# Adversarial review of H. Hironaka, "Resolution of Singularities in Positive Characteristics" (preprint dated March 23, 2017, 92 pp.) Date: 2026-10-23, America/New_York (−04:00). Status: completed research dump — a line-level reading of the whole manuscript, a reconstruction of its proof architecture, explicit counterexamples to several of its theorems (checked by computer), a literature/reception check, and an assessment of repairability. **Provenance: AI-generated (model claude-fable-5-1, Claude Code agent; one general-purpose subagent, same model family, used for the literature search), not human-reviewed; no formal verification.** The manuscript was read from /Users/tdupuy/repos/dumps/refs/hironaka-resolution-of-singularities-in-char-p.pdf via `pdftotext -layout`; page numbers below are the manuscript's printed ones. Working notes: math-audit.md. Computations: verify.sage (SageMath 10.8; output verify-output.txt), search_constant_ridge.py and search_constant_ridge2.py (outputs .out). User's request: "Perform an adversarial review of the manuscript. If the technique can be repaired, repair it. If the technique cannot be repaired, explain why." **Headline verdict.** The manuscript does not prove resolution of singularities in characteristic p. I found **specific, located, false statements among the load-bearing theorems**, each refuted by an explicit example in characteristic 2 in ambient dimension 2 (so the failures are not exotic): 1. The new algebraic object of the paper, the negative-degree extension ℘̃(E) of the characteristic algebra (§5), is degenerate: under the manuscript's own standing hypothesis (Eq. (37), p.26) its negative part ℘^{nega}(E,−a) is the **unit ideal** in every degree −a < 0, at every singular point. This follows in one line from the manuscript's Theorem 6.3 together with its Theorem 4.5 (§3.2 below). Consequently the "master key" Theorem 9.2 is vacuous, and the transformation rule for ℘^{nega} (Theorem 7.8, Eq. (59), p.37) is **false** — the explicit example E = (x²+y⁵, 2) in char 2 gives O on the left and the ideal (y) on the right. The "domino-fall" Lemma 16.8, the key to Main Theorem I, rests on that rule. 2. Main Theorem I, Theorem 16.6(2) and (4) (p.84) — the claim that blowing up the "terminal plat" strictly decreases the lexicographic string of edge invariants at every point of the exceptional divisor off the strict transform — is **false** for E = (x²+y⁵, 2) in char 2: the invariant string is identical before and after the blowup (verify.sage, V1). With it falls the termination argument of Main Theorem II (Theorem 16.13: "apply Theorem 16.6 repeatedly but finitely many times"). The only quantity that does decrease in the example (the residual order 5 → 3 → 1) is not a component of the manuscript's invariant, and it is precisely the quantity that is known to *increase* in general (Moh's bound; Hauser's kangaroo example, reproduced in verify.sage V3). 3. Theorem 15.10 Step 2 (p.80), which produces the smooth center ("terminal plat") for the induction, is **false** for the same example; the core-focusing Ě of §6.2 is ill-defined (its explicit formula (44) equals S(Ê) identically); Theorem 6.1's second sentence is false (Narasimhan's example); two proofs in §14 break off mid-sentence (Theorem 14.4, Lemma 14.5); the "Sing(Ě) ⊂ tail hypersurface" theorems of §10 have proofs that are non sequiturs. **Repairability.** The local defects (ill-typed definitions, garbled lemmas, the Γ-monomial lemma, Theorem ∆) are repairable and I indicate repairs. The central gap is not local: once ℘^{nega} is seen to be trivial, the LLUED chain reduces to "write g = y^q + ε and peel off the non-p-th-power part of ε", i.e. to a choice of hypersurface of weak maximal contact, and the termination claim reduces to the assertion that the edge invariants (n, n−r, q_1…q_r) — the ridge data of the tangent cone, refined by formal maximal-contact information — strictly decrease under blowup of the terminal plat. They do not (example above), and the natural secondary invariant (the order of the ambient reduction F(Y), i.e. the residual order) is exactly the one that Moh/Hauser show can increase. So the technique, as written, reduces to the known open problem; I cannot repair it, and I cannot prove that no refinement of its invariants works (that would be a proof that the problem is unsolvable by such invariants, which no one has). The user's dichotomy "repairable or demonstrably irreparable" is therefore a **false dichotomy** for this manuscript: the correct description is "the argument as written is refuted; what remains after removing the false steps is the open problem in its standard form." Reception (literature check, §7): the preprint was never posted to arXiv, never revised or withdrawn, and never publicly confirmed; the one on-record expert statement found (K. Matsuki, Kyoto colloquium, 8 May 2019) is that "nobody seems to have been able to confirm the details of the alleged proof so far." I found no published analysis of any specific step, so the refutations above appear to be new as public statements; they are elementary. ## 1. Question-to-answer map | Question | Answer and status | |---|---| | Does the manuscript prove embedded resolution in char p? | **No.** Theorems 16.6(2),(4), 15.10 (Step 2), 7.8, and the second sentence of 6.1 are false (located counterexamples, §3); the termination of the induction (Theorem 16.13) is unproved. | | What is the single most consequential defect? | ℘^{nega}(E,−a) = O for all a > 0 (§3.2). It voids §§5, 9 (master key), the ℘^{nega} transformation rule, and the domino-fall. Proof uses only the manuscript's own Theorems 4.5 and 6.3. | | Is there an explicit example where the strategy's invariant fails to decrease? | **Yes**: E = (x²+y⁵, 2), char 2, dim 2 (§3.4). Inv-string unchanged under the blowup of the terminal plat; residual order drops 5→3→1 but is not in the invariant. | | Does maximal contact enter, and does it fail? | The tail hypersurfaces {y(e)=0} play the role of hypersurfaces of maximal contact. Their one-step containment property for point blowups is the classical directrix theorem; the manuscript's "Sing(Ě) ⊂ tail" (Thm 10.3/10.4) has an invalid proof and is undetermined (§3.6); the stability of the tail under *sequences* of blowups is where Hauser's kangaroo example bites (verify.sage V3: residual order 2 → 3). | | Can the technique be repaired? | Local items yes (§5). The termination argument no, not by me: the needed strictly decreasing invariant is the open problem (§5). | | Is the user's framing (repairable vs. demonstrably irreparable) right? | **False dichotomy** (§5, §6). | | What does the field say about the preprint? | No public verification, refutation, revision or withdrawal found; MathOverflow 269651 (May 2017) has 0 answers; Matsuki 2019: unconfirmed; dim ≥ 4 char p described as open through 2024–2026 (§7). | | Which parts are correct? | §3 (NE theorem), Theorem 4.5, Theorem 6.3, Lemmas 7.3–7.5, Theorem 7.1 (classical, proof sketched), Theorem ∆ (16.4) and the Γ-monomial case as a strategy (§3.8, math-audit §12). | ## 2. Definitions used (as in the manuscript) - Ideal exponent E = (J, b); Sing(E) = {ord J ≥ b}; a blowup is permissible if its center is smooth and inside Sing(E); transform E' = (I(D)^{−b}JO', b) (Def. 2.1, p.5). S(E) = all finite sequences of permissible localized blowups over Z[t], t arbitrary indeterminates (Def. 2.5, p.6). S(E_1) = S(E_2) ⟺ ℘(E_1) = ℘(E_2) (Eq. (35), p.25). - ℘(E) = ⊕_{d≥0}℘(E,d): geometric definition ℘(E,a) = ∪{I : S(I,a) ⊃ S(E)}; algebraic: integral closure in the graded algebra Bl(Z)* of O[Diff^{(j)}J in degree b−j] (p.17; equivalence cited to [23] for perfect K). Diff^{(m)} = differential operators of order ≤ m (§3.1.1, p.7). - Edge data at ξ ∈ Sing(E): minimal generators ḡ_1,…,ḡ_r of the image of ℘(E)_ξ in gr = K[z̄]; they are additive polynomials, deg ḡ_j = q_j = p^{e_j}; Inv_ξ(E) = (n, n−r, q_1,…,q_r) (Eq. (34), p.24). Edge generators g_j = y_j^{q_j} + ε_j with ord ε_j > q_j (Eq. (27), p.21). - Ê = (J, d), d = max ord J (maximum base-hike, §6.1). Ě = core-edge focusing: Sing(Ě) = Inv-max stratum of Ê (§6.2, (43)/(44), p.30). - ℘̃(E,−a) = Σ_d Diff^{(dm+a)}℘^{posi}(E,dm) (Def. 5.1, Eq. (36), p.25), with m a common multiple of generator degrees and (Eq. (37), p.26) g ∈ ℘(E,m)_ξ, ord_ξ g = m assumed throughout. - LL-head g = y^q + ε ∈ ℘(Ě,q), q = p^e; LL-chain g(i) = g(i+1)^p + h(i), h(i) ∈ ℘^{nega}, ord h(i) > q(i) = p^{e−i}, tail y(e) = g(e) of order 1 (Eqs. (83)–(84), pp.54–55). Cot(Ě) = module generated by tails (Def. 13.2, p.67). CotAR-scheme Y(E): hypersurfaces Y_{ji} = {h_{ji}=0} for generators h of Cot, with ambient reductions F_j(Y_{ji}) (Def. 15.5, p.76). Chain Y(0) ⇝ Y(1) ⇝ … via S(E(s+1)) = S(Ě(s)) ∩ S(Cot(Ě(s)),1); terminal plat ∇ = Sing(Y(t)) at stabilization (Def. 15.12, p.80). - Ambient reduction (Thm 3.10, p.10): J♯ = Σ_{j 0, and ℘̃(E,0)_ξ = O_ξ. Proof (two independent ways). (i) From Def. 5.1: D(m,a,1) = Diff^{(m+a)}℘^{posi}(E,m) ⊇ Diff^{(m)}(g) since Diff^{(k)} is "operators of order ≤ k" (p.7). As ord_ξ g = m, some Hasse derivative ∂^{(β)}g with |β| = m is a unit at ξ. (ii) Without using the "≤": the manuscript's Thm 6.3 (p.30, proof correct) says f ∈ ℘^{nega}(E,−a)_ξ iff g^{p^ℓ}f ∈ ℘(E, qp^ℓ − a) for ℓ ≫ 1 (q = deg g). For f = 1 this is Thm 4.5 (p.18): ℘(E,qp^ℓ) ⊂ ℘(E,qp^ℓ−a). Thm 4.5 is geometrically true (S(I,a) ⊂ S(I,b) for a > b). ∎ Consequences, each located: Thm 9.2 "Master key" (p.47: ∃h ∈ ℘^{nega} with ord h ≥ ord ε and (g−h)^{1/p} ∈ O) is vacuous (h = ε works); Thm 9.3 likewise once y ∈ I(D) (Lemma 7.3); Thms 7.11, 7.12, 6.5–6.7 are trivially true; the degree bookkeeping of §9.7 (p.51) carries no information; Lemma 16.11 (p.86: "the only important point is h'(i) ∈ ℘^{nega}(F̌'(i))") is vacuous. **Theorem 7.8, Eq. (59) (p.37) is false.** It claims ℘^{nega}(E',−a)_{ξ'} = ΠI(D_i)^a·℘^{nega}(E,−a)O_{Z'} at ξ' ∈ Sing(E') with Inv_{ξ'}(E') = Inv_ξ(E). Example: char 2, E = (x²+y⁵, 2), D = origin, ξ' = origin of the chart y, E' = (x'²+y³, 2), a = 1. Both E at 0 and E' at ξ' satisfy (37) (ord = 2 = b) and have the same Inv (see 3.4). Left side = O_{ξ'} by the Proposition applied to E'; right side = (y)O_{ξ'} ≠ O_{ξ'}. The invalid step in the derivation (Rem. 7.9, Eq. (60), p.37) is "= z^{−a}(g')^{p^ℓ}℘^{nega}(E',−a)_{ξ'} (degree down by division)", which asserts ℘^{posi}(E', mp^ℓ−a) = (g')^{p^ℓ}·℘^{nega}(E',−a); Thm 6.3 gives only one inclusion. Why this matters: Lemma 16.8(1) (p.85) argues "g'(e−1) = g'(e)^p + h'(e−1) does not vanish at ξ' because h'(e−1) vanishes there by the rule of transformation of ℘^{nega}". But the chain identity g(e−1) = g(e)^p + h(e−1) pulls back to an identity in O' only if every summand is transformed with the same weight; the weight of g(e−1) is −p, so h'(e−1) = z^{−p}π*h(e−1), which vanishes on the exceptional divisor only if ord_D h(e−1) > p, while Thm 9.3/Lemma 16.7 give ord_D h ≥ p. Rem. 9.25 (p.58) instructs to apply "inherited" rules summand by summand; that is not a valid operation on an identity. The domino-fall is unsupported — and in the example of 3.4 it fails. ### 3.3 (C) Theorem 15.10 Step 2 (p.80) is false; Theorem 15.9's dimension chain is false Example: char 2, n = 2, E = (x²+y⁵, 2). Then Ê = Ě = E (Sing(E) = {0}; verify.sage V1). Edge data at 0: the only permissible centers of E and of all its transforms lie on the strict transforms of {x = 0} (the whole tree is computed in V1: 0 ↦ (x'²+y³) at the origin of the chart y ↦ (x''²+y) resolved; the other charts have no order-2 points; adding indeterminates t does not change this), so S(y⁵,2) ⊃ S(E), y⁵ ∈ ℘(E,2), x² ∈ ℘(E,2), and x ∈ ℘(E,1) by integral closedness of ℘ as a graded algebra. The edge algebra is (x̄) and Inv_0(E) = (2,1,1). Cot(Ě) = (x) (the LL-chain headed by x² has tail x; the manuscript's H♭ for ε = y⁵ = y·(y²)² is case (I): H♭ = ∂_y(ε)·∂_y^{(4)}(ε) = y⁴·y = y⁵, so h(0) = y⁵, g(1) = x). Y(0) consists of Y = {x = 0} with F(Y) = (J♯|_Y, 2) = (y⁵, 2) (V1: J♯|_{x=0} = (y⁵)); Sing(Y(0)) = {0}. E(1): S(E(1)) = S(E) ∩ S(x,1) = S(E), so Ě(1) ~ Ě(0), Y(1) = Y(0), Sing(Y(1)) = Sing(Y(0)): stabilization at s = 0, dim Y(0) = dim Y(1) = 1 (refuting Thm 15.9's "n > dim Y(0) > dim Y(1) > …", p.79). Step 2 of Thm 15.10 asserts that then J(0) = (x²+y⁵) has generators in (x)²ρ^ℓ(O) and that Sing(Ě(1)) = {x = 0} is smooth of dimension 1. Both are false: y⁵ ∉ (x), and Sing(Ě(1)) = {0}. (Here ∇ = {0} happens to be smooth; whether ∇ is always smooth is undetermined — the proof is wrong, I have no counterexample.) ### 3.4 (A) Main Theorem I, Theorem 16.6(2),(4) (p.84), and Corollary 16.9 (p.85) are false; the induction has no termination argument Same example. ∇ = {0}, D = ∇, ∇' = ∅, so every point of π^{-1}(0) is a point "ξ' ∈ π^{-1}(D)_cl ∖ D'" for (2) and a point "ζ' ∈ π^{-1}(∇)" for (4). Take ξ' = origin of the chart y (the only point of order 2 on π^{-1}(0); V1). - (4) says ℘^{posi}(F'(i;α)) is empty at ζ' for every member F(i;α) of every Y(i). For F = Ě: E' = (x'²+y³, 2) has ord_{ξ'} = 2, i.e. ξ' ∈ Sing(E'). For the AR member F(Y) = (y⁵,2) on Y ≅ A¹: its transform (y³,2) is singular at the origin of Y'. False. - (2) says the string (Inv_{ξ'}(Y'(0)), Inv_{ξ'}(Y'(1)),…) <_lex (Inv_0(Y(0)), Inv_0(Y(1)),…), "extending by (0) if necessary". Inv_{ξ'}(E') = (2,1,1) = Inv_0(E) (same argument: all infinitely near points of E' lie on x' = 0, so x' ∈ ℘(E',1)). The AR members: (y⁵,2) and (y³,2) on A¹ both have edge algebra 0 (℘(·,1) = (y³) resp. (y²), no order-1 element; their maximum hikes (y⁵,5),(y³,3) both have edge algebra (ȳ)): identical. So the strings are equal, term by term, however one reads "Inv_ξ(Y(i))". No drop. The same holds at the second blowup; the process ends only because ord(x''²+y) = 1 < 2, i.e. because the residual order ord ε went 5 → 3 → 1 — a quantity that is not a component of Inv. - Cor. 16.9 ("if ξ' ∈ π^{-1}(∇)∖∇' then g(0)' does not vanish at ξ' because ξ' is not in the strict transform ∇' of ∇") identifies ∇ with the tail hypersurface {g(e)=0} (Eq. (129), p.84: "∇ = {g(e) = 0}"). Here ∇ = {0} ≠ Y = {x=0}; ξ' ∈ π^{-1}(∇)∖∇', and g'(0) = x'²+y³ vanishes at ξ'. False. Theorem 16.13 (Main Theorem II, p.87) is "apply Th. 16.6 repeatedly but finitely many times"; with 16.6(2) false there is no termination argument. Theorem 16.14 (Main Theorem III) = 16.4 + 16.13. ### 3.5 Theorem 6.1, second sentence (p.29) is false "For every d > 0 and every ξ ∈ Sing(Ê)_cl we have d = ord_ξ ℘(Ê,d) provided ℘(Ê,d) ≠ 0." Narasimhan's example, d = 1, ξ = 0: ℘(Ê,1) ∋ ∂_y f ≠ 0, but an element of order 1 in ℘(Ê,1)_0 would define a smooth hypersurface containing Sing(Ê) = C near 0, and C has embedding dimension 4 at 0 (semigroup ⟨32,7,19,15⟩: all four are minimal generators; I(C) has no generator of order < 2 — V2). So ord_0 ℘(Ê,1) ≥ 2. (True for d a multiple of the hike degree, which is what Def. 5.1 uses, so Eq. (37) itself is safe.) ### 3.6 (D) §10: Sing(Ě) ⊂ tail hypersurface — proofs invalid, statement undetermined Lemma 10.1 (p.59) produces, for each η ∈ Sing(Ě) ∩ V, a *different* modified chain whose tail vanishes at η (y(i) − u(i,η)). Thm 10.4 (p.60) concludes "its scheme-tail Y contains V ∩ Sing(Ě)" — a non sequitur (one hypersurface per η, not one hypersurface for all η). Thm 10.6 (p.61) replaces y(j) by y(j) − c(j) and asserts the tail y(e) is unchanged — incoherent (the tail is y(e)). Thm 10.3 inherits these. The statement itself (does the Inv-max stratum lie in a smooth hypersurface, namely the tail?) I could not settle: - Narasimhan's example does not refute it (the Inv-max stratum is {0}, §3.1), and the manuscript's refinement of the top locus by edge data is exactly the kind of refinement Hauser mentions as the way around Narasimhan ("refining the usual stratification … through the local embedding dimension", Hauser, arXiv:0811.4151, §A). - **Lemma (proved, math-audit §8).** In char 2 there is no hypersurface f = x² + ε(y,z,w) whose order-2 locus contains a *monomial* curve C = (t^A,t^a,t^b,t^c) of embedding dimension 4 along which the tangent cone is everywhere a square of a linear form. (Parity analysis forces a,b,c even, A odd; then D_t^{(2)}(ε∘C) = 0 but D_t^{(2)}t^{2A} = A(2A−1)t^{2A−2} ≠ 0.) An exhaustive F₂-linear search (search_constant_ridge2.py, A,a,b,c < 24, degree ≤ 10) found none, consistent with the lemma. For non-monomial curves the necessary condition is a tangency condition x'(t)² = Σ_j ∂_j^{(2)}ε(C(t))·y_j'(t)² (the tangent of C must lie in the kernel of the varying linear form whose square is the tangent cone); if that form is constant along C then C ⊂ {ℓ = 0}. So a counterexample, if any, needs a varying ridge direction along a curve of embedding dimension ≥ 3 with constant ℘-edge data. Undetermined. ### 3.7 Further located defects (lower weight) - Thm 14.4 (p.69) and Lemma 14.5 (p.70): proofs break off mid-sentence ("It is important to note that the"; "and we want to prove ρ^e(cot(Ě)) ⊂ L_{e−i−1}(0)."). Thm 14.6 line 2 (p.70) is type-mismatched (a set of LSBs ⊂ a module). Cor. 14.7 is trivial or false depending on "transform". - Thm 8.6/8.7 (pp.44–45): the "lexicographical induction" extracting the unit coefficients u(θ) via ∂^{(θ)}h ignores the contributions of terms x^{θ'} with θ' ∈ θ + Z^n_{≥0} (Lucas: C(θ+p^ℓk, θ) ≠ 0 in general); incomplete. - Cor. 7.15 (p.40): "∂_j((x−ρ^N(a_N))^α) = ∂_j(x^α)" is false for values (char 2: ∂_x(x−c)³ = (x−c)² ≠ x²); true for the operators. - Lemma 5.6 (p.27): proof uses the Leibniz inclusion backwards (conclusion true, trivially, by 3.2). Lemma 5.9(4) (p.28), Lemma 3.14 (p.12), Def. 4.14 (p.22: max(O)Bl(Z,p^ℓ) should be max^{p^ℓ+1}), Lemma 16.5 (p.83: definition of s) are garbled. - Thm 15.9 (p.79): "Y(t+1) is smooth, i.e. F(t+1)_j is the zero ideal" vs. Thm 15.10(3) "Sing(Y(t+1)) is a smooth closed non-empty subset" — inconsistent. - Rem. 9.11(5)–(7) (p.51): "(h♭)^k" must be read as operator iteration, not as a power (otherwise the top monomial is not eliminated: example g = x²+y³z+z⁴ in math-audit §7). ### 3.8 What is correct (checked) Thm 3.15 with Prop. 3.26 (numerical exponent theorem; complete modulo typos); Thm 4.5; Thm 6.3 (its proof — Cartier extension, Krull intersection, faithful flatness — is correct and is what yields 1 ∈ ℘^{nega}); Lemmas 7.3–7.5; Thm 7.1 (non-increase of edge invariants; classical ridge non-decrease, proof only sketched; a proxy check on 375 near points of random order-2/4 hypersurfaces in char 2 found no violation, verify.sage V4); Prop. 9.15; Thm 9.18/Cor. 9.19; Thm 14.8; Lemmas 15.2–15.3; Thm 16.4 (Theorem ∆) and Lemma 16.5 as strategies (standard maximal-order and monomial-case reductions). ## 4. Classification of claims in this report - **Proved (by hand, elementary):** Proposition of §3.2 (℘^{nega} = O); vacuity of (44) in §3.1; falsity of Thm 7.8/Eq.(59), Thm 15.10 Step 2, Thm 15.9 dimension chain, Thm 16.6(2),(4), Cor. 16.9, Thm 6.1 second sentence, Cor. 7.15 as stated; the monomial-curve Lemma of §3.6; invalidity of the proofs of Thm 10.4/10.6, Lemma 16.8. - **Computational (verify.sage, reproducible):** the singular loci, orders, tangent cones and blowup charts of E = (x²+y⁵,2); Narasimhan's locus = I(C), dim 1, embedding dimension 4, ord_0 ∂f = 3, ord_η f_z = 1; Hauser's kangaroo sequence f0 → f3 with residual orders (5,2,2,2 w.r.t. x=0; 3 w.r.t. the corrected tail at a3); V4 proxy test of Thm 7.1. - **Conditional / argued but relying on the manuscript's cited inputs:** the identification x ∈ ℘((x²+y⁵,2),1) uses the geometric definition of ℘ and its equivalence with the algebraic one ([23], perfect base field); the S-theoretic verification covers the finite blowup tree over Z and, with indeterminates t as spectators, over Z[t]. - **Conjectural / undetermined:** whether the Inv-max stratum is always contained in a smooth hypersurface (Thm 10.3/10.4 statements); whether the terminal plat is always smooth; whether Thm 7.1 holds for the full ℘-based Inv (believed yes). ## 5. Repair **Locally repairable (with the repair):** - Def. 4.14: replace max(O_ξ)Bl(Z,p^ℓ) by max(O_ξ)^{p^ℓ+1}, so that cot(E)_ξ = {y : ȳ^{p^ℓ} ∈ edge algebra}; this is what §6.3 uses. - §6.2: define Ě := Ê ∩ (I(Σ),1) when the Inv-max stratum Σ is smooth (then S(Ě) = S(Ê) ∩ S(I(Σ),1), Sing(Ě) = Σ, ℘(Ě) finitely generated); for singular Σ I know no canonical finitely generated choice — this must be supplied before §§6.3–16 make sense. - Lemma 3.14: use the construction of Lemma 15.3. Lemma 5.6: not needed (℘^{nega} = O). Cor. 7.15: state it for operators. Lemma 16.5: define s via subsystems C with a(C) ≥ b; the monomial game is standard. - Thm 6.1: restrict the second sentence to d a multiple of the hike degree. - Thm 7.1: supply the height/degree argument (math-audit §5). - Thm 10.6 (smooth D): the correct route is ord_D h(e−1) ≥ p ⇒ g(e)^p ∈ I(D)^p ⇒ g(e) ∈ I(D); this needs the choice h = ε − φ^p with ord_D φ ≥ 1, which is available once y ∈ I(D) (Lemma 7.3). But this fixes the tail for one D at a time; a tail that works for all of Sing(Ě) simultaneously is the open point of §3.6. **Not repaired — and why the obstruction is structural.** After §3.2, the LLUED chain is: g = y^q + ε ↦ choose h ≡ ε mod ρ(O) with ord h > q (and ord_D h ≥ q for the intended center), g(1) = (g−h)^{1/p}, iterate; the tail is a hypersurface Y = {y + (higher terms) = 0} with T_ξY = {ȳ = 0}, i.e. a hypersurface of weak maximal contact chosen by removing the p-th-power part of ε in the chosen coordinates. The induction then consists of (i) the ambient reduction F(Y) (coefficient ideal, Thm 3.10), (ii) the claim that blowing up the terminal plat strictly lowers Inv = (n, n−r, q). (ii) is false (§3.4). The only candidate for a secondary invariant produced by the construction is the order of F(Y) along Sing, i.e. the residual order ("shade" in Hauser's terminology), and Moh's theorem (Hauser, arXiv:0811.4151 §B: shade_{a'}f' ≤ shade_a f + p^{e−1}, with increase actually occurring) shows that it can go up; Hauser's example x²+y⁷+yz⁴ realizes the increase 2 → 3 at the third point blowup, and the manuscript's own tail recipe at that point produces the hypersurface x + yz³(1+y) = 0 with respect to which the increase is visible (verify.sage V3). So appending the residual order to the Inv-string does not restore termination, and the manuscript offers no other quantity. Nothing in the manuscript addresses kangaroo points, Moh's bound, or any mechanism for a decrease at them (Hironaka's 2008 lectures reportedly did, via Hauser's classification — Hauser arXiv:0811.4151 §H — but that material is not in this preprint). Hence: the argument as written is refuted; what remains is the standard open problem (find an invariant that decreases through kangaroo points in all dimensions). I cannot prove that *no* refinement of edge data works; so "demonstrably irreparable" is not established either — the honest statement is that the manuscript contains no new idea that survives the removal of its false steps. ## 6. Refutations and corrected expectations 1. It was natural to expect that the negative-degree extension ℘̃(E) (the paper's advertised main novelty, Intro item (1), p.3) carries information about E. In fact ℘^{nega}(E,−a)_ξ = O_ξ for every a > 0 at every singular point, by the manuscript's own Thms 4.5 and 6.3 (§3.2, two proofs). Evidence: §3.2; verify.sage V1 (∂_x^{(2)}(x²+y⁵) = 1). 2. It was natural to expect that elements of ℘^{nega}(E,−a) transform by "adding a times the exceptional divisor" (Rem. 7.7, Thm 7.8). In fact Eq. (59) is false (E = (x²+y⁵,2), char 2: left side O, right side (y)), and the chain identity forces the opposite weight z^{−q(i)} on h(i). Evidence: §3.2. 3. It was natural to expect the "master key" Thm 9.2 (p.47) to be the technical heart of the paper. In fact it is vacuous (h = ε). 4. It was natural to expect that blowing up the terminal plat strictly decreases the edge-invariant string (Thm 16.6(2)), so that Thm 16.13 terminates. In fact the string is unchanged for E = (x²+y⁵,2) in char 2 under the blowup of ∇ = {0}; the resolution of this E proceeds only because the residual order 5→3→1 drops, which Inv does not record. Evidence: §3.4, verify.sage V1. 5. It was natural to expect that if D = ∇ then the transforms have no singular points on π^{-1}(∇) (Thm 16.6(4), Cor. 16.9). In fact (x'²+y³,2) has one. Evidence: §3.4. 6. It was natural to expect the stabilization Sing(Y(s)) = Sing(Y(s+1)) to force Sing(Ě(s+1)) = {f = 0} smooth (Thm 15.10 Step 2). In fact for E = (x²+y⁵,2) stabilization occurs at s = 0 with Sing = {0} ≠ {x=0}, and J(0) ⊄ (x)². Evidence: §3.3. 7. It was natural to expect the two definitions (43),(44) of Ě to agree. In fact (44) is identically S(Ê). Evidence: §3.1 (Narasimhan's example shows they differ). 8. It was natural to expect ord_ξ ℘(Ê,d) = d for all d (Thm 6.1). In fact it fails for d = 1 in Narasimhan's example (no smooth hypersurface contains the singular curve). Evidence: §3.5, verify.sage V2. 9. It was natural to expect Narasimhan's example to refute the manuscript's "Sing(Ě) ⊂ tail hypersurface" (Thm 10.3) directly. In fact it does not: the edge data jumps along the curve (f_z acquires order 1 away from 0), the Inv-max stratum is {0}, and the manuscript's refinement sidesteps Narasimhan; moreover no monomial-curve example of the needed kind exists (Lemma, §3.6). Evidence: §3.1, §3.6, search_constant_ridge2.py (0 solutions). 10. It was natural to expect Hauser's kangaroo example x²+y⁷+yz⁴ to give a singular terminal plat (its order-2 locus is the cusp x=0, z²=y³). In fact the ℘-edge data is not constant along the cusp (near η ≠ 0 both x and u = y³+z² lie in ℘(E,1), since every infinitely near point there lies on x = 0 and on u = 0), so again Sing(Ě) = {0}. The example instead shows that the natural secondary invariant (order of the ambient reduction) increases 2 → 3 at the kangaroo point. Evidence: verify.sage V3; math-audit §10–11. 11. It was natural to expect the manuscript's Inv to be computable from the tangent cone (ridge). In fact ℘(E,1) can contain linear forms not visible in the tangent cone (x ∈ ℘((x²+y⁵,2),1) since all infinitely near points lie on x = 0), so Inv_0 = (2,1,1), not (2,1,2). Evidence: §3.3. (This also means ridge-based tests such as verify.sage V4 are only proxies.) 12. It was natural to expect (user's framing) the technique to be either repairable or demonstrably irreparable. In fact the false steps are elementary and irreparable as stated, while what remains is the open problem itself, for which no impossibility proof exists: a false dichotomy. Evidence: §5. 13. It was natural to expect, from the acknowledgments, that the preprint had been checked ("multiple proof-reading and misprints-checking", p.4). In fact the reception record (§7) contains no confirmation by anyone, and two proofs in §14 end mid-sentence. Zero negative findings: no — thirteen are listed above. ## 7. Literature and reception context (From the literature subagent's report, same session; items marked OPENED were fetched and read by it, UNVERIFIED are search snippets or memory. I did not independently re-open these pages except Hauser arXiv:0811.4151, which I read in full via pdftotext.) - Hosting/versions: the preprint is at https://people.math.harvard.edu/~hironaka/pRes.pdf (OPENED); no arXiv posting, no later version, no withdrawal, no erratum found. Wikipedia (OPENED): "In 2017, he posted to his personal webpage a manuscript that claims to prove …". The subagent reports Hironaka died March 18, 2026 (NYT obituary via MacTutor mirror, Harvard notice, Nature obituary 28 Aug 2026 — all OPENED by it); none of the obituaries mentions the char-p claim. - MathOverflow 269651, "Hironaka's proof of resolution of singularities in positive characteristics", asked 2017-05-13, score 101, **0 answers** (OPENED via proxy). Comments: Libli (2017-05-13): "92 pages of 'hard' theory and not a single one example of a variety in char p to which the theory is applied"; Henry.L (2018-04-01): "most attempts to decipher his proof stopped at the globalization of LLUED … no serious flaw found … but no uniform conclusion"; Achinger (2019-06-20) on Abramovich–Temkin–Włodarczyk weighted blowups: relies on hypersurfaces of maximal contact, no light on char p. - Expert statements: K. Matsuki, Kyoto colloquium 2019-05-08, https://www.math.kyoto-u.ac.jp/en/node/3882 (OPENED): "Hironaka has announced a proof … To the best of my knowledge, however, nobody seems to have been able to confirm the details of the alleged proof so far." Cutkosky, SLMath talk 2024-03-20 (OPENED): credits positive-characteristic resolution only to Abhyankar (dim ≤ 3, char > 5) and Cossart–Piltant (dim 3). P. Woit blog 2017-05-10/14 (OPENED via proxy): "experts … are just starting to look at the proof"; no later verdict. Hauser 2010 (Bull. AMS 47, 1–30; OPENED via proxy) on the 2008 program: "no complete written proof … is available"; and on kangaroo points: Hironaka "asserts that this type of singularities can be resolved directly. There is no written proof of this available yet" (arXiv:0811.4151, §H, p.16; OPENED by me). - Hironaka's program papers: "Theory of infinitely near singular points", J. Korean Math. Soc. 40 (2003) 901–920 (Zbl 1055.14013, review descriptive; OPENED); "Three key theorems on infinitely near singularities", Sém. Congr. 10 (2005) 87–126 (Zbl 1093.14021; OPENED); ICTP 2006 notes not available online (UNVERIFIED). Neither review addresses the geometric/algebraic equivalence of ℘. - State of the art (all as reported, with the subagent's locators): Abhyankar 1956 (surfaces), 1966 (3-folds, char > 5); Cutkosky arXiv:math/0606530 (3-folds, char > 5, short proof; OPENED); Cossart–Piltant J. Algebra 320 (2008), 321 (2009) (3-folds, any char, [k:k^p] < ∞; UNVERIFIED) and arXiv:1412.0868 (arithmetic 3-folds; OPENED); Cossart–Jannsen–Saito arXiv:0905.2191 / LNM 2270 (excellent surfaces; OPENED/UNVERIFIED); Kawanoue–Matsuki idealistic filtration program, arXiv:math/0607009, 1205.4556, 1507.05195 (reduction to the monomial case in all dimensions, monomial case solved only in dim 3, invariant exhibits "Moh–Hauser jumping"; OPENED); Benito–Villamayor (elimination algebras; UNVERIFIED); Temkin arXiv:0804.1554 (inseparable local uniformization) and arXiv:1508.06255 (desingularization by p-alterations, Ann. Math. 2017; OPENED); de Jong 1996 alterations; Abramovich–Temkin–Włodarczyk arXiv:1906.07106 (weighted blowups, char 0 only; OPENED); Hauser–Perlega, Publ. RIMS 60 (2024) (surfaces in A³, any char; OPENED); Schober arXiv:1708.04784 and Perlega arXiv:2011.14443 (dim ≥ 4 / embedded dim 3 open; OPENED); newer unrefereed claims Yi Hu arXiv:2203.03842, 2507.21400 and C. Tian arXiv:2608.20066 (OPENED; no verification found). Moh's bound: Hauser arXiv:0811.4151 §B (quoted above). Narasimhan: Proc. AMS 87 (1983) 403–408 (abstract OPENED); the polynomial x²+yz³+zw³+y⁷w is from memory and verified computationally here (V2), not located in a source. - Terminology "domino fall", "AR-scheme", "terminal plat", "core-focusing": zero hits outside the PDF. No published analysis of any specific step was found. ## 8. Verification - verify.sage (SageMath 10.8), output verify-output.txt: V1 (x²+y⁵ example: Sing, Hasse derivative unit, blowup charts over GF(4), J♯|_Y), V2 (Narasimhan: f and ∂f vanish on C, critical scheme has dimension 1 and radical I(C), I(C) has no order-1 generator, semigroup minimal generators {32,7,19,15}, ord_η f_z = 1), V3 (Hauser's sequence f0→f3, residual orders, manuscript-style tail at a3, increase 2→3), V4 (375 near points, 0 ridge increases). - search_constant_ridge.py, search_constant_ridge2.py (python3 -I): 0 solutions in the searched ranges; the Lemma of §3.6 explains why. - All hand arguments of §3 are elementary; the two most important (§3.2 Proposition; vacuity of (44)) use only the manuscript's own definitions and Theorems 4.5/6.3. ## 9. Limitations - I did not re-derive the cited foundations ([22]–[24]: finite generation of ℘, Diff theorem, AR theorem, edge decomposition, equivalence of the two definitions of ℘ over perfect fields). The refutations do not depend on their failure; they take them as given. - The identification x ∈ ℘((x²+y⁵,2),1) relies on the geometric definition of ℘ (S over Z[t]); I verified the finite blowup tree over Z explicitly and argued the Z[t] case by the same computation (t is a spectator). If one insisted on the algebraic definition without integral closure, Inv_0 would be (2,1,2) instead — the refutations of §§3.3–3.4 are unchanged (both sides still equal). - "Inv_ξ(Y(i))" is not defined in the manuscript; I checked the counterexample under every reading I could think of (Inv of Ě(i) in Z; Inv of the AR-reduced F(Y) on Y; hiked or not). - The §10 statements and the smoothness of the terminal plat are left undetermined; §3.6 gives the partial results. - V4 tests the ridge, not the ℘-based Inv. - Reception facts are from a subagent's web search (locators given); the death date and obituaries are as it reported (marked OPENED by it); I did not re-open them. ## 10. Unresolved questions 1. Is the Inv-max stratum of a maximum base-hiked ideal exponent (with the manuscript's ℘-based edge data) always contained in a smooth hypersurface? (Thm 10.3/10.4 statements.) The monomial-curve case is excluded (§3.6); a counterexample would need a varying ridge direction tangent to the curve. 2. Does the Y-chain of §15 always stabilize with a smooth Sing(Y(t))? 3. Is the manuscript's Thm 7.1 (non-increase of ℘-edge data at near points) a theorem? It is classical for the ridge; the ℘-refinement presumably follows from S(E') ⊂ S(E)-type arguments, not checked. 4. Does a canonical finitely generated Ě exist when the Inv-max stratum is singular? 5. (The real one.) An invariant that strictly decreases through kangaroo points in all dimensions — the open problem; nothing in this manuscript bears on it.