NEW_WORK.md gains N35, which records the five retractions rather than only the corrections that stand, per the file's own rule that a hypothesis tested and failed earns the same space as one that held.

Gates

Build clean: 266 pages, 0 errors, 0 undefined references, 0 undefined citations, 0 rerun requests. readme-paths 100/100 · check_paths --strict 259/259 · git diff --name-only master..HEAD -- '*.mag' empty · ThesisPublished/ byte-identical.