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Review date: 10 September 2026. Reviewer: independent source compilation reviewer. Frozen native state: the repository as it stood on 2026-09-10T05:48:43Z. This is a native rendition of the completed report, not a new source reading, proof assessment or review cycle. First-person deductions and verdicts below remain attributed to the original reviewer.

Verdict and current filing boundary

Refutation-failed for the six reconstructed proofs and their exact consumed interfaces, relative to the Corvaja-Zannier-recalled unit bounds and Heath-Brown's journal Theorem 2. I found no material defect in the frozen native reconstruction. Every essential deduction in Lemma 1.4, Corollary 1.5, Proposition 1.6, Lemma 1.7, Proposition 1.8 and Theorem 1.1 survives the checks below. Relative to those identified literature premises, together they prove a unique additive complement for all integer thirteenth powers.

The external proofs are not reconstructed. The source context supports the recorded interpretation of Heath-Brown's families as nonconstant. I accept that interpretation with the existing qualification: the introductory definition does not expressly state it. This is neither an author-issued correction nor a verbatim nonconstancy clause attributed to the introduction.

The manuscript's whole-box restatement of the journal theorem is a source discrepancy, already identified in the native pages. The local fixed-shift shell summation and finite small-scale bound supply the needed deduction. This report does not endorse the stronger uniform whole-box restatement.

The distinct grade is pass with named corrections. The native transformation supplying the required disclosures and recommended presentation changes has been accepted. The seven source pages now record independent compilation review of their exact frozen statements, essential deductions and consumed interfaces, relative to the two identified literature premises. This same-checkpoint standing reconciliation does not assess new mathematics. No claim tier, consumer status, formal verification, external-proof coverage or public-acceptance finding changes.

Subject, independence and selection of attacks

The source-coverage record pins all nine mathematical Markdown subjects, the separately scoped consumer and the four PDF editions it identifies, identifies versions, and records actual text/visual reading, exclusions and reproducible rendering instructions.

I did not author or collaborate in constructing these native proofs and had not previously built on them. The supplied context contained the assignment and triage's assessment that the reconstruction was complete, not an independent proof verdict. That assessment was scope context, not mathematical evidence. The frozen subject's author-recorded standing and qualifications were read as part of the subject. No private proof plans, previous independent verdicts or excluded research narratives were read. The consumer was read only for formulation and scope; linked web acceptance history was not fetched or revalidated.

The commissioned read set also contained acceptance text about the conclusion, not the reconstruction: _index.md lines 56-75 (public acceptance via Bloom's exposition and Price's claim) and the consumer wiki/problems/diophantine_problems/E0477/_index.md line 7 (status: proved) and lines 21-23, 45-47, 53-55 and 66-67 as they stood on 2026-09-10T05:48:43Z; together with the author-recorded standing paragraphs of the seven frozen pages (_index.md lines 5-6 and 171-179, theorem_1_1.md lines 114-122, lemma_1_4.md lines 239-242, corollary_1_5.md lines 58-62, proposition_1_6.md lines 229-232, lemma_1_7.md lines 96-100, proposition_1_8.md lines 70-75), a separately spawned materiality grader (Claude Fable 5.1) ruled this exposure immaterial on 2026-09-18: the text states no verdict on the six deductions, the pages separate public acceptance from the reconstruction's verification state, and the verdict rests on the rederivations below and the grader's independent rederivation.

Selection disclosure. The supplied triage pre-directed attention to coordinate normalization, the fixed-shift shell repair and the family convention. In particular, it pre-identified two of the three weakest-step routes reported below. Their derivations were independently performed, but those attack selections were not independently identified by the reviewer. This disclosure qualifies the source of the attack selection; it does not attribute the derivations to the triage.

Exposure ruling. Besides the unretained triage note that assessed the reconstruction as complete and pre-named two of the three weakest steps, the commissioned read set as it stood on 2026-09-10T05:48:43Z carried the subject pages' own author-recorded standing (under library/diophantine_problems/pipeline_math_2026_tiling_complement/: _index.md lines 171–179, lemma_1_4.md 239–244, corollary_1_5.md 58–64, proposition_1_6.md 229–233, lemma_1_7.md 96–101, proposition_1_8.md 70–77, theorem_1_1.md 116–122) and public-acceptance text for the existence conclusion (_index.md lines 56–75; wiki/problems/diophantine_problems/E0477/_index.md line 7, lines 21–26 and 44–57); a separately commissioned materiality grader acting in the corpus grader role (model: Claude Fable 5.1) ruled on 18 September 2026 by the content test that all three exposures are immaterial, because none states or implies the premise-relative verdict asked of the reviewer and the review's reasoning rests on independent rederivation corroborated by the subreview and the grader, so the pass with named corrections stands unchanged.

I read the applicable organization/repository instructions, wiki anatomy, verification and evidence guidance, and the PDF skill. All nine native mathematical pages and all six manuscript pages were read. The exact external-interface pages and source context in the coverage record were checked in text and complete rendered images. The manuscript's extracted text corrupts mathematical glyphs, so rendered pages controlled formulas.

A separate fresh-context subreview checked only the Heath-Brown interface and fixed-shift shell deduction. My own derivation and verdict were reached and communicated before it returned. Its bounded verdict was also refutation-failed, with Corollary 1.5 assumed and the full determinant proof excluded. It separately checked the journal/v1 pages, source hashes, signed witness bound, shell endpoints, remainder and exponent. Its findings were compared after the primary checks and supply supplementary coverage, not a substitute for the whole six-result review. Both reviewers remain distinct from the reconstruction author and the grader.

No source fetch or mathematical computation was performed in that reading. PDF rendering, text extraction and hashing were source inspection only. No new source or PDF reading is claimed by this native filing.

Exact conclusion and premises

Let $B=\{m^{13}:m\in\mathbb Z\}$ and $D=B-B$. There exists one $A\subseteq\mathbb Z$ such that, for every $n\in\mathbb Z$, there is exactly one pair $(a,m)\in A\times\mathbb Z$ with $n=a+m^{13}$. Equivalently, the translates $a+B$ partition $\mathbb Z$. Negative inputs and zero are included. Injectivity of the odd-power map equates uniqueness of the value $b$ with uniqueness of $m$.

The dependency order is the recalled genus-zero unit bounds to Lemma 1.4 to Corollary 1.5; that corollary and the journal theorem to Proposition 1.6; then Proposition 1.8 supplies finite avoidance. The separately proved Lemma 1.7 converts avoidance to a tiling; Theorem 1.1 converts uniqueness of $b$ to uniqueness of $m$. No native L-claim is a premise. All six proofs were author-recorded and awaiting compilation review at the frozen revision. There is no circular dependency and no conjectural antecedent introduced into the final existence statement. Reliance on the identified literature premises is not local proof coverage of those external results.

Per-result deductions

Lemma 1.4: rational curves and the three lines

For every $N\in\mathbb Q\setminus\{0\}$, the statement concerns maps defined over $\mathbb Q$ from $\mathbb P^1_{\mathbb Q}$ to $X^{13}+Y^{13}+Z^{13}=NW^{13}$. If $N$ is not a rational thirteenth power, there is no nonconstant rational map. If $N=d^{13}$ with $d\in\mathbb Q\setminus\{0\}$, the geometric images of nonconstant morphisms are exactly the three lines $X=-Y,\ Z=dW$; $X=-Z,\ Y=dW$; and $Y=-Z,\ X=dW$. This does not exclude maps defined only over an extension or assert that every rational point in an image has a rational parameter preimage.

The normalization is valid. Generic rational coordinates over $\mathbb Q$ can be cleared of denominators, homogenized to their maximum degree and divided by their common homogeneous divisor. All active coordinates retain a common degree $e$. A common geometric zero would be a common linear factor over the algebraic closure; its conjugates would give a common factor over $\mathbb Q$. Therefore the reduced tuple has no base point, including infinity, and extends to a morphism. The surface relation is a homogeneous polynomial identity and survives cancellation. A degree-zero tuple is constant, so nonconstancy gives $e\ge1$.

The height and support calculation needs all these normalization facts. For the active forms $P_i$, the polynomials $p_i(s)=P_i(s,1)$ have minimum order zero at every finite point. At least one $p_i$ has degree $e$, or infinity would be a common zero of the homogeneous tuple. The minimum order of the thirteenth powers at infinity is thus $-13e$, giving projective height $13e$. A nonzero constant coefficient such as $-N$ changes no order; a common rational factor changes the sum by its principal divisor, of degree zero.

A dehomogenized polynomial can have a pole at infinity even if its homogeneous form does not vanish there. The applied $S$-units are instead ratios of forms of equal degree. Their divisors are supported on the union of the homogeneous zero divisors: the equal-degree dehomogenization terms cancel. Thus this union $\Sigma$, of at most $re$ distinct points for $r$ active forms, is the correct support for the normalized units. Infinity is included whenever it is a homogeneous root.

If all normalized thirteenth-power ratios were constant, every original coordinate ratio would have zero order everywhere and be constant, making the map constant. At least one normalized unit is therefore nonconstant. Its zero and pole occur at distinct points, so $|\Sigma|\ge2$ as required. This checks normalized nonconstancy, not merely nonconstancy of an unreduced coordinate.

The active-coordinate and zero-subsum cases exhaust the possibilities:

• With four active terms and no proper vanishing subsum, the four-term bound gives $13e\le3(|\Sigma|-2)\le12e-6$, impossible for $e\ge1$.
• With three active terms and no proper vanishing subsum, the three-term bound gives $13e\le|\Sigma|-2\le3e-2$, also impossible. A proper vanishing subsum would have two terms and leave a single nonzero complementary term equal to zero.
• Zero active terms are impossible in projective coordinates; one active term cannot sum to zero. With exactly two, the coordinate ratio has constant thirteenth power and hence is constant, so the map is constant.
• The remaining case is four active terms split into two vanishing pairs. The term $-NR^{13}$ must pair with one of the other three.

For example, $F^{13}+G^{13}=0$ makes $F/G$ constant and rational: it lies in $\mathbb Q(s)$, whose constant field is $\mathbb Q$. The only rational thirteenth root of $-1$ is $-1$. The complementary pair $H^{13}=NR^{13}$ similarly gives $H/R=d\in\mathbb Q^\times$ and $N=d^{13}$. The other pairings give exactly the other two lines. Roots of unity in an extension cannot produce additional lines for these $\mathbb Q$-defined maps.

Containment in a line is upgraded to equality of its geometric image. On the first line the map is $[F:R]$, with no common zero. For any geometric $[a:b]$, the nonzero homogeneous form $bF-aR$ has positive degree and a root; the no-common-zero property makes that root map to $[a:b]$. Identically zero $bF-aR$ would make the map constant. The three explicit maps conversely parametrize the lines and satisfy the surface equation.

Reviewer verdict: every case and deduction is accepted relative to the two identified unit inequalities, including coordinate normalization, complete support, height, rational constants and full image classification.

Corollary 1.5: affine rational-family exclusion

For every $c\in\mathbb Z\setminus B$, there is no nonconstant rational one-parameter map over $\mathbb Q$ into $u^{13}-v^{13}-t^{13}=-c$. In particular, there is no nonconstant polynomial triple over $\mathbb Q$ satisfying that identity.

For an integer $c\notin B$, $c\ne0$. If a reduced fraction $(p/q)^{13}$ is an integer, $q^{13}$ divides $p^{13}$ and coprimality forces $q=1$. Hence an integer is a rational thirteenth power exactly when it is in $B$. Oddness preserves this property under negation, so $-c$ is not a rational thirteenth power.

The invertible substitution $(X,Y,Z)=(u,-v,-t)$ gives the projective surface with $N=-c$. A nonconstant rational one-parameter map to the affine surface stays nonconstant in the projective closure and extends as established in Lemma 1.4. That lemma excludes it. Polynomial triples over $\mathbb Q$ are a special case. No dominance hypothesis for parametrizing the whole surface is inserted.

Reviewer verdict: the entire stated exclusion is accepted for precisely $c\in\mathbb Z\setminus B$.

Proposition 1.6: the fixed-shift counting estimate

Define $S_c(T)=\{t\in\mathbb Z:|t|\le T,\ t^{13}-c\in D\}$. For each fixed $c\in\mathbb Z\setminus B$ and every real $T\ge1$, the claim is $|S_c(T)|\le K_cT^{5/6}$, with $K_c$ independent of $T$, and consequently $|S_c(T)|/T\to0$. No uniformity in $c$ is required.

Every witness $t^{13}-c=u^{13}-v^{13}$ has $u\ne v$. The quotient polynomial $Q(u,v)$ is positive for every real nonzero pair: strict increase of the odd-power map proves this off the diagonal; on the nonzero diagonal, $Q(u,u)=13u^{12}>0$. Its minimum $\kappa$ on the compact max-norm unit boundary is positive. Homogeneity and integer separation $|u-v|\ge1$ yield

$$\kappa\max(|u|,|v|)^{12}\le |t^{13}-c|\le(1+|c|)T^{13}.$$

This bounds every witness, not just a selected representative. It includes opposite signs and zero coordinates. With $C_c\ge1$, $X=C_cT^{13/12}$ also bounds $|t|$.

Set $(x,y,z)=(u,-v,-t)$. The equation is $x^{13}+y^{13}+z^{13}=-c$. Multiplication by $\operatorname{sgn}(-c)$ gives a homogeneous integral ternary form $F_c$ with positive right side $N_c=|c|$. Its first derivatives are nonzero constants times $x_i^{12}$, whose only common geometric zero is the origin. Thus the required projective nonsingularity holds.

The degree threshold is $\lfloor13/10\rfloor=1$. Under the disclosed source convention, a removed family has positive maximum degree. Any such integer-polynomial identity would give a nonconstant rational map to Corollary 1.5's affine surface after the inverse sign change. None exists. This excludes every such family before its parameter-value domain matters.

For the local whole-box deduction, fix $c$ and choose $R_c\ge1$ above which the source range $N_c\ll_{F_c}R$ holds. The theorem's shell constant is uniform in $R$ above this threshold. If $X\ge R_c$, take shells with upper radii $X/2^j$, where $0\le j\le J$ and $J$ is maximal with $X/2^J\ge R_c$. They are disjoint and cover the interval of heights $(X/2^{J+1},X]$. Their strict lower and inclusive upper endpoints place each shared boundary in exactly the smaller shell. Their bounds total at most

$$\frac{K_{F_c}X^{10/13}}{1-2^{-10/13}}.$$

The remaining box has height less than $R_c$ and no more than $(2\lceil R_c\rceil+1)^3$ integer points. The same constant bounds the whole box when $1\le X<R_c$. Since $X^{10/13}\ge1$, this gives the claimed $O_c(X^{10/13})$ bound for every $X\ge1$. No small shell outside the source range is used, and no logarithmic loss appears.

Projection of the finite set of witnessing triples onto $-z$ covers $S_c(T)$. Different parameters have different third coordinates, so there are no more parameters than triples. Finally, $(13/12)(10/13)=5/6$, and division by $T$ leaves a fixed multiple of $T^{-1/6}$, which tends to zero.

Reviewer verdict: the entire estimate is accepted, including the witness bound for all integer signs, theorem application, exceptional-family exclusion, shell sum, finite lower scales, projection and limit. The box estimate is a local fixed-shift deduction, not the journal's statement as printed or a uniform estimate in $c$.

Lemma 1.7: the greedy criterion for arbitrary B

For arbitrary $B\subseteq\mathbb Z$, the hypothesis is that, for every finite $C\subseteq\mathbb Z\setminus B$, there exists $b\in B$ such that $(C-b)\cap(B-B)=\varnothing$. The conclusion is one $A\subseteq\mathbb Z$ such that, for every $n\in\mathbb Z$, exactly one pair $(a,b)\in A\times B$ satisfies $n=a+b$. The explicit pair domain restores a quantifier omitted in the original report's abbreviated restatement; it does not change the frozen lemma.

The hypothesis includes the empty finite obstruction set, so it implies $B\ne\varnothing$. Enumerate all integers. At an uncovered $n$, let $A'$ be the finite set of previously selected indices and use $C=n-A'$. The uncovered condition ensures $C\subseteq\mathbb Z\setminus B$. The chosen $b\in B$ permits $a_{\mathrm{new}}=n-b$ and covers $n$. For every old $a$,

$$a_{\mathrm{new}}-a=(n-a)-b\notin B-B.$$

An intersection of the old and new translates would force that same difference into $B-B$. The new index cannot already be old, because $n$ was uncovered. The finite-stage induction therefore preserves coverage and disjointness.

At the union, every integer is covered at its own finite stage. Any two final indices coexist in one finite stage, so their translates remain disjoint. Once $a$ is fixed, $b=n-a$ is unique. Selecting the first available $b$ in an enumeration suffices for existence without asserting an effective decision procedure.

Reviewer verdict: accepted at the stated generality for arbitrary $B\subseteq\mathbb Z$, including the empty-$C$ edge case. No additional symmetry or polynomial premise on $B$ is used.

Proposition 1.8: finite avoidance

For the full integer thirteenth-power set $B$, every finite $C\subseteq\mathbb Z\setminus B$ admits $b\in B$ with $(C-b)\cap D=\varnothing$, where $D=B-B$.

Fix the finite set $C$ before choosing the height. For $C=\varnothing$, use $b=0$. Otherwise a bad parameter at $c$ satisfies $c-t^{13}\in D$, or equivalently $t^{13}-c\in D$, since every difference set is symmetric. Each $c$ has its own finite constant $K_c$ from Proposition 1.6. Their finite sum $K_C$ bounds the union of bad sets by $K_CT^{5/6}$.

Choose integral $T$. There are exactly $2T+1$ candidates, and the ratio of this union bound to $2T+1$ tends to zero. Some candidate therefore escapes every obstruction for sufficiently large $T$. Its thirteenth power is the required $b$. Neither an infinite union nor a constant uniform over all shifts is used.

Reviewer verdict: accepted. Using integral $T$ makes the exact $2T+1$ count valid; it suffices for the source's existence deduction.

Theorem 1.1: composition and formulation

Proposition 1.8 supplies exactly Lemma 1.7's hypothesis for the full image of integer thirteenth powers. The lemma supplies a unique pair $(a,b)$ for every integer. The odd-power map is injective, so each $b$ has exactly one integer input $m$. Thus one $A$ works for every integer with no finite exceptional set and no positivity restriction. The polynomial $X^{13}$ has integer coefficients and degree at least two, so the consumer's existence question follows.

Reviewer verdict: accepted. This is an existence and uniqueness conclusion, not effectiveness or an exponent classification.

External premises and source fidelity

Corvaja-Zannier, printed p. 438, PDF p. 3. The actual source assumes an algebraically closed characteristic-zero field, a smooth complete curve and a finite set of at least two points. It defines the valuation projective height used here. Its recalled Mason-Stothers bound requires $u,v$ not both constant when $1+u+v=0$ and gives $H\le2g-2+|S|$. Its recalled Brownawell-Masser consequence has the unit $z=1+u+v$, no vanishing subsum on the right, and gives $H\le3(2g-2+|S|)$. It prints no additional nonconstancy assumption in the four-term paragraph.

Dividing by one active term and negating the fourth yields those exact normalized equations. The no-proper-subsum conditions agree: a zero subsum involving the fourth term has a zero complementary subsum among the first three. Genus zero gives the factors used in Lemma 1.4. The interface is claims checked, and the local normalization deductions are independently checked. The original classical proofs and Corvaja-Zannier's own main theorems are not independently proved here.

Heath-Brown, journal printed p. 1580, PDF p. 2. The native interface records a shell count, positive integer $N$ in the stated range, a nonsingular homogeneous integral ternary form, the threshold $\lfloor k/10\rfloor$, and exponent $10/k$ without epsilon. For each fixed such form $F$ of degree $k\ge3$ and positive integer $N\ll_F R$, the solutions $F(\mathbf x)=N$ in $R/2<\max_i|x_i|\le R$ outside the indicated polynomial families number $O_F(R^{10/k})$. It is a claims-checked literature premise with limited proof-context reading. The determinant argument and its dependencies remain outside the review. The additional clauses about counts of parametrizations and diagonal special solutions were compared with the printed statement but are not used here.

Neither introductory definition literally excludes degree-zero triples. Permitting them would place every integral solution in the removed set and give no bound on all solutions. The paper's curve-family discussion and $O(B^{1/d})$ family count on printed p. 1589 support positive-degree families. V1 pp. 10–11 supply the same curve context. I accept the qualified interpretation already recorded in both native pages. I do not attribute a nonexistent express nonconstancy clause to the source. The absence of any nonconstant rational family makes the parameter-value convention immaterial in this use.

Grader-supplied context. The distinct grader additionally identified same-page support on journal p. 1580: Theorem 1 calls the $d=1$ case “essentially different linear parameterizations.” This is attributed to the grader's source inspection, not a new reading by the filing author or a detail independently identified in the primary report. It supports the existing interpretation without adding an express nonconstancy clause to the introductory definition.

V1 p. 1 defines a whole-box count; journal p. 1580 defines a shell count. Identical-looking Theorem 2 displays do not identify these different counts. V1 was read for comparison, not silently substituted as the primary premise. The manuscript's Theorem 1.3 cannot supply the claimed uniform whole-box journal formulation. The local proof supplies exactly the weaker fixed-$c$ bound needed by the argument. No author-issued erratum is claimed. The manuscript's general-$r$ Theorem 1.2 and full external proofs receive no independent proof coverage here.

Weakest steps and strongest attempted refutation

The three most vulnerable connections were independently rederived. As disclosed above, coordinate normalization and the shell application were pre-directed by the supplied triage; their selection was not independently identified by the reviewer.

1. Rational-map normalization, exact height $13e$ and complete-line support. Omitting infinity, confusing unnormalized polynomial poles with homogeneous roots, or testing only unreduced coordinate nonconstancy would invalidate this application. The reduced equal-degree tuple and normalized ratios prevent all three errors.
2. The fixed-$c$ journal-shell application. A direct whole-box citation or a theorem call below its allowed scale would leave a gap. The geometric series and finite residual box close precisely the estimate used.
3. The move from separate estimates to one tiling of all integers. An infinite union of sparse sets need not be sparse. Here each finite stage fixes only finitely many obstructions, followed by a separate union argument for covering and disjointness.

The strongest attempted refutation combined the shell discrepancy with the degree-zero-family ambiguity. The first is resolved by the valid local deduction; the second remains disclosed and supported by source context. Neither defeats the reconstruction under its stated reading. Further attacks by sign cancellation, an omitted zero coordinate, a proportional nonconstant tuple or extra rational root-of-unity lines also fail for the reasons above.

Checklist

Exact obligation and retention boundary

The obligation addressed is the fresh independent compilation review of the actual statements, every essential deduction and consumed external interfaces recorded on the seven frozen pages: the source digest and Lemma 1.4, Corollary 1.5, Proposition 1.6, Lemma 1.7, Proposition 1.8 and Theorem 1.1. It includes the local coordinate normalization, journal-shell summation and small-scale bound. The exact subject is the repository as it stood on 2026-09-10T05:48:43Z, with the paths in the coverage record.

The recorded independent compilation review covers only those frozen reconstructed statements and proofs, relative to the Corvaja-Zannier p. 438 recalled unit bounds and Heath-Brown's journal p. 1580 Theorem 2 under the explicitly qualified nonconstant-family interpretation. It does not reconstruct or independently prove either external literature premise. It says nothing new about catalog status, public acceptance, the manuscript's unused general-$r$ Theorem 1.2 or stronger Theorem 1.3, other exponents, effectiveness, positive/nonnegative variants or Lean.

The accepted native transformation and same-checkpoint standing reconciliation discharge that compilation-review obligation only in this premise-relative sense. The seven pages link this final report; its subject pins remain those of the frozen state (2026-09-10T05:48:43Z), not the later standing edits. No bounded mathematical correction to their frozen statements or proofs was requested by the reviewer or grader. Pre-directed attack selection remains disclosed; the derivations, not those selections, were independently performed.

Transformation provenance

The original report remains unchanged in working storage. The original coverage receipt and grade are described in their respective native records. This rendition retains the full per-result deductions, external interfaces, attacks, checklist and limits. It converts ASCII notation to LaTeX, supplies the premise rider and $A\times B$ domain, discloses pre-directed selection and the exact obligation, and explicitly attributes the added p. 1580 family context to the grader.

The original reports are preserved unchanged in working storage; native mathematical use depends on the complete content retained here and on ordinary repository provenance, not on private paths or runtime products. The frozen native pages are named by path and date, and the text they carried then is not retained as copies. The current pages differ from it in their verification wording and the source digest's description, updated: field and generated navigation (the standing reconciliation of 2026-09-10T07:43:28Z, rewrapped on 2026-09-17), in American spellings (2026-09-17: "canceling" in the Corollary 1.5 proof, "catalog" in the source digest and Theorem 1.1) and in the source digest's provenance wording (its sentence on the Heath-Brown entry's separate reading, reworded on 2026-09-17; its PDF digest lines replaced on 2026-09-18 by links to the Corvaja-Zannier and Heath-Brown source cards), not in any statement or proof step. The four PDFs are identified by edition on their source cards; no file of the manuscript is held. A new location or edited presentation supplies no new mathematical verdict.