# V4 report: second independent reading of lanes E (fold-arithmetic-bridge 3a/4a) and A2 (fixed-endpoint-discrepancy 2/4.1)

Reader: V4 (fresh session, no authorship of either candidate). Repo at 1285d47, tree clean, no repo file edited. Both candidates read at 30f9698 text as present in 1285d47. Own reconstruction was completed and written to this file BEFORE research-round-validation.md sections 11 and 12 were opened.

    Lane / stable question id: V4 (second reader of E) / Q-fold-arithmetic-bridge
    Starting commit / report commit or shared-checkout paths: 1285d47 / scratchpad/round-0908b/V4/V4-report.md, prop5-pieces.js, dk-check.js, wu-p6-06.png
    Disposition / exact claim / unproved hypotheses: Prop. 3 CONDITIONAL on the uniformity-in-f reading of Wu Lemma 2.3, otherwise verified within stated scope; Prop. 4 same condition, verified; Prop. 5 verified within stated scope (elementary, all five pieces recomputed). Unproved: (Cov_u), (Dec_1); Pan-Ding / Pan-Pan Cor. 8.12 UNREAD
    Changed step compared with the reviewed baseline: none proposed; two wording items only
    Source theorem and first unmatched hypothesis, if any: Wu 0705.1652v1 p. 6 Lemma 2.3 (page image read, PDF SHA-256 matches custody); no unmatched hypothesis; the constant's dependence on f is not displayed, and Wu's own p. 40 application (f = indicator of an N-dependent set M) uses the same reading
    Validation command, falsifier, result and compute used: node research/fold-arithmetic-bridge-validation.js (about 1 s); own prop5-pieces.js and dk-check.js (seconds); every displayed piece bound reproduced to 4 decimals; D_k recursion matched against the simplex integral and u*omega(u)
    Independent reviewer / disposition (PENDING until actually reviewed): this reading is the second; handler's section 11 agrees on all points checked here
    Full-consumer payoff and unpaid complement: none; two sufficient tests closed in union-bound form, hypotheses untouched
    Proposed shared-record changes / next bounded obligation: none required for E beyond the two wording items below

    Lane / stable question id: V4 (second reader of A2) / Q-fixed-endpoint-discrepancy
    Starting commit / report commit or shared-checkout paths: 1285d47 / same report
    Disposition / exact claim / unproved hypotheses: exact split (2.2)-(2.3), flip (2.1), Vaughan (2.5)-(2.7) and the cancellation of 2C_2M VERIFIED within stated scope; section 4.1 (T_I^low=O(x/log^A x)) CORRECTION REQUIRED: Step 1's modulus bound q=e[r,g]<=e_0UV is false for g>1, so (BV*) is applied outside its level in Steps 5-6; consequently S=C_2x+B+O_A(x/log^A x) is CONDITIONAL on a repaired (4.1), not derived
    Changed step compared with the reviewed baseline: the accepted line "the level q=e[r,g]<=x^(1/2-eps'/3)" in validation section 12 is withdrawn by this reading
    Source theorem and first unmatched hypothesis, if any: (BV*) as accepted; unmatched: its level Q_0 does not cover the moduli e[r,g] with g|e large (q reaches e_0^2 ~ x^(1-2eps'))
    Validation command, falsifier, result and compute used: node research/fixed-endpoint-discrepancy-validation.js (about 0.5 s, passes; it cannot see the defect, its multiplicity check is over q<=3000); own one-liner: at x=2^16 with the note's e_0=212, U=V=3, Q_0=1908, the triple e=g=209, r=1 gives q=43681
    Independent reviewer / disposition (PENDING until actually reviewed): this is the second reading; it disagrees with section 12 on one line
    Full-consumer payoff and unpaid complement: none; B unestimated as before; T_I^low returns to the unpaid side until repaired
    Proposed shared-record changes / next bounded obligation: verbatim text in section D below; obligation: lane A (owner) writes the g-truncation repair or records the failed step

## A. Dispositions, one line each

1. Proposition 3 (BV for 1_{set_k} at level sqrt x/(log x)^B): verified within stated scope, CONDITIONAL on the uniformity reading of Wu Lemma 2.3 (constant depending on f only through its bound). No defect in (3a.3)-(3a.7).
2. Proposition 4 (contamination constant 4 uniform in the tuple): verified within stated scope, same condition; one wording item (edge case D_k(u)=0).
3. Proposition 5 (Q_cov<1, c*_real<4 for every u>4): verified within stated scope (elementary; all five pieces recomputed independently; definitions of section 2 unchanged in 4a).
4. Candidate 2 exact split D^(e_1)=2C_2M+P_low+P_band+O_A(x log^(1-A)x), the flip, Vaughan with vanishing linear terms, and the sign bookkeeping to S=C_2x+T_I^low+B+O: verified within stated scope.
5. Candidate 2 section 4.1 (T_I^low=O_(A,eps')(x/log^A x)): correction required. Defective line: 4.1 Step 1, "q:=e[r,g] odd, ... q<=e_0UV<=x^(1/2-eps'/3)=:Q_0". The consequence S=C_2x+B+O_A(x/log^A x) is conditional on a repaired (4.1).

Calibration: item 5 is a proof defect in a derived (not proven) step; it does not refute the statement (4.1), which I consider likely repairable by the device already used in centered-discrepancy-estimate 3a.3 (see C.3). Nothing here moves any sufficient margin; twin-prime infinitude and D^(e_1)>=-4x/25+o(x) remain OPEN.

## B. Candidate 1: what was checked

Source custody. scratchpad/round-0908b/E/wu-0705.1652v1.pdf, SHA-256 41d432dd63da6d1fe7836ba3beda8b601ce6e64420e50501d26d79f7d971043e, equal to the note's record. Page 6 rendered to PNG (pdftoppm, 160 dpi) and read as an image; page 5 text layer read for (2.1)-(2.3) and the level/order definitions. The note's transcription of Lemma 2.2 (2.4)-(2.6) and Lemma 2.3 (three estimates) matches the page. Wu defines "order k" as |lambda(q)|<=tau_k(q), so order 1 gives |lambda^+|<=1 as the note uses.

(a) Hypotheses of Lemma 2.3 against the three applications. f(m)<<1: each f is an indicator, values in {0,1}. alpha in (0,1]: alpha=1/u in (0,1/2). r_1(y) positive, depending on x, r_1<<x^alpha: r_1=Y=X^(1/u)<=(4X)^(1/u) exactly. m<=x^(1-alpha): every pair (m,p) with p>Y, mp<=y<=2X has m<2X^(1-1/u)<=(4X)^(1-1/u) iff 4^(1-1/u)>=2 iff u>=2; the note states this. Reduced a and the weight mu^2 3^nu: carried verbatim into (3a.2). max over y<=x: the lemma is applied at a fixed f (fixed y_0 in {X,2X} inside the cutoff 1_{m<=y_0/Y}) and then evaluated at y=y_0, which is legitimate for each fixed f. The li(y/m) terms for m>y/Y cancel identically between the first and third applications, so the lemma's behaviour at small y/m never enters; for y in {X,2X} and m<=x^(1-alpha), y/m>=X^(1/u)/4. No hypothesis is left undischarged except the dependence of the implied constant on f.

(b) The uniformity reading. My view, with evidence: the reading "constant depends on A, alpha and the bound of f only" is the standard meaning of the statement and is the one the source itself uses. Wu p. 40 (text layer, around (10.7)) applies Lemma 2.3 with "f(m) the characteristic function of M", where M is a set of cofactors defined in terms of N (m<=N^(3/4) for m in M); p. 24 applies it to remainders R_1^(j) built from N-dependent ranges. A theorem whose constant depended on f beyond its bound would not give Wu's (10.7). This is evidence from the source's usage, not a reading of Pan-Ding 1979 or Pan-Pan Corollary 8.12, which remain UNREAD; the conditional label should stay in the record, but I assess the risk that it fails as low. Note the lemma's structure: the m-sum is a bilinear form with arbitrary bounded coefficients, which is the content of Pan-Ding's theorem as a generalisation of Bombieri-Vinogradov.

(c) Identity (3a.3) and the split. For n in set_k the pairs (m,p) with mp=n, p>Y, m in set_{k-1} are in bijection with the distinct prime factors of n (n/p has Omega=k-1 and all factors >Y; set_0={1} covers k=1); conversely such a pair always has mp in set_k. So the count is omega(n)1_{set_k}(n), equal to k exactly on squarefree n. Summing over n<=y, n=a (q) with (a,q)=1 forces (m,q)=1. The subtracted small-prime count: for m<=y/Y every p<=Y satisfies mp<=y, giving pi(mY;q,a,m); for m>y/Y every p with mp<=y has p<Y, giving pi(y;q,a,m). Both checked. Boundary p=Y is not a prime for u>4 and X=2^j.

(d) Non-squarefree corrections. (3a.6): the main-term side sums (k-omega(n)) over non-squarefree n in set_k, at most k B^(q)(y)<=k y/Y; J_q<=nu(q)y/Y; the PNT step needs |pi(t)-li(t)|<<t/(log x)^(A+k+4) for t>=Y=X^(1/u), which holds with a u-dependent constant since log t>=(1/u)log X; sum_{m<=y/Y}1/m<=log x+1; the weights mu^2 3^nu/phi(q) sum to O(log^3 x). (3a.7): for p<=(y/q)^(1/2) and p not dividing q (forced by (a,q)=1), one class mod p^2 q, count <=2y/(p^2 q), summed with sum_{p>Y}p^-2<<1/Y and sum_q 3^nu(q)/q<<log^3 x. For p>(y/q)^(1/2), n=p^2 m with m<q, and for each m the congruence p^2 m=a (q) has at most 2^nu(q) roots mod squarefree q, so at most 2^nu(q)((y/m)^(1/2)/q+1) primes; sum_{m<q}m^(-1/2)<=2q^(1/2) gives 2^nu(q)(2(y/q)^(1/2)+q); with sum_{q<=Q}6^nu(q)<<Q log^5 x the two terms are y^(1/2)Q^(1/2)log^5 x and Q^2 log^5 x, the latter <=x/log^A x iff B>=(A+5)/2. All re-derived; no defect.

(e) Proposition 4. r(A,d) decomposition re-derived: X_A/phi(d)=(1/phi(d))[N_k^(d)(2X)-N_k^(d)(X)+#{(n,d)>1}], so the three-term r(A,d) is exact. Every n in set_k is odd for X>=2^u, d|P(z) is odd squarefree, so n=2 (d) is reduced. w(p)=p/(p-1)<p. (2.3) with absolute K: V(z_1)/V(z_2)=prod_{z_1<=p<z_2}(p-1)/(p-2), Mertens. V(z)=prod_{2<p<z}(1-1/(p-1)^2)(1-1/p) since (1-1/(p-1)^2)(1-1/p)=(p-2)/(p-1); removing p=2 from Mertens gives the factor 2; V(z)=2C_2e^-gamma(1+o(1))/log z. z=Q^(1/2) satisfies 2<=z<=Q^(1/2); log Q/log z=2, F(2)=2e^gamma/2=e^gamma; log z=(1/4)log x-(B/2)loglog x. The (n,d)>1 term: sum_{d<=Q}mu^2(d)/phi(d) sum_{p|d,p>Y}2X/p<<X log x/Y, re-derived by writing d=pd'. Remainder: |lambda_l^+|<=1 (order 1), support q<=Q, q|P(z) squarefree, L fixed by eps, so it is at most L sum_{q<=Q}mu^2(q)|r(A,q)|, and Proposition 3 at y=X and y=2X (weight 3^nu>=1 dropped) bounds it by X/(log X)^A. Primes n-2>z are counted by S(A;P,z). Conclusion (4+O(eps)+o(1))2C_2X_A/log X reproduced.
Wording item 1: (3a.8) as displayed, "<=(4+o(1))2C_2 #(set_k cap (X,2X])/log X", absorbs the O(X/log^A X) remainder only when D_k(u)>0, i.e. k<u; for k>=u the set is empty or o(X/log X) and the second display "(4+o(1))2C_2D_k(u)M" should read (4D_k(u)+o(1))2C_2M. Harmless for the tests (only k<u contribute).

(f) Proposition 5. Section 2's definitions re-derived from the input bounds: under (Cov_u), T>=P_odd P'_odd/S-N_odd3 with the union bound N_odd3<=2F_1(u/2)(rho-1)2C_2e^-gamma uM gives exactly Q_cov=f_1^2 rho^2/(2e^-gamma u F_1F_2(rho-1)); under (Dec_1) the threshold is c*_real=f_1^2 rho/(F_2(rho-1)) against c_eff. Section 4a uses the same two expressions (e^gamma/(2u) is 1/(2e^-gamma u)); no silent change. Inputs (i)-(vi) each re-derived: f_1 increasing on [2,4] iff 4/3>log 3 (0.2347 margin); D_k(u)<=log^(k-1)(u-1)/(k-1)! by the induction with 1/v<=1/(v-1) and lower limit extended to 2; rho_odd<=cosh(log(u-1))<u/2 for u>2; rho^2/(rho-1)=rho+1+1/(rho-1); D_3(u)>=((u-2)log(u-2)-(u-3))/(u-1) from 1/v>=1/(u-1). D_k itself: the note's smallest-prime recursion D_k(u)=int_{k-1}^{u-1}D_{k-1}(v)dv/v was cross-checked (dk-check.js) at k=3 against the direct simplex integral and against the any-prime recursion kD_k=int D_{k-1}(v)(1/v+1/(u-v))dv to 5 decimals at six depths, and sum_k D_k(u)=u*omega(u) to 4 decimals at five depths. All five pieces recomputed independently (prop5-pieces.js, closed forms plus lower Riemann sums with cell 0.01): Q_cov<=0.2089, 0.4063, 0.7862, 0.7765, 0.6480 and c*_real<=0.7639, 1.4588, 2.6119, 2.3601, 1.8213, matching the note and validator to the printed digits; the true D_3(u_0) values (0.1472, 0.2455, 0.3515, 0.6837, 1.2191) lie above each lower bound used. Tail (8,infinity): with f_1^2 replaced by 1, the bracket decreases in u because D_3 increases and 1/(2u) decreases, so the value at 8 bounds the tail.
Wording item 2: "On (8,infinity) both right sides decrease in u" is true only after f_1(u/2)^2 (increasing) is replaced by 1, which the proof does; the sentence should say so.

Scope of the closure: as the note and section 11 state, the two union-bound sufficient tests are closed for every u>4; (Cov_u), (Dec_1) and any joint N_odd3 bound retaining the partner's parity are untouched. I concur with that scope and with the OUTCOMES closed-route row as worded, subject to keeping "conditional on the uniformity reading".

## C. Candidate 2: what was checked and the defect

C.1 Exact split and sign bookkeeping (verified). D^(e_1) is D_y of moving-cutoff (9) with the odd e-sum truncated at e_1 and a_e=x/2 (the Stieltjes sum over (a_e,x] against atoms at n gives exactly the bracket (1_{e|n}-1/phi(e))f(n)log(e/n)); same sign convention. D_R=P_R-Q_R with Q_R=sum mu(e)(M log e-F_1)/phi(e) re-derived. Q_low=M S_1(e_0)-F_1 S_0(e_0) with S_0(1)=S_1(1)=0; from (11), S_1(e_0)=-2C_2+O_A(log^-A x). I re-derived the sign of (11): the odd-e series sum mu(e)e^-s/phi(e)=H_0(s)/zeta(1+s) with H_0(0)=2 prod_{p>2}p(p-2)/(p-1)^2=2C_2, and sum mu(e)log e/phi(e)=-(d/ds)[H_0/zeta(1+s)]_{s=0}=-H_0(0)=-2C_2. Hence Q_low=-2C_2M+O_A(x log^(1-A)x) (the F_1 term costs one log), Q_band=O_A(x log^(1-A)x), and D^(e_1)=2C_2M+P_low+P_band+O_A(x log^(1-A)x). With (12) S=C_2x-2C_2M+D_y+O and (3a.1) D_y=D^(e_1)+O, the 2C_2M cancels exactly and S=C_2x+P_low+P_band+O_A(x log^(1-A)x). Flip (2.1): mu(e)mu(em)=mu(m)1_((e,m)=1) for squarefree e, log(e/n)=-log m, n in J iff m in I_e. Vaughan (V) re-derived from mu_{>U}*mu_{>V}*1=(mu-mu_{<=U})*(mu-mu_{<=V})*1 and mu*1=delta; the linear terms vanish since m>x/(2e_0)>=x^(1/2+eps')/2>U=V=floor(x^(eps'/3)); T_I^low, T_II^low as displayed, with gamma_V(b)=0 unless b>V. e_1<=x/(2y) for eps<1/50. The band arrangements (2.4) and the consumer restatements in 2.5 check. All of section 2: verified within stated scope, up to the accepted inputs (11), (12), (3a.1).

C.2 Section 4.1: correction required. Step 1 expands 1_((m,e)=1)=sum_{g|(m,e)}mu(g) over ALL divisors g of e, sets q:=e[r,g], and asserts q<=e_0UV<=x^(1/2-eps'/3)=:Q_0. Since [r,g]>=g and g runs up to e, q reaches e^2 (g=e, r=1), i.e. e_0^2 ~ x^(1-2eps'), beyond the level of (BV*) and beyond any BV level. Counterexample at the validator's own scale: x=2^16, e_0=212, U=V=3, Q_0=1908; e=g=209 (=11*19, odd squarefree, <e_0), r=1 gives q=209*209=43681>Q_0. A one-line count at x=2^16 with U=V=1 found 160 of 261 triples (e,g,r) with q>e_0UV. Consequences inside the proof: E_BV in Step 5 sums only q<=Q_0 and so omits the large-g terms entirely (they are neither bounded nor mentioned); Step 6's count of triples is unaffected; (4.3) itself holds for every odd q, so the omission is only in the BV averaging. The sentence in 4.1 "No expansion of mu^2(e) is needed in this piece, so there are no b,g tails: ... the finite coprimality expansion [is] exact" is where the omission was rationalised: the expansion is exact but its moduli are not bounded by Q_0. (4.1) is therefore not proved as written, and (2.8)'s reduction to "S=C_2x+B+O_A(x/log^A x)" is conditional on a repaired (4.1).
What still holds in 4.1: the power-of-two atom (one value n_0=x/2+2, bound O(x^(1/4))); (4.3) for each odd q; the density (4.4), re-derived prime by prime (at p|(e,r) the p-part of e[r,g] is p^(a+1) with p^a||r, the same for both choices of g, so the pair cancels; the note's "p^2" is the a=1 case, wording only); the identification of the d-sum with (3a.9) at (b,g)=(k,e) including h(p)=(p-1)/p at p|(d,k) from phi(dk)=phi(d)phi(k)(d,k)/phi((d,k)), with k odd and e odd squarefree so (3a.9)'s hypotheses hold and its constant is uniform in k; log U=(eps'/3)log x-O(1) so log^-A U is a constant times log^-A x; the multiplicity c'(q)<=tau(q)^4 (for e|q and g|e, r ranges over divisors of q/e, so at most sum_{e|q}tau(e)tau(q/e)tau(q)<=tau(q)^4; I found no q exceeding it, and the bound is attained at q=1); the Cauchy device with sum tau^8/phi<<log^256 x; the parameter choices A_3=A+5, A_1=2A+260, A+6 in (4.6). The validator's finite checks all pass and are consistent with the defect: they test (4.4) and c'(q) on q<=3000, not the range of q.

C.3 Apparent repair route (NOT executed, NOT verified; for the owning lane). Truncate g<=G:=(log x)^L as centered-discrepancy-estimate 3a.3 does for its g. Body moduli are then q<=e_0UVG=x^(1/2-eps'/3)(log x)^L, inside (BV*) as derived (its rounding argument allows Q_0=x^(1/2-eps)(log x)^(3L)). Both tails need [r,g]=rg/(r,g) rather than [r,g]>=g: progression side, sum_{r<=UV}tau(r)(r,g)/r<<tau(g)^2 log^2 x (write d=(r,g)), then sum_{g|e,g>G}tau(g)^2/g<=tau(e)^3/G and sum_{e<e_0}tau(e)^3/e<<log^8 x, giving O(x log^12 x/G) plus O(x^(1/2-eps'/3+o(1))) from the +1 terms; main-term side, the same with phi in place of the variables, O(x log^12 x (loglog x)^2/G). With L>=A+13 both are O(x log^-A x). The truncated density then equals 1_((r,e)=1)/(e phi(r)) minus that tail, so (4.5)-(4.6) survive. A trivial tail using only [r,g]>=g does NOT close (the r-sum is a power of x). This sketch is calibrated as plausible and unchecked; the owner must write and validate it.

C.4 Effect on records. The OUTCOMES entry "Fixed-endpoint discrepancy — split at x^(1/2-eps'), Type I paid, remainder exhibited", handoff section 3's sentence "with the below-level Type I piece paid ... S=C_2x+B+o(x)", the board row A ("pays the below-level Type I piece") and validation section 12's "verified within stated scope ... the level q=e[r,g]<=x^(1/2-eps'/3)" all rest on the defective line. None of them affects the sufficient consumer's OPEN status, which is unchanged either way.

## D. Proposed record updates (verbatim, for the handler)

1. research-round-validation.md, new section after 12, "12a. Second reading of the fixed-endpoint Type I estimate (reader V4, 2026-09-08)": "Section 4.1 Step 1 asserts q=e[r,g]<=e_0UV. The expansion 1_((m,e)=1)=sum_{g|(m,e)}mu(g) runs over every divisor g of e, and [r,g]>=g, so q reaches e^2 (g=e, r=1), up to e_0^2 ~ x^(1-2eps'); at x=2^16 with e_0=212, U=V=3, the triple e=g=209, r=1 gives q=43681>Q_0=1908. Steps 5-6 therefore apply (BV*) to moduli outside its level, and the large-g terms are not bounded. (4.1) is HELD: correction required, not refuted. The exact split (2.2)-(2.5), the density (4.4), the multiplicity bound, the identification with (3a.9) and the sign bookkeeping to S=C_2x+T_I^low+B+O_A(x log^(1-A)x) stand. The line 'the level q=e[r,g]<=x^(1/2-eps'/3)' in section 12 is withdrawn. A repair by truncating g<=(log x)^L with both tails paid through [r,g]=rg/(r,g) appears available and is the owner's next obligation."
2. OUTCOMES.md, entry "Fixed-endpoint discrepancy": change the title to "split at x^(1/2-eps'), Type I estimate HELD, remainder exhibited"; in the grade replace "Type I paid" by "T_I^low=O_(A,eps')(x/log^A x) DERIVED with a modulus-range defect found on second reading (validation 12a): HELD"; add to Failed step: "4.1 Step 1 bounds q=e[r,g] by e_0UV, false for g>1; BV applied outside level"; add to Reuse/revisit: "reuse (2.2)-(2.5), (4.4), c'(q)<=tau(q)^4 and the (3a.9) identification; revisit (4.1) only with the g-truncation and paid tails".
3. RESEARCH-HANDOFF.md section 3, centered alternative: replace "with the below-level Type I piece paid" by "with the below-level Type I piece estimated in a derivation that is HELD for a modulus-range defect (validation 12a)"; replace "S=C_2x+B+o(x)" by "S=C_2x+T_I^low+B+o(x), with T_I^low=o(x) conditional on that repair".
4. RESEARCH-EXECUTION.md board row A: replace "pays the below-level Type I piece (handler reading, section 12)" by "estimates the below-level Type I piece; HELD after second reading (section 12a: modulus range); repair is lane A's next obligation".
5. fold-arithmetic-bridge.md (lane E owner): wording items 1 and 2 of section B; no change of status.
6. CHANGELOG: "2026-09-08 V4: second reading of E (concur, conditional on Lemma 2.3 uniformity) and A2 (exact split verified; 4.1 held for a modulus-range defect)."

## E. What was not checked

- Pan-Ding 1979 and Pan-Pan Corollary 8.12 remain UNREAD; the uniformity reading is supported by Wu's own usage, not by the original statement.
- (BV*), (3a.9), (11), (12) and (3a.1) were used as accepted inputs; I re-derived the sign of (11) and the structure of (3a.9)'s hypotheses but did not re-review their proofs.
- The dimension-2 upper sieve constant F_2 and the exact Rosser-Iwaniec functions beyond the closed forms on [2,4] were not re-derived; only F_2>=1, f_1<=1<=F_1 are used and those are standard.
- The repair route in C.3 was not executed; it is a sketch with its own unverified inequalities.
- No repo file was edited; no QC index run; no enumeration.

## F. Files and compute

Written (all under /private/tmp/claude-501/-Users-benjaminsen-Files-Git-primeoire/9058ffd0-271d-4adc-bdd5-dabb0916b242/scratchpad/round-0908b/V4/): V4-report.md (this file), prop5-pieces.js, dk-check.js (left by the earlier V4 attempt, run here), wu-p5.txt, wu-p6.txt, wu-all.txt, wu-p6-06.png. Compute: three repo validators (about 2 s total) and two scratch scripts (under 5 s). No compute allocation consumed.