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AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as its Third Constraint Block
Kenan Xu, Xiangfeng Wang
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 8/23/2026, 3:07:13 AM
Summary
This paper addresses the open problem of whether direct three-block Alternating Direction Method of Multipliers (ADMM) converges when the third constraint block is the identity matrix. The authors provide a negative answer by constructing an explicit rational counterexample using AI-assisted methods (Codex with GPT-5.6 Sol and Kimi Code with Kimi K3). They demonstrate that direct three-block ADMM can fail to converge, producing a bounded nonconvergent orbit of period 66, even when the first two blocks are strongly convex quadratics. The study also investigates multiplier relaxation, showing that a problem-dependent small dual step can restore convergence, but no uniform positive relative step works for the entire class.
Entities (13)
Relation Signals (7)
Three-Block ADMM → failstoconvergefor → Identity Matrix Constraint
confidence 95% · direct three-block ADMM may fail even when the first two blocks are strongly convex quadratics... the case in which the third constraint block is the identity remains unresolved
GPT-5.6 Sol → pairedwith → Codex
confidence 90% · Using Codex with GPT-5.6 Sol
Kimi K3 → pairedwith → Kimi Code
confidence 90% · test the recent Kimi Code with Kimi K3 model
Codex → usedtoconstruct → Period-66 Counterexample
confidence 90% · Using Codex with GPT-5.6 Sol, we construct an explicit rational counterexample candidate
Kimi Code → usedtoconstruct → Period-23 Certificate
confidence 90% · test the recent Kimi Code with Kimi K3 model... produces an exact locally attracting period-23 certificate
Bingsheng He → acknowledgedby → Kenan Xu
confidence 85% · We would like to express our sincere gratitude to Professor Bingsheng He
Multiplier Relaxation → restoresconvergencefor → Three-Block ADMM
confidence 85% · a problem-dependent small dual step can restore convergence
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Abstract
Abstract:The alternating direction method of multipliers (ADMM), as a landmark algorithm, has attracted tremendous research attention and extensive practical applications over the past two decades. It is well known that, although the two-block ADMM enjoys well-established theoretical convergence guarantees, its direct extension to the three-block case may fail to converge, as demonstrated by existing counterexamples [5]. However, to the best of our knowledge, the case in which the third constraint block is the identity remains unresolved: the existing literature gives neither a general convergence proof nor a counterexample for this subclass. In this paper, we give a negative answer: direct three-block ADMM may fail even when the first two blocks are strongly convex quadratics. Using Codex with GPT-5.6 Sol, we construct an explicit rational counterexample candidate and verify it along a piecewise-affine reduction path; exact checks show that direct three-block ADMM on this instance produces a bounded nonconvergent orbit of period 66. Within the same Codex workflow, we further guide a study of multiplier relaxation and clarify when convergence can be restored at the fixed-instance and class levels: a problem-dependent small dual step can restore convergence, whereas no positive relative step works uniformly over the whole class. Furthermore, we also test the recent Kimi Code with Kimi K3 model without the Codex candidate or project-specific route guidance; along a different path it produces an exact locally attracting period-23 certificate, convertible to an equivalent all-identity instance. The comparison suggests that different research-harness configurations can shape the mathematical objects explored and the certificates pursued.
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- Source: https://arxiv.org/abs/2608.14396v1
- Canonical: https://arxiv.org/abs/2608.14396v1
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AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as Its Third Constraint Block Thanks: Codes, certificates, prompts, and workspace manifests are available at https://github.com/ConanXu-math/identity-slack-admm-cycle-certificate.Thanks: We would like to express our sincere gratitude to Professor Bingsheng He for encouraging me to explore the problem addressed in this paper using AI, and for the in-depth discussions we had throughout the process, from which we have benefited immensely. Kenan Xu Thanks: School of Mathematical Sciences, East China Normal University, Shanghai 200241, P.R. China. Xiangfeng Wang Thanks: Key Laboratory of Mathematics and Engineering Applications (MoE) and School of Mathematical Sciences, East China Normal University, Shanghai 200241, P.R. China. Abstract The alternating direction method of multipliers (ADMM), as a landmark algorithm, has attracted tremendous research attention and extensive practical applications over the past two decades [3]. It is well known that, although the two-block ADMM enjoys well-established theoretical convergence guarantees, its direct extension to the three-block case may fail to converge, as demonstrated by existing counterexamples [5]. However, to the best of our knowledge, the case in which the third constraint block is the identity remains unresolved: the existing literature gives neither a general convergence proof nor a counterexample for this subclass. In this paper, we give a negative answer: direct three-block ADMM may fail even when the first two blocks are strongly convex quadratics. Using Codex with GPT-5.6 Sol, we construct an explicit rational counterexample candidate and verify it along a piecewise-affine reduction path; exact checks show that direct three-block ADMM on this instance produces a bounded nonconvergent orbit of period 6666. Within the same Codex workflow, we further guide a study of multiplier relaxation and clarify when convergence can be restored at the fixed-instance and class levels: a problem-dependent small dual step can restore convergence, whereas no positive relative step works uniformly over the whole class. Furthermore, we also test the recent Kimi Code with Kimi K3 model without the Codex candidate or project-specific route guidance; along a different path it produces an exact locally attracting period-2323 certificate, convertible to an equivalent all-identity instance. The comparison suggests that different research-harness configurations can shape the mathematical objects explored and the certificates pursued. 1 The Open Problem 1.1 The Identity-Slack Question Consider the linearly constrained three-block convex program minx1,x2,x3 _x_1,x_2,x_3 θ1(x1)+θ2(x2)+θ3(x3) _1(x_1)+ _2(x_2)+ _3(x_3) (1.1) s.t. .t. A1x1+A2x2+A3x3=b, A_1x_1+A_2x_2+A_3x_3=b, where, for i=1,2,3i=1,2,3, xi∈ℝnix_i ^n_i, Ai∈ℝm×niA_i ^m× n_i, b∈ℝmb ^m, and θi:ℝni→(−∞,+∞] _i:R^n_i→(-∞,+∞] is proper, closed, and convex. Block constraints may be absorbed into θi _i through indicator functions, and the solution set is assumed nonempty. Although two-block ADMM is convergent under standard assumptions [3, 6], direct three-block ADMM can fail for general coefficient matrices. Both the zero-objective feasibility example and the strongly convex example in [5] lack an identity third block. We study the distinct subclass n3=mn_3=m, A3=ImA_3=I_m. Typically, this can be considered as the closest case compared with classical two-block problem. This structure usually arises from the inequality-constrained problem minx,y _x,y F(x)+G(y) F(x)+G(y) (1.2) s.t. .t. Ax+By≤b, Ax+By≤ b, where F:ℝnx→(−∞,+∞]F:R^n_x→(-∞,+∞] and G:ℝny→(−∞,+∞]G:R^n_y→(-∞,+∞] are proper, closed, and convex, A∈ℝm×nxA ^m× n_x, B∈ℝm×nyB ^m× n_y, and b∈ℝmb ^m. All vector inequalities are understood componentwise, and ℝ+m:=u∈ℝm:u≥0R_+^m:=\u ^m:u≥ 0\. For a nonempty closed convex set C, let ιC _C denote its indicator: ιC(u)=0 _C(u)=0 for u∈Cu∈ C and ιC(u)=+∞ _C(u)=+∞ otherwise. Introducing a nonnegative slack variable gives minx,y,z _x,y,z F(x)+G(y)+ιℝ+m(z) F(x)+G(y)+ _R_+^m(z) (1.3) s.t. .t. Ax+By+z=b. Ax+By+z=b. This is (1.1) with (x1,x2,x3)=(x,y,z)(x_1,x_2,x_3)=(x,y,z) and (A1,A2,A3)=(A,B,Im)(A_1,A_2,A_3)=(A,B,I_m); the last subproblem is projection onto ℝ+mR_+^m. We consider the direct slack-last sweep, rather than a grouped two-block reformulation [14]. For β>0β>0 and λ∈ℝmλ ^m, the augmented Lagrangian function is defined as ℒβ(x,y,z,λ)=F(x)+G(y)+ιℝ+m(z)−λ⊤r+β2∥r∥2,r=Ax+By+z−b∈ℝm.L_β(x,y,z,λ)=F(x)+G(y)+ _R_+^m(z)-λ r+ β2 r ^2, r=Ax+By+z-b ^m. (1.4) Assuming that the subproblems attain solutions, direct three-block ADMM in the order x→y→z→λx→ y→ z→λ is xk+1 x^k+1 ∈argminxℒβ(x,yk,zk,λk), ∈ *arg\,min_xL_β(x,y^k,z^k,λ^k), (1.5a) yk+1 y^k+1 ∈argminyℒβ(xk+1,y,zk,λk), ∈ *arg\,min_yL_β(x^k+1,y,z^k,λ^k), (1.5b) zk+1 z^k+1 ∈argminzℒβ(xk+1,yk+1,z,λk), ∈ *arg\,min_zL_β(x^k+1,y^k+1,z,λ^k), (1.5c) λk+1 λ^k+1 =λk−β(Axk+1+Byk+1+zk+1−b). =λ^k-β(Ax^k+1+By^k+1+z^k+1-b). (1.5d) we can employ the normal-cone convention NC(u)=v:⟨v,w−u⟩≤0for every w∈C,u∈C.N_C(u)=\v: v,w-u ≤ 0\ for every w∈ C\, u∈ C. With the sign convention in (1.4), a KKT point of (1.3) satisfies A⊤λ⋆∈∂F(x⋆),B⊤λ⋆∈∂G(y⋆),Ax⋆+By⋆+z⋆=b,z⋆∈ℝ+m,λ⋆∈Nℝ+m(z⋆). gatheredA λ ∈∂ F(x ), B λ ∈∂ G(y ),\\ Ax +By +z =b, z _+^m, λ ∈ N_R_+^m(z ). gathered (1.6) As mentioned above, it is well known that, although the two-block ADMM enjoys well-established theoretical convergence guarantees, its direct extension to the three-block case may fail to converge, as demonstrated by existing counterexamples [5]. However, available multi-block convergence results impose additional regularity or parameter assumptions [4, 17, 18, 19], or modify the iteration by correction, proximal, or back-substitution steps [9, 10, 11, 15]. None of these results establishes convergence from A3=ImA_3=I_m alone, leaving the identity-slack case unresolved: the existing literature gives neither a general convergence proof nor a counterexample for this subclass. Professor Bingsheng He from Nanjing University also mentioned this open problem in his talk [12]. 1.2 Main Results In this paper, we give a negative answer to the above open problem through an AI-assisted research process, i.e., direct three-block ADMM may fail even when the first two blocks are strongly convex quadratics. The main contributions are as follows: • Mathematically. An exact rational period-6666 identity-slack counterexample (Theorem 2.1); problem-dependent multiplier-step convergence and the exclusion of any class-uniform relative step (Section 3); and a locally attracting period-2323 certificate (Section 4). • AI-assisted research process. We document how human-guided reasoning-and-coding agents contributed to representation discovery, structured counterexample construction, exact certification, and theorem continuation. Floating-point divergence or cycling is treated only as a clue; theorem-level claims require exact certificates. Prompts, workspace records, human interventions, and verification artifacts are archived in Appendix C. 1.3 Paper Organization Section 2 presents the period-6666 counterexample constructed on the Codex (GPT 5.6 Sol) route, together with its analysis and exact certification; Section 3 then studies multiplier relaxation, including when convergence is restored and when it is not. Section 4 presents the period-2323 attracting counterexample from the Kimi Code (Kimi K3) route and compares the different discovery mechanisms of the two AI-assisted research routes. Section 5 summarizes the main findings and discusses implications for AI for optimization; the evidence workflow, prompts, interventions, and verification records are collected in Appendix C. 2 A Period-66 Counterexample This section presents a rational period-6666 counterexample and its piecewise-affine certificate. The construction proceeds through four mathematical steps: rejecting fixed-branch instability as insufficient without a realizable projection itinerary; deriving the signed piecewise-affine realization on s=(y,q)s=(y,q), q=z+λq=z+λ (Section 2.3); shifting the search from (Q1,Q2)(Q_1,Q_2) to resolvents (M,N)(M,N) and reachable active-set itineraries; and reconstructing the candidate over ℚQ with exact checks of closure, branch admissibility, the original ADMM iteration, and minimality. AI-assisted process records are collected in Appendix C. 2.1 Problem Data and Main Theorem Set A=B=I2A=B=I_2, β=1β=1, and consider minx,y,z∈ℝ2 _x,y,z ^2 12x⊤Q1x+12y⊤Q2y+ιℝ+2(z) 12x Q_1x+ 12y Q_2y+ _R_+^2(z) (2.1) s.t. .t. x+y+z=b¯. x+y+z= b. Define M=(Q1+I2)−1,N=(Q2+I2)−1.M=(Q_1+I_2)^-1, N=(Q_2+I_2)^-1. Then algorithm (1.5) becomes xk+1 x^k+1 =M(b¯−yk−zk+λk), =M( b-y^k-z^k+λ^k), (2.2a) yk+1 y^k+1 =N(b¯−xk+1−zk+λk), =N( b-x^k+1-z^k+λ^k), (2.2b) zk+1 z^k+1 =[b¯−xk+1−yk+1+λk]+, = [ b-x^k+1-y^k+1+λ^k ]_+, (2.2c) λk+1 λ^k+1 =λk−(xk+1+yk+1+zk+1−b¯). =λ^k-(x^k+1+y^k+1+z^k+1- b). (2.2d) Let ε=11000,μ=895710000,ν=9991000,d1=(−1,20)⊤,d2=(−1,10)⊤, = 11000, μ= 895710000, ν= 9991000, d_1=(-1,20) , d_2=(-1,10) , (2.3) and define M M =εI2+(μ−ε)d1d1⊤d1⊤d1, = I_2+(μ- ) d_1d_1 d_1 d_1, Q1 Q_1 =M−1−I2, =M^-1-I_2, (2.4) N N =εI2+(ν−ε)d2d2⊤d2⊤d2, = I_2+(ν- ) d_2d_2 d_2 d_2, Q2 Q_2 =N−1−I2. =N^-1-I_2. To define the right-hand side, prescribe a KKT point. Choose z⋆=(0,1)⊤,λ⋆=(−1,0)⊤,z =(0,1) , λ =(-1,0) , and set x⋆=Q1−1λ⋆,y⋆=Q2−1λ⋆,b¯=x⋆+y⋆+z⋆.x =Q_1^-1λ , y =Q_2^-1λ , b=x +y +z . (2.5) By construction, Q1x⋆=λ⋆,Q2y⋆=λ⋆,x⋆+y⋆+z⋆=b¯,λ⋆∈Nℝ+2(z⋆),Q_1x =λ , Q_2y =λ , x +y +z = b, λ ∈ N_R_+^2(z ), so (x⋆,y⋆,z⋆,λ⋆)(x ,y ,z ,λ ) satisfies the KKT conditions (1.6). Every entry of Q1,Q2,b¯Q_1,Q_2, b is rational. Moreover, σ(Q1)=999,10438957,σ(Q2)=999,1999,σ(Q_1)= \999, 10438957 \, σ(Q_2)= \999, 1999 \, (2.6) where σ(Q)σ(Q) denotes the spectrum. Thus Q1,Q2≻0Q_1,Q_2 0. Strong convexity in (x,y)(x,y) and the constraint determine a unique primal solution, while Q1x⋆=λ⋆Q_1x =λ fixes the multiplier. Hence the KKT point is unique. All three ADMM subproblems also have unique minimizers, so the iteration is single-valued. Theorem 2.1 (Exact rational period-6666 counterexample). For the rational problem (2.1)–(2.5), there is a rational initialization for which the sequence generated by the unmodified direct three-block ADMM (1.5) is bounded, non-KKT, and periodic with minimal period 6666. Its strict projection-sign sequence is =(00)2(01)64,W=(00)^2(01)^64, (2.7) where the i-th bit is 11 if the corresponding projection input is positive and 00 if it is negative. All 132132 projection inequalities are strict with a common margin greater than 10−310^-3. Therefore an identity third constraint block does not guarantee unconditional global convergence of direct three-block ADMM. 2.2 Orbit Geometry Figure 1 is a decimal visualization of the exact rational sequence in Theorem 2.1. The values qk=zk+λkq^k=z^k+λ^k form a closed loop that does not contain the unique KKT value q⋆=(−1,1)q =(-1,1). Figure 1: Decimal visualization of the exact rational period-6666 sequence. Red squares mark the two 0000 iterates; the remaining iterates have pattern 0101. The KKT coordinate q⋆=(−1,1)q =(-1,1) is not on the periodic sequence. Numerically, the dominant 0101-branch pair has modulus 1.000181.00018 and angle 0.0951955≈2π/660.0951955≈ 2π/66, while ρ(T00)≈0.97058ρ(T_00)≈ 0.97058. Thus 6464 near-rotational 0101 steps followed by two resetting 0000 steps explain the word (00)2(01)64(00)^2(01)^64. This is only intuition; the proof follows. 2.3 Exact Period Certificate We now reduce the ADMM iteration to verify the periodic sequence in Theorem 2.1. For quadratic programs, local ADMM behavior and parameter selection can often be reduced to linear or affine iterations after active-set identification [2, 7, 16]. Here we instead compose an entire finite sign itinerary in exact rational arithmetic. Define the orthant-projection input qk+1=b¯−xk+1−yk+1+λk.q^k+1= b-x^k+1-y^k+1+λ^k. (2.8) The z- and multiplier updates give zk+1=[qk+1]+,λk+1=[qk+1]−,qk+1=zk+1+λk+1.z^k+1=[q^k+1]_+, λ^k+1=[q^k+1]_-, q^k+1=z^k+1+λ^k+1. (2.9) For a∈ℝ2a ^2, write ([a]+)i=maxai,0([a]_+)_i= \a_i,0\, ([a]−)i=minai,0([a]_-)_i= \a_i,0\, and (|a|)i=|ai|(|a|)_i=|a_i| for i=1,2i=1,2. Thus (2.9) shows that the single vector qk+1q^k+1 uniquely determines both zk+1z^k+1 and λk+1λ^k+1. Since zk+1=Πℝ+2(qk+1)z^k+1= _R_+^2(q^k+1), the projection optimality condition gives λk+1=qk+1−zk+1∈Nℝ+2(zk+1).λ^k+1=q^k+1-z^k+1∈ N_R_+^2(z^k+1). We index the recurrence after the projection–multiplier update and set sk=((yk)⊤,(qk)⊤)⊤∈ℝ4.s^k=((y^k) ,(q^k) ) ^4. Since z−λ=|q|z-λ=|q|, the eliminated variables are recovered by z=[q]+,λ=[q]−,x+=M(b¯−y−z+λ)=M(b¯−y−|q|).z=[q]_+, λ=[q]_-, x^+=M( b-y-z+λ)=M( b-y-|q|). (2.10) Define p:=b¯−x+−z+λp:= b-x^+-z+λ and g:=(I2−M)b¯g:=(I_2-M) b. Substitution gives the reduced recurrence p p =My−(I2−M)|q|+g, =My-(I_2-M)|q|+g, (2.11a) y+ y^+ =Np, =Np, (2.11b) q+ q^+ =b¯−x+−y++λ=p+z−y+=(I2−N)p+[q]+. = b-x^+-y^++λ=p+z-y^+=(I_2-N)p+[q]_+. (2.11c) Together with (2.10), this recurrence is equivalent to the original ADMM after the first projection–multiplier update. For a strict projection pattern D=diag(q1>0,q2>0)D=diag(1_\q_1>0\,1_\q_2>0\), one has [q]+=Dq[q]_+=Dq and |q|=SDq|q|=S_Dq, where SD=2D−I2S_D=2D-I_2. Hence, on the corresponding polyhedral projection region, (2.11) is the affine update s+ s^+ =TDs+h, =T_Ds+h, (2.12) TD T_D =(NM−N(I2−M)SD(I2−N)MD−(I2−N)(I2−M)SD), = pmatrixNM&-N(I_2-M)S_D\\ (I_2-N)M&D-(I_2-N)(I_2-M)S_D pmatrix, h h =(Ng(I2−N)g). = Ng(I_2-N)g. Only two projection patterns are needed: D00=diag(0,0)D_00=diag(0,0) and D01=diag(0,1)D_01=diag(0,1). Write T00:=TD00T_00:=T_D_00 and T01:=TD01T_01:=T_D_01, and represent each affine update in homogeneous coordinates by Lω=(Tωh01)∈ℝ5×5,ω∈00,01.L_ω= pmatrixT_ω&h\\ 0&1 pmatrix ^5× 5, ω∈\00,01\. (2.13) Composing the affine updates in the order prescribed by (2.7) gives L0164L002=(a01).L_01^64L_00^2= pmatrixP&a\\ 0&1 pmatrix. (2.14) Here ∈ℚ4×4P ^4× 4 and a∈ℚ4a ^4. Exact rational elimination verifies det(I4−)≠0 (I_4-P)≠ 0. Hence s66=s0s^66=s^0 has the unique rational solution s0=(I4−)−1a∈ℚ4.s^0=(I_4-P)^-1a ^4. (2.15) Thus the cycle is an affine fixed point selected by the nonzero offset a, which comes from b¯ b, rather than a unit-eigenvalue orbit of P or an effect of linear objective terms. Prescribing a branch itinerary and then solving the fixed-point equation of its return map is related to constructive counterexample methodology [8]; the projection admissibility of every step must still be verified here. It remains to verify that the fixed point (2.15) follows the prescribed projection branches. Exact rational iteration of (2.11) gives q1k<0(0≤k<66),q2k<0(k=0,1),q2k>0(2≤k<66).q_1^k<0 (0≤ k<66), q_2^k<0 (k=0,1), q_2^k>0 (2≤ k<66). (2.16) More precisely, if δk=0 _k=0 for k=0,1k=0,1 and δk=1 _k=1 otherwise, then min0≤k<66min−q1k,(2δk−1)q2k>11000. _0≤ k<66 \-q_1^k,(2 _k-1)q_2^k\> 11000. (2.17) Hence every branch is admissible and the orthant projection realizes (00)2(01)64(00)^2(01)^64. Equation (2.10) recovers the full ADMM sequence. Define x0=M(b¯−y65−|q65|).x^0=M( b-y^65-|q^65|). Together with z0=[q0]+z^0=[q^0]_+ and λ0=[q0]−λ^0=[q^0]_-, this gives (x0,y0,z0,λ0)∈ℚ8(x^0,y^0,z^0,λ^0) ^8, and the step-6565 update returns the full state. If the ADMM sequence had a proper subperiod, its projection-sign sequence would be a proper repetition. The only possible repetition factor is two, because the sequence has length 6666 and contains exactly two occurrences of 0000. Its two length-3333 halves, however, contain different numbers of 0000 symbols. The first return therefore occurs at step 6666. The periodic sequence cannot contain the unique KKT point, which is a fixed point of this single-valued ADMM iteration. It is consequently non-KKT, nonconvergent, and bounded. This proves the theorem. The strict projection margin and nonsingularity of the return equation also make the cycle persist under sufficiently small perturbations of the problem data; see Corollary A.1. 3 Multiplier Relaxation With the period-6666 certificate established, a natural next question is whether a smaller multiplier step restores convergence on the same instance. We kept the rational QP, β, and initialization fixed, and had the Codex route explore only τ. The resulting branch maps, candidate regimes, and certificate proposals were then checked by exact replay; the authors separated the claims into local KKT stability, convergence from the specified initialization, and a problem-dependent class-level extension (Appendix C). Keep the primal penalty β>0β>0 and replace only the multiplier update by λk+1=λk−τ(Axk+1+Byk+1+zk+1−b),τ>0.λ^k+1=λ^k-τ(Ax^k+1+By^k+1+z^k+1-b), τ>0. (3.1) The standard method corresponds to τ=βτ=β. In the counterexample below β=1β=1, so the certified choice τ≈1/2τ≈ 1/2 is a substantial multiplier relaxation, not a small perturbation of the standard step. We first state a class-level result and its limitation, and then give sharper ranges for the period-6666 instance. 3.1 Small-Step Convergence and Its Limits The small-dual-step contraction principle is due to Hong and Luo [13, Theorem 3.1 and (3.16)–(3.17)]. Their argument uses primal and dual error bounds obtained under compact-polyhedral assumptions. The contribution needed here is to verify those bounds globally for the noncompact slack-last model under global smooth strong convexity and the full-row-rank condition on [AB][A\ B]. Thus the next proposition is an assumption-replacement specialization of the Hong–Luo framework, rather than a new general small-step principle. Proposition 3.1 (Hong–Luo-type slack-last specialization). In (1.3), suppose that F:ℝnx→ℝF:R^n_x and G:ℝny→ℝG:R^n_y are continuously differentiable, respectively μF _F- and μG _G-strongly convex, and have LFL_F- and LGL_G-Lipschitz gradients, where μF,μG>0 _F, _G>0 and LF,LG<∞L_F,L_G<∞. Assume also that [AB][A\ B] has full row rank. For every fixed β>0β>0, there exists a problem-dependent τ¯>0 τ>0 such that, for every 0<τ<τ¯0<τ< τ, the direct x→y→z→λx→ y→ z→λ iteration with (3.1) converges R-linearly from every finite initialization satisfying z0∈ℝ+mz^0 _+^m to the unique KKT point. Proof. Let u=(x,y,z)u=(x,y,z), let u¯(λ) u(λ) minimize the augmented Lagrangian at fixed λ, and set dβ(λ)=minuℒβ(u,λ)d_β(λ)= _uL_β(u;λ). Consider Vk=ℒβ(uk+1,λk)−dβ(λk)+dβ(λ⋆)−dβ(λk),V^k=L_β(u^k+1;λ^k)-d_β(λ^k)+d_β(λ )-d_β(λ^k), and write Sk=‖uk+1−uk‖2S_k=\|u^k+1-u^k\|^2 and Rk=‖∇dβ(λk)‖2R_k=\|∇ d_β(λ^k)\|^2. Combining the Hong–Luo primal–dual gap calculation with the global primal and dual error bounds verified here for the noncompact slack-last model gives finite positive constants γ,C,Cp,Cdγ,C,C_p,C_d, depending only on the problem data and β, such that Vk−Vk−1≤−(γ−τC)Sk−τRk,Vk≤CpSk+CdRk.V^k-V^k-1≤-(γ-τ C)S_k-τ R_k, V^k≤ C_pS_k+C_dR_k. (3.2) Hence 0<τ<τ¯:=γ/C0<τ< τ:=γ/C implies Vk≤ρVk−1,ρ=(1+min(γ−τC)/Cp,τ/Cd)−1<1.V^k≤ρ V^k-1, ρ= (1+ \(γ-τ C)/C_p,τ/C_d\ )^-1<1. The iterate error is bounded by a constant multiple of VkV^k, which gives R-linear convergence. The full derivation is given in Appendix A.1. ∎ The threshold in Proposition 3.1 cannot be chosen independently of the problem; the obstruction already occurs in the strongly convex quadratic subclass. With ϑ=τ/β =τ/β, the following result gives a quantitative obstruction. Theorem 3.2 (No problem-independent positive relative-step interval). For every 0<ϑ≤80119,0< ≤ 80119, there is an m=3m=3 slack-last QP with Q1,Q2≻0Q_1,Q_2 0 and [AB][A\ B] of full row rank, together with a finite feasible initialization, for which direct x→y→z→λx→ y→ z→λ ADMM with relative multiplier step ϑ generates a bounded nonconvergent sequence. Consequently, there is no ϑ¯>0 >0 such that, for every problem in the class, the method converges for every 0<ϑ<ϑ¯0< < . Proof. Fix ϑ . Appendices A.3 and A.4 construct a continuous one-parameter branch matrix U(α):=UD(r,α)U(α):=U_D(r,α), where D=diag(1,0,0)D=diag(1,0,0) and r=7ϑ/(80−7ϑ)r= 7 /(80-7 ). The branch is unstable at the degenerate endpoint α=0α=0, while the other endpoint α=2−1α= 2-1 is strictly stable. Continuity gives an interior αc _c with ρ(U(αc))=1ρ(U( _c))=1; at this parameter the associated problem is already strongly convex and [AB][A\ B] has full row rank. Uniqueness of the KKT point excludes the eigenvalue +1+1, so the remaining unit-circle mode generates a bounded nonconvergent error sequence. Finally, the strict projection margin keeps every sufficiently small perturbation on the same affine branch; the appendix also constructs a feasible initialization and verifies boundedness of the full ADMM orbit. ∎ When τ¯ τ is unavailable, the same global error bounds yield an exact look-ahead backtracking rule with one additional trial primal sweep. We make no model-independent novelty claim for this rule; it is recorded as a consequence of the slack-last bounds rather than a primary contribution. Corollary 3.3 (General adaptive dual step with backtracking). Under the assumptions of Proposition 3.1, write u=(x,y,z)u=(x,y,z), E=[A,B,Im]E=[A,B,I_m], and set κp _p :=1+maxLF,LG+β‖E‖2λmin(diag(μFI,μGI,0)+βE⊤E), := 1+ \L_F,L_G\+β\|E\|^2 _ \! (diag( _FI, _GI,0)+β E E ), PGλ(u) _λ(u) :=u−proxh(u−∇sλ(u)),BE(v):=∥E∥2κp2∥v∥2, :=u-prox_h\! (u-∇ s_λ(u) ), B_E(v):=\|E\|^2 _p^2\|v\|^2, where h(u)=ιℝ+m(z)h(u)= _R_+^m(z) and sλs_λ is the smooth part of ℒβ(⋅,λ)L_β(·;λ), while λG_λ denotes one exact x→y→zx→ y→ z Gauss–Seidel primal sweep at fixed multiplier. Fix χ,δ∈(0,1)χ,δ∈(0,1) and τmax>0 _ >0. Algorithm 1 accepts a trial only when its (v^,D^)( v, D) satisfy τBE(v^)≤(1−χ)D^.τ B_E( v)≤(1-χ) D. (3.3) Under exact block solves and exact evaluation of the acceptance test, backtracking terminates finitely, the accepted steps have a positive lower bound independent of k, and (uk+1,λk)k≥1\(u^k+1,λ^k)\_k≥ 1 converges globally R-linearly. Algorithm 1 Backtracking dual-step ADMM with an exact trial primal sweep 1: uinit=(xinit,yinit,zinit)u^init=(x^init,y^init,z^init) with zinit∈ℝ+mz^init _+^m, λ0∈ℝmλ^0 ^m; χ,δ∈(0,1)χ,δ∈(0,1), τmax>0 _ >0 2: u1←λ0(uinit)u^1 _λ^0(u^init) 3: for k=1,2,…k=1,2,… do 4: rk←Euk−br^k← Eu^k-b 5: for j=0,1,2,…j=0,1,2,… do 6: τ←τmaxδjτ← _ δ^j; λ^←λk−1−τrk λ←λ^k-1-τ r^k; u^←λ^(uk) u _ λ(u^k) 7: v^←PGλ^(uk) v _ λ(u^k); D^←ℒβ(uk,λ^)−ℒβ(u^,λ^) D _β(u^k; λ)-L_β( u; λ) 8: if τBE(v^)≤(1−χ)D^τ B_E( v)≤(1-χ) D then 9: (τk,λk,uk+1)←(τ,λ^,u^)( _k,λ^k,u^k+1)←(τ, λ, u); exit inner loop 10: else 11: retain (uk,λk−1)(u^k,λ^k-1) 12: end if 13: end for 14: end for If F(x)=12x⊤Q1x+c1⊤x,G(y)=12y⊤Q2y+c2⊤y,Q1,Q2≻0,F(x)= 12x Q_1x+c_1 x, G(y)= 12y Q_2y+c_2 y, Q_1,Q_2 0, set K=diag(Q1,Q2,0)+βE⊤E,BK(v):=β−1v⊤(K−1+2I+K)v.K=diag(Q_1,Q_2,0)+β E E, B_K(v):=β^-1v (K^-1+2I+K)v. Then BK(v)B_K(v) is also an upper bound for ‖E(u−u¯(λ))‖2\|E(u- u(λ))\|^2, so one may replace (3.3) by τBK(v^)≤(1−χ)D^,τ B_K( v)≤(1-χ) D, (3.4) with the same finite-backtracking and convergence conclusions. The proof of Corollary 3.3 and of this quadratic refinement is given in Appendix A.2. 3.2 Stability of the Period-66 Example For the rational QP (2.1), set β=1β=1 and use the six-dimensional essential state w=(y,z,λ)∈ℝ6w=(y,z,λ) ^6. On a strict projection branch with mask D, one complete update is affine: w+=TD(τ)w+aD(τ),TD(τ)=TD(0)+τTD(1).w^+=T_D(τ)w+a_D(τ), T_D(τ)=T_D^(0)+τ T_D^(1). (3.5) Here τ enters only the multiplier block row. The explicit block matrices are given in Appendix A.7. At the unique KKT point, q⋆=(−1,1)q =(-1,1), so the local branch is D01D_01. Figure 2 separates the three scopes below. Figure 2: Certified dual-step ranges for the fixed rational instance: pointwise local attraction on the strict D01D_01 branch, a uniform common-Lyapunov interval, and convergence of the initialization in Theorem 2.1. For H≻0H 0, write ‖e‖H:=(e⊤He)1/2.\|e\|_H:=(e He)^1/2. For a fixed τ, we say that the KKT state w⋆w is locally Q-linearly attracting if there exist a neighborhood τU_τ of w⋆w , a norm ∥⋅∥τ\|·\|_τ, and a constant κτ∈(0,1) _τ∈(0,1) such that every w0∈τw^0 _τ remains in τU_τ and satisfies ‖wk+1−w⋆‖τ≤κτ‖wk−w⋆‖τ,k≥0.\|w^k+1-w \|_τ≤ _τ\|w^k-w \|_τ, k≥ 0. Theorem 3.4 (Dual step length for the period-6666 instance). For the rational QP (2.1) and (3.1): 1. there exist a rational matrix H≻0H 0, a radius r>0r>0, and a constant κ∈(0,1)κ∈(0,1), all independent of τ∈[49/100,51/100]τ∈[49/100,51/100], such that ‖w0−w⋆‖H≤r⟹‖wk+1−w⋆‖H≤κ‖wk−w⋆‖H\|w^0-w \|_H≤ r \|w^k+1-w \|_H≤κ\|w^k-w \|_H for every k≥0k≥ 0. In particular, the KKT point is uniformly locally Q-linearly attracting throughout 49100≤τ≤51100; 49100≤τ≤ 51100; 2. the initialization of Theorem 2.1 converges to that KKT point whenever |τ−12|≤10−10; |τ- 12 |≤ 10^-10; 3. for each fixed τ∈(0,1)τ∈(0,1), the KKT state w⋆w is locally Q-linearly attracting if and only if 0<τ<τc,0<τ< _c, where 0.9366061114<τc<0.9366061115.0.9366061114< _c<0.9366061115. Equivalently, T01(τ)T_01(τ) is Schur stable exactly for 0<τ<τc0<τ< _c. For every fixed τ in this interval, the neighborhood, Lyapunov metric, and contraction factor may depend on τ. Proof. At τ=1/2τ=1/2, an exact rational Lyapunov equation produces a common metric. Exact endpoint tests and a quadratic chord identity extend contraction to [0.49,0.51][0.49,0.51]; strict complementarity supplies a projection-safe neighborhood. For the specified period-6666 initialization, an exact 232232-step interval enclosure for |τ−1/2|≤10−10|τ-1/2|≤ 10^-10 enters that neighborhood. Finally, exact Schur recursion and a Sturm count identify the unique boundary τc _c; inside the strict D01D_01 branch, the resulting Lyapunov metric and the Banach fixed-point theorem give local linear convergence. The exact derivation, polynomial, and verification commands are in Appendices A.8 and A.9. ∎ The quantifiers in the three assertions are intentionally different. The first is uniform in τ but local in the initial state; the second concerns only the initial state used in Theorem 2.1; and the third is a pointwise local branch criterion whose metric and neighborhood may depend on τ. Only the second assertion says anything about the former period-6666 initialization. None of the three assertions gives arbitrary-initial global convergence over the displayed fixed-instance ranges. Proposition 3.1 has a fourth, class-level quantifier: for every fixed problem satisfying its assumptions, there is a problem-dependent τ¯ τ that works from every finite initialization. 4 A Period-23 Attracting Cycle This section presents a separate Kimi Code K3 route that was not supplied with the period-6666 candidate or its certificate. 4.1 Exact Period-23 Certificate The route received the same mathematical question without the Codex matrices, projection word, initialization, certificate, or project-specific route guidance (Appendix C.1). It produced an m=3m=3 non-KKT sequence of minimal period 2323. More strongly, Proposition 4.1 certifies an open invariant ellipsoid of reduced initializations attracted phasewise to that sequence. All 6969 projection inputs have common strict margin greater than 1/2501/250, and an exact Lyapunov inequality certifies local attraction. The retained run record and starting condition are summarized in Appendix C. The route reduced each fixed projection branch to an affine map on v=(y,t)v=(y,t), t=z+λt=z+λ, screened projection words with KKT-compatible QP data, and promoted a candidate only after exact rational replay. An exact Lyapunov search then certified local attraction. Consider the rational m=3m=3 QP with β=1β=1 and A^=[5/361/6025/971/605/775/9125/975/9111/12],B^=[4/611/545/831/5413/9223/885/8323/8853/58],Q^x=[85/93−1/57−3/11−1/5799/100−5/86−3/11−5/867/78],Q^y=[99/100−1/51−3/47−1/5172/79−18/65−3/47−18/654/43],b^=[−2/17−1/1456/73],c^1=[33/98−33/25−24/55],c^2=[−31/9919/36−11/20]. gathered A= bmatrix5/36&1/60&25/97\\ 1/60&5/77&5/91\\ 25/97&5/91&11/12 bmatrix, B= bmatrix4/61&1/54&5/83\\ 1/54&13/92&23/88\\ 5/83&23/88&53/58 bmatrix,\\[4.0pt] Q_x= bmatrix85/93&-1/57&-3/11\\ -1/57&99/100&-5/86\\ -3/11&-5/86&7/78 bmatrix, Q_y= bmatrix99/100&-1/51&-3/47\\ -1/51&72/79&-18/65\\ -3/47&-18/65&4/43 bmatrix,\\[4.0pt] b= bmatrix-2/17\\ -1/14\\ 56/73 bmatrix, c_1= bmatrix33/98\\ -33/25\\ -24/55 bmatrix, c_2= bmatrix-31/99\\ 19/36\\ -11/20 bmatrix. gathered (4.1) Composing the 2323 branch maps of word 23W_23 yields the return map Φper(v)=Mperv+cper _ per(v)=M_ perv+c_ per (B.2) with exact rational fixed point v^0∈ℚ6 v^0 ^6. The certified projection-sign word is 23=(+,−,+)5(−,+,+)7(−,−,+)2(−,−,−)(−,−,+)8.W_23=(+,-,+)^5\,(-,+,+)^7\,(-,-,+)^2\,(-,-,-)\,(-,-,+)^8. (4.2) Proposition 4.1 (Exact period-2323 certificate for nearby initializations). For the rational data (4.1), Q^x,Q^y≻0 Q_x, Q_y 0 and det(A^)det(B^)≠0 ( A) ( B)≠ 0. The direct iteration (1.5), with β=1β=1, applied to minx,y,z∈ℝ3 _x,y,z ^3 12x⊤Q^xx+c^1⊤x+12y⊤Q^yy+c^2⊤y+ιℝ+3(z) 12x Q_xx+ c_1 x+ 12y Q_yy+ c_2 y+ _R_+^3(z) (4.3) s.t. .t. A^x+B^y+z=b^, Ax+ By+z= b, has the following properties. (i) It has a unique primal–dual KKT point. (i) Direct ADMM has a non-KKT sequence of minimal period 2323. Its strict projection-sign word is (4.2); its exponents denote consecutive repetitions and sum to 2323. All 23×3=6923× 3=69 projection inputs are separated from zero: min0≤k<23min1≤i≤3|t^ik|>1250. _0≤ k<23 _1≤ i≤ 3| t_i^k|> 1250. (i) There is a rational matrix P≻0P 0 for which ℰcert:=v^0+e:e⊤Pe<14000E_ cert:= \ v^0+e:e Pe< 14000 \ (4.4) is an open invariant set for the 2323-step return map (B.2). Every v0∈ℰcertv^0 _ cert follows the repeating pattern 23W_23 and converges phasewise to the period-2323 sequence. Consequently, the reconstructed ADMM sequence is nonconvergent for every reduced initialization v0∈ℰcertv^0 _ cert. The proof, together with the reduced branch maps, the return-map construction, and the invariant-ellipsoid data, is given in Appendix B. Because A A and B B are nonsingular, the example converts to an all-identity instance. Corollary 4.2 (Identity-slack equivalence). Since A A and B B are nonsingular, the change of variables u=A^xu= Ax, w=B^yw= By converts (4.3) into an equivalent rational strongly convex [I3,I3,I3][I_3,I_3,I_3] identity-slack QP. The transformed Hessians remain positive definite, so Proposition 4.1 yields a locally attracting period-2323 counterexample in the all-identity model as well. The change-of-variables formulas defining the equivalent rational all-identity instance are given in Appendix B. Figure 3 summarizes the certified word, projection margin, invariant-ellipsoid slice, and spectrum of the period map. Figure 3: Kimi period-2323 certificate: projection word and margin; invariant-ellipsoid slice; spectrum of MperM_ per. 4.2 Different Discovery Mechanisms of the Two AI-Assisted Routes This paper compares two research routes that actually occurred and were fully recorded, rather than the intrinsic capabilities of two base models under controlled conditions. To describe the research environment beyond the model, we write a research harness as H=(M,W,T,S,A,G,V),H=(M,W,T,S,A,G,V), where M denotes the base model, W the versioned workspace, T the tools, S the prompts, skills, and instructions, A the persistent research artifacts, G the human decision and claim-promotion gates, and V the exact verifiers. Human direction is a control mechanism inside G, not an independent component outside H. A research route is one concrete run of a harness configuration together with the mathematical trajectory it produces. The present H is used only to describe the research process for the identity-slack question in this paper; it does not constitute a general-purpose AI4Math platform. The two routes address the same mathematical problem, but under different harness configurations. The Codex route ran in a long-maintained project workspace with access to accumulated failure records, reduced representations, experiment scripts, and exact certificates, and with iterative human steering. The Kimi route ran in a frozen blind workspace without access to the period-6666 candidate, certificate, or project history, and received only a few continue-search instructions. Both routes share the same evidence principles: floating-point trajectories and model-generated derivations nominate candidates only; periodicity claims must pass exact problem validation, strict branch admissibility, raw ADMM replay, period closure, and minimal-period checks; stability claims further require the corresponding Lyapunov, Schur, Sturm, or interval tests. Detailed process dossiers appear in Appendix C. Mathematical brief identity-slack AI exploration reduction screening Numerical or symbolic candidate Exact rational reconstruction Claim promotion exact gates Theorem-level claim Explorationcandidate evidence onlyCertificationexact verification + human audit Gate failure: revise the search object, candidate, or evidence requirement Codex route Long-running project workspace; iterative human steering Kimi route Frozen blind workspace; no Codex candidate or certificate Figure 4: Candidate-generation and claim-promotion workflow shared by the two AI-assisted research routes. Models may propose representations, search objects, and candidates in the exploration stage, but only conclusions that pass exact verification and a human claim-scope audit enter the final mathematical record. The two routes share the same mathematical question and evidence principles, but use different harness configurations. Search space and construction style. The two routes first differ markedly in their search objects. The Codex route reduced the original ADMM iteration to a piecewise-affine dynamical system and then shifted from searching Hessian data (Q1,Q2,b¯)(Q_1,Q_2, b) to searching resolvent parameters (M,N)(M,N) and a realizable active-set itinerary. Its core object is a switching system formed by distinct projection branches, culminating in the itinerary (00)2(01)64.(00)^2(01)^64. Thus the route is closer to a construction driven by dynamical mechanism: first identify a near-rotation/reset structure, then recover an optimization problem that realizes it. The Kimi route is closer to inverse design of an optimization instance. It jointly searched over KKT-compatible quadratic-program data (A^,B^,Q^x,Q^y,c^1,c^2)( A, B, Q_x, Q_y, c_1, c_2) and a realizable projection itinerary 23W_23, and used a fixed point of the period return map to construct an ADMM instance with a prescribed dynamical property. State representation and analysis object. The Codex route used the four-dimensional signed piecewise-affine state s=(y,q),q=z+λ,s=(y,q), q=z+λ, whose main role is to expose active-set switching. The period-6666 itinerary can be interpreted as a comparatively long near-rotation phase, followed by a short reset through a few 0000 branches. The Kimi route used the six-dimensional projector-aligned state v=(y,t),t=z+λ,v=(y,t), t=z+λ, and analyzed the one-period return map Φper(v)=Mperv+cper _ per(v)=M_ perv+c_ per directly. This representation unifies existence and local stability of the periodic orbit as questions about a return-map fixed point and contraction. Certificate goals and dynamical conclusions. The primary goal of the Codex certificate is a strict existence proof for a non-KKT periodic orbit. Its evidence includes exact period closure, strict branch admissibility, raw ADMM replay, and minimal-period verification. The resulting period-6666 counterexample moreover retains the same branch itinerary under small data perturbations. The Kimi certificate further seeks local attraction of the periodic orbit. In addition to period closure and strict projection conditions, it constructs a rational Lyapunov matrix satisfying P−Mper⊤PMper≻0P-M_ per PM_ per 0 and an invariant ellipsoid preserved by the return map. Thus period-2323 nonconvergence is not the phenomenon of a single special initialization, but occurs on an open set of reduced initializations. Continuation in optimization theory. The Codex route not only produced a counterexample, but also carried the active-set switching instability into multiplier-relaxation analysis: after freezing the problem and initialization and varying only the multiplier step τ, it further studied problem-dependent small-step convergence, the impossibility of a class-uniform step, and a fixed-instance local stability boundary. The theoretical emphasis of the Kimi route is instead on return-map contraction and basin stability under reduced-initialization perturbations. Table 1 summarizes the main differences between the two routes in harness configuration, mathematical search objects, and final certificates. Table 1: Comparison of the two realized AI-assisted research routes. Aspect Codex / GPT-5.6 Sol route Kimi Code / Kimi K3 route Harness configuration Long-running project workspace; project-specific ADMM skill and experiment toolchain; persistent research artifacts; iterative human steering Frozen blind workspace; generic research skills with shell/Python tools; isolated route state; few continue-search instructions Main search object Resolvent parameters (M,N)(M,N) and realizable active-set itinerary (00)2(01)64(00)^2(01)^64 KKT-compatible QP data and projection itinerary 23W_23 Reduced state s=(y,q)∈ℝ4s=(y,q) ^4, q=z+λq=z+λ; direct m=2m=2 all-identity model v=(y,t)∈ℝ6v=(y,t) ^6, t=z+λt=z+λ; m=3m=3 model, equivalently convertible to all-identity form Construction style Mechanism-driven: state reduction, branch-spectrum analysis, and near-rotation/reset itinerary design Inverse design: jointly synthesize problem data, projection itinerary, and return-map fixed point Main certificate Exact period closure, strict branch checks, raw ADMM replay, and minimal-period verification Exact period closure, strict projection checks, discrete Lyapunov inequality, and invariant ellipsoid Dynamical conclusion Minimal period-6666 non-KKT orbit; exposes active-set switching instability Minimal period-2323 locally attracting non-KKT orbit; certified open attracting basin Robustness type Structural persistence under problem-data perturbations Basin stability under reduced-initialization perturbations Theorem continuation Multiplier relaxation, step-size sensitivity, and class-level step conclusions Return-map contraction and local attracting-basin certification Recorded process use ≈40≈ 40 h wall-clock; ≈1.704×109≈ 1.704× 10^9 recorded tokens ≈9≈ 9 h wall-clock; ≈6.327×107≈ 6.327× 10^7 recorded tokens Scope note. The times and token counts in the table record only the process use of two realized runs. Because the routes differ in base model, workspace, tools, accumulated context, token accounting, and human steering, these numbers do not constitute a controlled model-efficiency comparison. Full prompts, workspace manifests, intervention records, and verification commands appear in Appendix C. The two routes illustrate complementary modes of AI-assisted optimization research. The Codex route emphasizes discovery of a hidden state-space structure and an algorithmic instability mechanism; the Kimi route emphasizes inverse construction of an optimization instance with prescribed dynamical behavior, together with certification of its local stability. These differences are consistent with the respective harness configurations and historical paths, suggesting that persistent workspaces, prior representations, project skills, evidence gates, and human decisions may jointly shape the mathematical space a model actually explores. Nevertheless, the present records cannot uniquely attribute the divergence to the base model, workspace memory, project skills, or human steering alone. Identifying their separate effects would require controlled model–harness ablations under matched tools, budgets, and prompts. Accordingly, the comparison in this paper should be understood as one between two realized model–harness routes, not as a ranking of base-model capabilities. 5 Discussion 5.1 What the Results Show An identity last coefficient matrix does not by itself restore convergence of unmodified direct three-block ADMM. The period-6666 example rules out explanations based on nonconvexity, nonunique block solves, projection ties, or floating-point drift, and its itinerary persists under small data perturbations. The period-2323 example adds a certified open attracting basin and, by nonsingularity of A A and B B, an equivalent all-identity [I3,I3,I3][I_3,I_3,I_3] instance (Corollary 4.2). Multiplier relaxation further shows that a problem-dependent small step can restore convergence, but no positive relative step is class-uniform. 5.2 AI for Optimization: From Problem Solving to Theory Discovery Broadly, AI for optimization concerns not only using AI to solve a given optimization model, but also using AI in the formation of optimization models, algorithms, and theory. Relative to the object of study, this role can be organized into three interrelated levels. At the problem-solving level, AI may assist formulation and reformulation, variable and constraint decomposition, algorithm and solver selection, initialization, parameter configuration, and runtime diagnosis or intervention. The usual goal is to improve efficiency, robustness, or automation on a fixed problem. At the algorithm-design level, AI may further participate in the design of update rules, relaxation parameters, penalty terms, correction steps, and adaptive strategies. The task is no longer only to execute an existing algorithm, but to diagnose failure modes and propose analyzable modifications suggested by the discovered structure. The multiplier- relaxation results in this paper illustrate this level: after the period-6666 counterexample had been certified, the research continued from “why the algorithm fails” to “which part of the update restores stability”, and then distinguished problem-dependent convergent steps, fixed-instance local stability boundaries, and the impossibility of class-uniform guarantees. At the theory-discovery level, AI may participate in mathematical representation discovery, structured instance design, counterexample and extreme-instance construction, theorem conjecture, proof-route search, and exact certificate generation. The present work primarily studies this theory-facing role. The two routes show that AI can reshape the search space of optimization research in different ways: the Codex route reduced the original ADMM iteration to a low-dimensional signed piecewise-affine system and turned counterexample search into the design of resolvent parameters and reachable active-set itineraries; the Kimi route jointly searched over KKT-compatible problem data and projection itineraries, turning periodic counterexample construction into inverse instance design with prescribed return-map properties. The AI-for-optimization capabilities illustrated here can therefore be summarized in the following stages. Representation discovery. The raw iteration of a complicated optimization algorithm often contains redundant variables, implicit projections, and multilayered dependencies. AI can search for more analysis-friendly state variables, coordinate changes, operator splittings, or piecewise structures, thereby transforming the original problem into a finite-dimensional linear, affine, or operator dynamical system. A change of representation affects not only proof difficulty, but also which mathematical objects can subsequently be searched and certified. Structured construction and inverse design. AI can convert open questions such as “does a counterexample exist?” into construction problems with explicit parameterizations, branch conditions, and algebraic constraints. The goal may be to design an optimization instance on which an algorithm exhibits a prescribed behavior—for example a periodic orbit, boundary stability, slow convergence, active-set switching, or parameter sensitivity. The same capability can also generate stress-test instances and robustness benchmarks, not only theoretical counterexamples. Failure diagnosis and algorithmic intervention. When an algorithm exhibits nonconvergence, oscillation, or numerical instability, AI can help separate sources such as objective geometry, constraint degeneracy, block ordering, active-set switching, or parameter choice. The resulting dynamical structure can then guide algorithmic modifications, including relaxation, damping, proximal regularization, correction steps, and adaptive parameter rules. Counterexample construction is therefore not only a negative result, but also an input to algorithm design. Proof search and exact certification. Numerical experiments and floating-point computation may discover candidates, but they cannot independently carry theorem-level evidence. AI can organize symbolic derivation, branch enumeration, rational reconstruction, interval arithmetic, Lyapunov search, Schur stability tests, and Sturm root counting into a reproducible certification pipeline. In this paper, agents nominated candidate representations, derivations, and certificate designs; period closure, strict branch conditions, raw ADMM replay, and the corresponding stability predicates were supported by independently executable exact checks. Analytic conclusions remain established by the mathematical arguments in the main text and appendices. Theorem continuation. The role of AI can extend beyond “finding an answer” to a single question. After a candidate or theorem has been certified, representations, code, failure records, and proof artifacts in a persistent workspace can be reused to pose neighboring questions—changing a parameter, relaxing an assumption, seeking a boundary case, or establishing an impossibility result. The progression in this paper from the period-6666 counterexample to local multiplier-step stability, convergence from a specified initialization, problem-dependent global guarantees, and the impossibility of class-uniform guarantees is an instance of such theorem continuation. The overall process can be summarized as problem formulation⟶representation discovery⟶candidate or algorithm design formulation discovery or algorithm design ⟶analysis⟶exact certification⟶theorem or algorithm continuation⟶knowledge reuse. certification or algorithm continuation reuse. The final reuse step may include distilling effective state reductions, proof templates, certificate structures, and experimental workflows into research skills that can be invoked on other optimization problems. The role of the research harness. These capabilities are not produced by the base model alone; they depend on the research environment in which the model operates. A versioned workspace provides long-term research memory; tools and skills delimit executable operations; persistent artifacts allow earlier results to be reused; exact verifiers prescribe the evidentiary standard by which candidates enter the mathematical record; and human checkpoints control problem choice, claim scope, literature boundaries, and final acceptance. Thus a research harness does more than store research logs: it also shapes the space of representations, candidates, and proofs that a model can explore. This paper presents one case study of that research pattern on the identity-slack ADMM question. It does not prove that the same capabilities already transfer reliably to other optimization domains, nor can it separate the causal contributions of the base model, workspace memory, project skills, and human steering from the difference between the two routes. Future work in AI for optimization may further explore Lyapunov-function construction, error-bound discovery, splitting design, adaptive update rules, extreme-instance generation, and reusable proof patterns, and may evaluate transferability through frozen workspaces, matched tool budgets, and controlled ablation experiments. 5.3 Limits and Outlook Several mathematical questions remain open. First, it is still unclear whether the two-dimensional pure-quadratic subclass admits a shorter counterexample whose projection branches are all strictly admissible. Second, quantitative conditions excluding active-set switching instability are still needed, as is a sharper characterization of the exact global multiplier-step range for a fixed problem. Finite-precision computation, inexact block solves, alternative block update orders, nonorthant cone constraints, and more general nonsmooth models also require further analysis. The present certificates rely mainly on exact rational arithmetic, finite symbolic derivation, and independently executable algebraic and spectral checks; they have not yet been fully translated into formal proofs inside a proof assistant. A natural next step is to encode the critical ingredients— period closure, branch admissibility, Lyapunov inequalities, and Schur–Sturm tests—in Lean, Isabelle, or another formal verification system, thereby narrowing the gap between computer-assisted certificates and fully formalized proofs. On the AI-evaluation side, this paper compares two realized model–harness routes rather than base-model capabilities under controlled conditions. Future studies should use frozen workspaces, preregistered success criteria, matched tool and compute budgets, and controlled human or model baselines. Systematic ablations of the base model, workspace memory, project-specific skills, persistent research artifacts, and human steering are also needed to identify each component’s contribution to representation discovery, candidate construction, certificate design, and theorem continuation. Another important question is whether the state-reduction, structured-search, and exact-certification patterns developed here transfer to Douglas–Rachford splitting, primal–dual splitting, and other operator splitting methods, rather than merely rediscovering or reproducing the ADMM examples of this paper. References [1] H. H. Bauschke and P. L. Combettes (2017) Convex analysis and monotone operator theory in Hilbert spaces. 2 edition, CMS Books in Mathematics, Springer, Cham. External Links: Document Cited by: §A.1, §A.1. [2] D. Boley (2013) Local linear convergence of the alternating direction method of multipliers on quadratic or linear programs. SIAM Journal on Optimization 23 (4), p. 2183–2207. 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Yuan (2018) Convergence analysis of the direct extension of ADMM for multiple-block separable convex minimization. Advances in Computational Mathematics 44 (3), p. 773–813. External Links: Document Cited by: §1.1. Appendix A Multiplier-Relaxation Proofs A.1 Proof of Proposition 3.1 Proof. Write u=(x,y,z)u=(x,y,z) and E=[A,B,Im]E=[A,B,I_m]. Decompose the augmented Lagrangian as ℒβ(u,λ)=sλ(u)+h(u),L_β(u;λ)=s_λ(u)+h(u), where sλ(u):=F(x)+G(y)−λ⊤(Eu−b)+β2‖Eu−b‖2,h(u):=ιℝ+m(z).s_λ(u):=F(x)+G(y)-λ (Eu-b)+ β2\|Eu-b\|^2, 18.49988pth(u):= _R_+^m(z). Set dβ(λ)=minuℒβ(u,λ).d_β(λ)= _uL_β(u;λ). We first verify uniqueness of the KKT point. Full row rank of [AB][A\ B] implies strict feasibility: for any z>0z>0, the equation [AB](x,y)=b−z[A\ B](x,y)=b-z is solvable. Strong convexity gives a unique primal solution. In the present multiplier convention, the unaugmented negative dual objective is ϕ0(λ)=F∗(A⊤λ)+G∗(B⊤λ)−b⊤λ+ιℝ−m(λ). _0(λ)=F^*(A λ)+G^*(B λ)-b λ+ _R_-^m(λ). Standard conjugacy results imply that an L-smooth convex function has a 1/L1/L-strongly convex conjugate [1]. Hence F∗F^* and G∗G^* are 1/LF1/L_F- and 1/LG1/L_G-strongly convex, respectively. Consequently, m0=λmin(AA⊤LF+BB⊤LG)>0,m_0= _ \! ( A L_F+ B L_G )>0, where strict positivity follows from the full-row-rank assumption. The multiplier is therefore unique. We next establish the global primal error bound at fixed multiplier. Set Kμ=diag(μFI,μGI,0)+βE⊤E,μp=λmin(Kμ)>0,K_μ=diag( _FI, _GI,0)+β E E, 18.49988pt _p= _ (K_μ)>0, Lp=maxLF,LG+β‖E‖2.L_p= \L_F,L_G\+β\|E\|^2. For all u,vu,v, ⟨∇sλ(u)−∇sλ(v),u−v⟩ ∇ s_λ(u)-∇ s_λ(v),u-v ≥(u−v)⊤Kμ(u−v)≥μp‖u−v‖2, ≥(u-v) K_μ(u-v)≥ _p\|u-v\|^2, ‖∇sλ(u)−∇sλ(v)‖ \|∇ s_λ(u)-∇ s_λ(v)\| ≤Lp‖u−v‖. ≤ L_p\|u-v\|. Let u¯=u¯(λ) u= u(λ) denote the unique minimizer of ℒβ(⋅,λ)L_β(·;λ), and set p=proxh(u−∇sλ(u)),r=u−p=PGλ(u).p=prox_h (u-∇ s_λ(u) ), 18.49988ptr=u-p=PG_λ(u). Proximal optimality gives r−∇sλ(u)∈∂h(p),−∇sλ(u¯)∈∂h(u¯).r-∇ s_λ(u)∈∂ h(p), 18.49988pt-∇ s_λ( u)∈∂ h( u). By monotonicity of ∂h∂ h, ⟨∇sλ(u)−∇sλ(u¯),p−u¯⟩≤⟨r,p−u¯⟩. ∇ s_λ(u)-∇ s_λ( u),p- u ≤ r,p- u . Using the μp _p-strong monotonicity and LpL_p-Lipschitz continuity of ∇sλ∇ s_λ, we obtain μp‖u−u¯‖2 _p\|u- u\|^2 ≤⟨∇sλ(u)−∇sλ(u¯),u−u¯⟩ ≤ ∇ s_λ(u)-∇ s_λ( u),u- u =⟨∇sλ(u)−∇sλ(u¯),p−u¯⟩+⟨∇sλ(u)−∇sλ(u¯),r⟩ = ∇ s_λ(u)-∇ s_λ( u),p- u + ∇ s_λ(u)-∇ s_λ( u),r ≤⟨r,p−u¯⟩+Lp‖u−u¯‖‖r‖ ≤ r,p- u +L_p\|u- u\|\,\|r\| ≤(1+Lp)‖u−u¯‖‖r‖. ≤(1+L_p)\|u- u\|\,\|r\|. Therefore ‖u−u¯(λ)‖≤κp‖PGλ(u)‖,κp=1+Lpμp,\|u- u(λ)\|≤ _p\|PG_λ(u)\|, 18.49988pt _p= 1+L_p _p, globally, with a constant independent of λ. Exact x→y→zx→ y→ z block minimization gives γ=12minλmin(μFI+βA⊤A),λmin(μGI+βB⊤B),β>0γ= 12 \! \ _ ( _FI+β A A), _ ( _GI+β B B),β \>0 and ℒβ(uk,λk)−ℒβ(uk+1,λk)≥γ‖uk+1−uk‖2.L_β(u^k;λ^k)-L_β(u^k+1;λ^k)≥γ\|u^k+1-u^k\|^2. Let ax=LF+β‖A⊤A‖,ay=LG+β‖B⊤B‖,a_x=L_F+β\|A A\|, 18.49988pta_y=L_G+β\|B B\|, σ2=ax2+β2‖B⊤A‖2+ay2+β2‖A‖2+β2‖B‖2+(β+2)2.σ^2=a_x^2+β^2\|B A\|^2+a_y^2+β^2\|A\|^2+β^2\|B\|^2+(β+2)^2. For Δx=xk−xk+1 x=x^k-x^k+1, Δy=yk−yk+1 y=y^k-y^k+1, and Δz=zk−zk+1 z=z^k-z^k+1, write the three blocks of PGλk(uk)PG_λ^k(u^k) as (PGx,k,PGy,k,PGz,k)(PG_x,k,PG_y,k,PG_z,k). The block optimality conditions and nonexpansiveness of the orthant projection then give ‖PGx,k‖ \|PG_x,k\| ≤ax‖Δx‖, ≤ a_x\| x\|, ‖PGy,k‖ \|PG_y,k\| ≤β‖B⊤A‖‖Δx‖+ay‖Δy‖, ≤β\|B A\|\| x\|+a_y\| y\|, ‖PGz,k‖ \|PG_z,k\| ≤β‖A‖‖Δx‖+β‖B‖‖Δy‖+(β+2)‖Δz‖. ≤β\|A\|\| x\|+β\|B\|\| y\|+(β+2)\| z\|. Consequently, ‖PGλk(uk)‖≤σ‖uk+1−uk‖.\|PG_λ^k(u^k)\|≤σ\|u^k+1-u^k\|. For the x,yx,y blocks this uses only global Lipschitz continuity of ∇F,∇G∇ F,∇ G; for z, it compares the projection fixed-point identities at the current point and the block minimizer. Completing the square in the quadratic penalty and applying Fenchel duality gives the explicit Moreau-envelope representation −dβ(λ)=minη∈ℝmϕ0(η)+12β‖η−λ‖2.-d_β(λ)= _η ^m \ _0(η)+ 12β\|η-λ\|^2 \. (A.1) Since ϕ0 _0 is m0m_0-strongly convex, its Moreau envelope in (A.1) is strongly convex with modulus md=m01+βm0>0m_d= m_01+β m_0>0 and has 1/β1/β-Lipschitz gradient [1]. It follows that ‖λ−λ⋆‖≤md−1‖∇dβ(λ)‖,∇dβ(λ)=b−Eu¯(λ).\|λ-λ \|≤ m_d^-1\|∇ d_β(λ)\|, 18.49988pt∇ d_β(λ)=b-E u(λ). Define Δpk _p^k =ℒβ(uk+1,λk)−dβ(λk), =L_β(u^k+1;λ^k)-d_β(λ^k), Δdk _d^k =dβ(λ⋆)−dβ(λk), =d_β(λ )-d_β(λ^k), Vk V^k =Δpk+Δdk. = _p^k+ _d^k. The gap calculation of Hong and Luo [13, Theorem 3.1, Lemma 3.1, and (3.16)–(3.17)], in the present sign convention, yields Vk−Vk−1≤−a‖uk+1−uk‖2−τ‖∇dβ(λk)‖2,a=γ−τ‖E‖2κp2σ2.V^k-V^k-1≤-a\|u^k+1-u^k\|^2-τ\|∇ d_β(λ^k)\|^2, 18.49988pta=γ-τ\|E\|^2 _p^2σ^2. Thus a>0a>0 whenever 0<τ<γ‖E‖2κp2σ2.0<τ< γ\|E\|^2 _p^2σ^2. The cost-to-go estimate on the noncompact orthant can be verified directly. Set Δx=xk−xk+1 x=x^k-x^k+1, Δy=yk−yk+1 y=y^k-y^k+1, and Δz=zk−zk+1 z=z^k-z^k+1, and define HGS=β(0A⊤BA⊤00B⊤000),Cp=‖HGS‖22μp.H_ GS=β pmatrix0&A B&A \\ 0&0&B \\ 0&0&0 pmatrix, 18.49988ptC_p= \|H_ GS\|^22 _p. The three exact block optimality conditions, including the normal-cone condition for z, give wk+1=(−βA⊤(BΔy+Δz)−βB⊤Δz0)∈∂uℒβ(uk+1,λk).w^k+1= pmatrix-β A (B y+ z)\\ -β B z\\ 0 pmatrix∈ _uL_β(u^k+1;λ^k). Uniform μp _p-strong convexity at fixed multiplier therefore yields Δpk≤‖wk+1‖22μp≤Cp‖uk+1−uk‖2. _p^k≤ \|w^k+1\|^22 _p≤ C_p\|u^k+1-u^k\|^2. Moreover, qβ=−dβq_β=-d_β is mdm_d-strongly convex and 1/β1/β-smooth, so one may take Cd=12βmd2,Δdk≤Cd‖∇dβ(λk)‖2.C_d= 12β m_d^2, 18.49988pt _d^k≤ C_d\|∇ d_β(λ^k)\|^2. These constants are independent of k,λk,λ^k, and the dual step. Consequently, η=mina/Cp,τ/Cd>0,ρ=(1+η)−1∈(0,1)η= \a/C_p,τ/C_d\>0, 18.49988ptρ=(1+η)^-1∈(0,1) imply Vk−Vk−1≤−ηVk,Vk≤ρVk−1.V^k-V^k-1≤-η V^k, 18.49988ptV^k≤ρ V^k-1. Finally, strong convexity and Lipschitz continuity of the fixed-multiplier minimizer give ‖uk+1−u¯(λk)‖2 \|u^k+1- u(λ^k)\|^2 ≤2Δpk/μp, ≤ 2 _p^k/ _p, ‖λk−λ⋆‖2 \|λ^k-λ \|^2 ≤2Δdk/md, ≤ 2 _d^k/m_d, ‖u¯(λk)−u⋆‖ \| u(λ^k)-u \| ≤(‖E‖/μp)‖λk−λ⋆‖. ≤(\|E\|/ _p)\|λ^k-λ \|. Hence there is a constant C>0C>0 for which ‖(uk+1,λk)−(u⋆,λ⋆)‖≤Cρk/2.\|(u^k+1,λ^k)-(u ,λ )\|≤ Cρ^k/2. The assumption z0∈ℝ+mz^0 _+^m in Proposition 3.1 ensures that the initial augmented-Lagrangian value is finite. The z-update then preserves zk∈ℝ+mz^k _+^m for every k≥0k≥ 0. The compact-polyhedral assumption in the original Hong–Luo theorem is not invoked directly here: the global smooth strongly convex structure above replaces the error bounds for which that assumption is used. Accordingly, the inherited component is the gap-contraction architecture; the model-specific component proved here is the verification of the required global bounds and explicit constants on the noncompact slack-last class. ∎ A.2 Backtracking Rule This appendix proves Corollary 3.3 and the quadratic refinement stated in Section 3. Proof of Corollary 3.3. Fix an arbitrary candidate τ, and abbreviate its trial values by λ=λ^,u=uk,u+=u^,D=ℒβ(u,λ)−ℒβ(u+,λ).λ= λ, 18.49988ptu=u^k, 18.49988ptu^+= u, 18.49988ptD=L_β(u;λ)-L_β(u^+;λ). Let e e =u−u¯(λ), =u- u(λ), v v =PGλ(u). =PG_λ(u). The global primal error bound in Proposition 3.1 directly gives ‖Ee‖2≤‖E‖2‖e‖2≤‖E‖2κp2‖v‖2=BE(v).\|Ee\|^2≤\|E\|^2\|e\|^2≤\|E\|^2 _p^2\|v\|^2=B_E(v). (A.2) The trial decrease can be evaluated from the block optimality conditions, without subtracting two nearby augmented-Lagrangian values. After the initial primal sweep, z,z+∈ℝ+mz,z^+ _+^m. If Δx=x−x+ x=x-x^+, Δy=y−y+ y=y-y^+, and q=b−Ax+−By++λ/β,z+=[q]+,q=b-Ax^+-By^++λ/β, 18.49988ptz^+=[q]_+, and if DF(a,c)=F(a)−F(c)−⟨∇F(c),a−c⟩D_F(a,c)=F(a)-F(c)- ∇ F(c),a-c with DGD_G defined analogously, cancellation of the first-order terms in the block optimality conditions gives D= D= DF(x,x+)+β2‖AΔx‖2 D_F(x,x^+)+ β2\|A x\|^2 (A.3) +DG(y,y+)+β2‖BΔy‖2 +D_G(y,y^+)+ β2\|B y\|^2 +β2(‖z−q‖2−‖z+−q‖2). + β2 (\|z-q\|^2-\|z^+-q\|^2 ). Exact block minimization gives uniform γ,σ>0γ,σ>0 such that D≥γ‖u+−u‖2,‖v‖≤σ‖u+−u‖.D≥γ\|u^+-u\|^2, 18.49988pt\|v\|≤σ\|u^+-u\|. Consequently every 0<τ≤τsafe,E:=(1−χ)γ‖E‖2κp2σ20<τ≤ _safe,E:= (1-χ)γ\|E\|^2 _p^2σ^2 satisfies (3.3). A geometric search starting from τmax _ therefore terminates and accepts τk≥τ¯E:=minτmax,δτsafe,E>0. _k≥ τ_E:= \ _ ,δ _safe,E\>0. For the accepted sequence, set Dk=ℒβ(uk,λk)−ℒβ(uk+1,λk)D_k=L_β(u^k;λ^k)-L_β(u^k+1;λ^k). With u¯k=argminuℒβ(u,λk) u^k= *arg\,min_uL_β(u;λ^k), rk=Euk−br^k=Eu^k-b, r¯k=Eu¯k−b r^k=E u^k-b, and dk=dβ(λk)d_k=d_β(λ^k), direct expansion gives Vk−Vk−1=−Dk+τk‖rk‖2−2(dk−dk−1).V^k-V^k-1=-D_k+ _k\|r^k\|^2-2(d_k-d_k-1). Concavity of dβd_β and λk−1−λk=τkrkλ^k-1-λ^k= _kr^k imply dk−dk−1≥τk⟨r¯k,rk⟩.d_k-d_k-1≥ _k r^k,r^k . Completing the square yields Vk−Vk−1≤τk‖E(uk−u¯k)‖2−Dk−τk‖∇dβ(λk)‖2.V^k-V^k-1≤ _k\|E(u^k- u^k)\|^2-D_k- _k\|∇ d_β(λ^k)\|^2. (A.4) Equations (3.3) and (A.2) imply Vk−Vk−1≤−χDk−τk‖∇dβ(λk)‖2.V^k-V^k-1≤-χ D_k- _k\|∇ d_β(λ^k)\|^2. The proof of Proposition 3.1 explicitly constructs step-sequence-independent constants Cp,Cd>0C_p,C_d>0 such that Δpk≤(Cp/γ)Dk,Δdk≤Cd‖∇dβ(λk)‖2. _p^k≤(C_p/γ)D_k, 18.49988pt _d^k≤ C_d\|∇ d_β(λ^k)\|^2. Using τk≥τ¯E _k≥ τ_E gives Vk≤(1+η)−1Vk−1,η=minχγCp,τ¯ECd>0.V^k≤(1+η)^-1V^k-1, 18.49988ptη= \ χγC_p, τ_EC_d \>0. (A.5) This closes the proof for the general smooth strongly convex class. Quadratic weighted refinement. If F,GF,G are the strongly convex quadratics stated in the main text, set K=diag(Q1,Q2,0)+βE⊤E,p=proxh(u−∇sλ(u))=u−v.K=diag(Q_1,Q_2,0)+β E E, 18.49988ptp=prox_h(u-∇ s_λ(u))=u-v. Proximal optimality and monotonicity of ∂h∂ h give e⊤Ke+‖v‖2≤e⊤(I+K)v.e Ke+\|v\|^2≤ e (I+K)v. Cauchy–Schwarz in the K-metric, together with K⪰βE⊤EK β E E, yields e⊤(I+K)v e (I+K)v ≤e⊤Kev⊤(K−1+2I+K)v, ≤ e Ke\, v (K^-1+2I+K)v, (A.6) ‖Ee‖2 \|Ee\|^2 ≤1βe⊤Ke≤1βv⊤(K−1+2I+K)v=BK(v). ≤ 1βe Ke≤ 1βv (K^-1+2I+K)v=B_K(v). This step uses the quadratic identity ∇sλ(u)−∇sλ(u¯)=Ke∇ s_λ(u)-∇ s_λ( u)=Ke. The majorant can be evaluated without forming K−1K^-1. For v=(vx,vy,vz)v=(v_x,v_y,v_z), v⊤K−1v= v K^-1v= (vx−A⊤vz)⊤Q1−1(vx−A⊤vz) (v_x-A v_z) Q_1^-1(v_x-A v_z) +(vy−B⊤vz)⊤Q2−1(vy−B⊤vz)+β−1‖vz‖2, +(v_y-B v_z) Q_2^-1(v_y-B v_z)+β^-1\|v_z\|^2, (A.7) v⊤Kv= v Kv= vx⊤Q1vx+vy⊤Q2vy+β‖Avx+Bvy+vz‖2. v_x Q_1v_x+v_y Q_2v_y+β\|Av_x+Bv_y+v_z\|^2. (A.8) Thus two pre-factorized solves with Q1Q_1 and Q2Q_2 suffice. The two Bregman terms in (A.3) reduce to 12‖Δx‖Q12 12\| x\|_Q_1^2 and 12‖Δy‖Q22 12\| y\|_Q_2^2. Set cK=1βmaxξ∈σ(K)(ξ+ξ−1+2).c_K= 1β _ξ∈σ(K)(ξ+ξ^-1+2). Since BK(v)≤cK‖v‖2B_K(v)≤ c_K\|v\|^2, every 0<τ≤τsafe,K:=(1−χ)γcKσ20<τ≤ _safe,K:= (1-χ)γc_Kσ^2 satisfies (3.4), and geometric backtracking accepts τk≥τ¯K:=minτmax,δτsafe,K>0. _k≥ τ_K:= \ _ ,δ _safe,K\>0. Replacing (3.3)–(A.2) by (3.4)–(A.6) in (A.4) gives the same contraction, with τ¯K τ_K in place of τ¯E τ_E. The trial-point evaluation order is essential: the trial residual v v and decrease D D must use the same trial dual iterate. If a candidate step is rejected, both trial iterates must be discarded, and the next candidate must be constructed from the same (uk,λk−1)(u^k,λ^k-1). The theorem assumes exact solves of the block subproblems and exact evaluation of the acceptance test; the strict margin needed for a finite-precision acceptance decision requires separate analysis. A.3 Proof of Theorem 3.2 Proof. It suffices to take β=1β=1. For a prescribed 0<ϑ≤80/1190< ≤ 80/119, set r=7ϑ80−7ϑ∈(0,1/4],u=(1,r,0)⊤,v=(1,3r/5,4r/5)⊤,r= 7 80-7 ∈(0,1/4], u=(1,r,0) , v=(1,3r/5,4r/5) , and define PA=uu⊤1+r2,PB=vv⊤1+r2.P_A= u 1+r^2, P_B= v 1+r^2. For α∈[0,2−1]α∈[0, 2-1], let s s =2α1+α2,t=1−α21+α2, = 2α1+α^2, t= 1-α^21+α^2, Aα A_α =tPA+s(I−PA), =tP_A+s(I-P_A), Q1,α Q_1,α =s2PA+t2(I−PA), =s^2P_A+t^2(I-P_A), Bα B_α =tPB+s(I−PB), =tP_B+s(I-P_B), Q2,α Q_2,α =s2PB+t2(I−PB). =s^2P_B+t^2(I-P_B). Then Q1,α+Aα⊤Aα=Q2,α+Bα⊤Bα=IQ_1,α+A_α A_α=Q_2,α+B_α B_α=I. For α>0α>0, both Hessians are positive definite and Aα,BαA_α,B_α are nonsingular. Take x⋆=y⋆=0,z⋆=(1,0,0)⊤,λ⋆=(0,−1,−1)⊤,x =y =0, z =(1,0,0) , λ =(0,-1,-1) , and set b=z⋆,c1=Aα⊤λ⋆,c2=Bα⊤λ⋆.b=z , c_1=A_α λ , c_2=B_α λ . This is the unique strictly complementary KKT point. Fix D=diag(1,0,0)D=diag(1,0,0). The exact Schur recursion in Appendix A.4 proves ρ(UD(r,0))>1,ϑ=80r27(1+r2).ρ(U_D(r,0))>1, = 80r^27(1+r^2). At α=2−1α= 2-1, Aα=Bα=I/2,Q1,α=Q2,α=I/2,A_α=B_α=I/ 2, Q_1,α=Q_2,α=I/2, and the coordinates decouple. The scalar characteristic polynomials for a positive-slack and a zero-slack coordinate are 14(2ζ−1)2(ζ+ϑ−1),ζ4[4ζ2+(3ϑ−5)ζ+1−ϑ]. 14(2ζ-1)^2(ζ+ -1), ζ4 [4ζ^2+(3 -5)ζ+1- ]. Write p(ζ)=4ζ2+(3ϑ−5)ζ+1−ϑp(ζ)=4ζ^2+(3 -5)ζ+1- . The Jury inequalities for this quadratic factor are 4>|1−ϑ|,p(1)=2ϑ>0,p(−1)=10−4ϑ>0,4>|1- |, p(1)=2 >0, p(-1)=10-4 >0, so this endpoint is strictly Schur stable throughout the stated interval. By continuity, some αc∈(0,2−1) _c∈(0, 2-1) satisfies ρ(UD(r,αc))=1ρ(U_D(r, _c))=1. The corresponding problem remains strongly convex and full row rank. The branch cannot have eigenvalue +1+1: a sufficiently small perturbation along its eigenvector would remain in the strict affine branch and create a second ADMM fixed point, contradicting uniqueness of the KKT point. The unit-circle eigenvalue is therefore −1-1 or a nonreal conjugate pair. Choose a genuine eigenmode: a real eigenvector for −1-1, or the real invariant subspace spanned by the real and imaginary parts of a complex eigenvector. For a nonzero e in that mode, UD(r,αc)ke\U_D(r, _c)^ke\ alternates or rotates. It is therefore bounded but nonconvergent. Scaling e below the strict KKT projection margin keeps every iterate in the same branch, so the branch orbit is a bounded nonconvergent raw ADMM orbit. Because the unit-circle eigenvalue is nonzero, the two inactive components of the z-part of e vanish; taking e still smaller preserves the positive active component and gives z0∈ℝ+3z^0 _+^3. Since AαcA_ _c is nonsingular, set x0=Aαc−1(b−Bαcy0−z0).x^0=A_ _c^-1(b-B_ _cy^0-z^0). Then Aαcx0+Bαcy0+z0=bA_ _cx^0+B_ _cy^0+z^0=b, so the initial state is fully feasible. The stored x0x^0 does not enter the right-hand side of the next x-subproblem and therefore does not alter the (y,z,λ)(y,z,λ) orbit above. Each subsequent x-update is an affine function of that bounded reduced state, so the full (x,y,z,λ)(x,y,z,λ) orbit is bounded as well. ∎ A.4 Uniform-Step Obstruction For e=(δy,δz,δλ)e=(δ y,δ z,δλ), RA=AαAα⊤R_A=A_αA_α , and RB=BαBα⊤R_B=B_αB_α , define Yy Y_y =Bα⊤RABα, =B_α R_AB_α, Yz Y_z =−Bα⊤(I−RA), =-B_α (I-R_A), Yλ Y_λ =Bα⊤(I−RA), =B_α (I-R_A), Cy C_y =(I−RB)RABα, =(I-R_B)R_AB_α, Cz C_z =RA−RBRA+RB, =R_A-R_BR_A+R_B, Cλ C_λ =I−RA−RB+RBRA. =I-R_A-R_B+R_BR_A. On a fixed projection mask D, the full error matrix is UD(r,α)=(YyYzYλDCyDCzDCλϑ(I−D)Cyϑ(I−D)Cz(1−ϑ)I+ϑ(I−D)Cλ).U_D(r,α)= pmatrixY_y&Y_z&Y_λ\\ DC_y&DC_z&DC_λ\\ (I-D)C_y& (I-D)C_z&(1- )I+ (I-D)C_λ pmatrix. For completeness, we recall the real Schur recursion used below. Let pj(ζ)=∑ℓ=0djaj,ℓζdj−ℓ,pj♯(ζ):=ζdjpj(ζ−1),p_j(ζ)= _ =0^d_ja_j, ζ^d_j- , p_j (ζ):=ζ^d_jp_j(ζ^-1), where aj,0a_j,0 and aj,dja_j,d_j are respectively the leading and constant coefficients of pjp_j. Define Δj:=aj,02−aj,dj2 _j:=a_j,0^2-a_j,d_j^2 and, whenever Δj>0 _j>0, pj+1(ζ):=aj,0pj(ζ)−aj,djpj♯(ζ)ζ.p_j+1(ζ):= a_j,0p_j(ζ)-a_j,d_jp_j (ζ)ζ. A necessary Schur–Cohn condition is that Δj>0 _j>0 at every recursion stage. Consequently, once the preceding recursions are well defined, a negative Δj _j certifies that the original polynomial has a zero outside the unit disk. At α=0α=0, D=diag(1,0,0)D=diag(1,0,0), and ϑ=80r2/[7(1+r2)] =80r^2/[7(1+r^2)], remove the four zero roots and the root 1−ϑ1- from the characteristic polynomial. The first three exact Schur deltas of the remaining quartic satisfy Δ0>0,Δ1>0,Δ2<0(0<r≤1/4), _0>0, _1>0, _2<0 (0<r≤ 1/4), where, after positive factors are removed, the signs of Δ1 _1 and Δ2 _2 are determined by −h1(r),h1(r) -h_1(r), h_1(r) =1205r8+1977r6−4643r4−7385r2−2450, =1205r^8+1977r^6-4643r^4-7385r^2-2450, h2(r),h2(r) h_2(r), h_2(r) =14530r10+57137r8−131931r6 =14530r^10+57137r^8-131931r^6 −449323r4−329735r2−82950. -449323r^4-329735r^2-82950. For 0<r≤10<r≤ 1, h1(r)≤−1461r4−7385r2−2450<0,h_1(r)≤-1461r^4-7385r^2-2450<0, and h2(r)≤−60264r6−449323r4−329735r2−82950<0.h_2(r)≤-60264r^6-449323r^4-329735r^2-82950<0. Thus the signs are strict and a root lies outside the unit disk. The exact polynomial, factorization, and endpoint Jury checks are regenerated by python python/verify_universal_step_obstruction.py --check from the GitHub certificate repository root. The underlying exact script is python/analyze_identity_slack_universal_step_obstruction.py. A.5 Period-66 Verification From the GitHub certificate repository https://github.com/ConanXu-math/identity-slack-admm-cycle-certificate run python python/verify_certificate_pair.py. One-shot acceptance of both frozen certificates is python python/verify_all.py. The command invokes separately implemented four- and six-dimensional rational replays and verifies strong convexity, the unique KKT point, every ADMM update, 6666-step closure, all 132132 strict projection inequalities, and minimality of the period. A.6 Perturbation Robustness Corollary A.1 (Persistence under data perturbations). Let ++2S_++^2 denote the cone of real 2×22× 2 symmetric positive-definite matrices. With A=B=I2A=B=I_2, β=1β=1, and the update order x→y→z→λx→ y→ z→λ fixed, there is an open neighborhood of (Q1,Q2,b¯)(Q_1,Q_2, b) in ++2×++2×ℝ2S_++^2×S_++^2×R^2 such that each resulting problem has a non-KKT periodic ADMM sequence of minimal period 6666, with the same projection-sign sequence and every projection inequality strict. Proof. For the fixed projection-sign sequence, P and a in (2.14) depend analytically on the problem data. The period system remains nonsingular near the displayed instance, so (I4−)−1a(I_4-P)^-1a and its 6666 reduced iterates vary continuously. The signs in (2.17) have a positive common margin and remain strict under sufficiently small perturbations. The primitive word is unchanged. ∎ A.7 Relaxed Branch Map Set β=1β=1, M=(Q1+I2)−1,N=(Q2+I2)−1,I=I2,M=(Q_1+I_2)^-1, N=(Q_2+I_2)^-1, I=I_2, and use the six-dimensional essential state w=(y,z,λ)∈ℝ6w=(y,z,λ) ^6. Eliminating x+x^+ from the primal updates gives Cy=(I−N)M,Cz=N+(I−N)M,Cλ=(I−N)(I−M),d=Cλb¯,C_y=(I-N)M, C_z=N+(I-N)M, C_λ=(I-N)(I-M), d=C_λ b, and q+=Cyy+Czz+Cλλ+d.q^+=C_yy+C_zz+C_λ+d. For a strict projection mask D=diag(q1+>0,q2+>0),z+=Dq+,D=diag (1_\q_1^+>0\,1_\q_2^+>0\ ), z^+=Dq^+, the identity x++y++z+−b¯=λ−q++z+x^++y^++z^+- b=λ-q^++z^+ yields λ+=(1−τ)λ+τ(I−D)q+.λ^+=(1-τ)λ+τ(I-D)q^+. Thus the matrices in (3.5) are TD(0) T_D^(0) =(NM−N(I−M)N(I−M)DCyDCzDCλ00I), = pmatrixNM&-N(I-M)&N(I-M)\\ DC_y&DC_z&DC_λ\\ 0&0&I pmatrix, TD(1) T_D^(1) =(000000(I−D)Cy(I−D)Cz(I−D)Cλ−I), = pmatrix0&0&0\\ 0&0&0\\ (I-D)C_y&(I-D)C_z&(I-D)C_λ-I pmatrix, and aD(τ)=(N(I−M)b¯Dd0)+τ(00(I−D)d).a_D(τ)= pmatrixN(I-M) b\\ Dd\\ 0 pmatrix+τ pmatrix0\\ 0\\ (I-D)d pmatrix. This also shows directly that τ affects only the multiplier block row. A.8 Proof of Theorem 3.4 Proof. Set T0:=T01(0)T_0:=T_01^(0), T1:=T01(1)T_1:=T_01^(1), and T(τ)=T01(τ)=T0+τT1T(τ)=T_01(τ)=T_0+τ T_1. Define ΦH(τ)=H−T(τ)⊤HT(τ) _H(τ)=H-T(τ) HT(τ). If τ=ητ−+(1−η)τ+τ=η _-+(1-η) _+, direct expansion gives ΦH(τ)−ηΦH(τ−)−(1−η)ΦH(τ+)=η(1−η)(τ+−τ−)2T1⊤HT1⪰0. _H(τ)-η _H( _-)-(1-η) _H( _+)=η(1-η)( _+- _-)^2T_1 HT_1 0. (A.9) At τ=1/2τ=1/2, let H∈ℚ6×6H ^6× 6 solve H−T01(1/2)⊤HT01(1/2)=I6.H-T_01(1/2) HT_01(1/2)=I_6. Exact Sylvester tests give H≻0H 0, ΦH(49/100)≻0 _H(49/100) 0, and ΦH(51/100)≻0 _H(51/100) 0. Equation (A.9) therefore gives uniform contraction throughout [49/100,51/100][49/100,51/100]. Strict complementarity places a sufficiently small closed H-ellipsoid around the KKT point entirely inside the D01D_01 projection region, proving the first assertion. For the initialization in Theorem 2.1, a 232232-step rational componentwise enclosure is propagated uniformly on |τ−1/2|≤10−10|τ-1/2|≤ 10^-10. Every projection sign remains strict, and the entire enclosure at step 232232 lies in the preceding projection-safe ellipsoid. The first assertion then proves convergence. For the third assertion, exact factorization of the D01D_01-branch characteristic polynomial, followed by a real Schur recursion and a Sturm root count, reduces the stability boundary to the unique root in (0,1)(0,1) of pstabp_ stab in (A.10); exact endpoint evaluation gives the stated bracket. If 0<τ<τc0<τ< _c, Schur stability gives the unique Hτ≻0H_τ 0 satisfying Hτ−T01(τ)⊤HτT01(τ)=I6.H_τ-T_01(τ) H_τT_01(τ)=I_6. In the induced norm there is a κτ<1 _τ<1 such that ‖T01(τ)e‖Hτ≤κτ‖e‖Hτ.\|T_01(τ)e\|_H_τ≤ _τ\|e\|_H_τ. Since q⋆=(−1,1)q =(-1,1) is in the interior of the strict branch, a sufficiently small closed HτH_τ-ellipsoid is invariant under the full projected map. Therefore ‖wk+1−w⋆‖Hτ≤κτ‖wk−w⋆‖Hτ,\|w^k+1-w \|_H_τ≤ _τ\|w^k-w \|_H_τ, so w⋆w is locally Q-linearly attracting. Conversely, suppose that τc≤τ<1 _c≤τ<1. Since w⋆w lies in the interior of the strict D01D_01 branch, the full projected map agrees with its affine branch map on a neighborhood of w⋆w . Local Q-linear attraction on that neighborhood would imply an induced norm in which the linear part T01(τ)T_01(τ) has operator norm strictly smaller than one, and hence ρ(T01(τ))<1ρ(T_01(τ))<1. This contradicts ρ(T01(τ))≥1ρ(T_01(τ))≥ 1. Thus the KKT point is not locally Q-linearly attracting for τc≤τ<1 _c≤τ<1. ∎ A.9 Verification of Theorem 3.4 The dual-step certificate is verified by python python/certify_relaxed_multiplier_interval_theory.py. It re-derives the essential-state projection-region matrices from the original updates, checks the endpoint Sylvester minors and chord identity, propagates the exact 232232-step finite-entry enclosure, and performs the Schur–Sturm boundary test. The boundary polynomial is pstab(τ)= p_ stab(τ)= 111794210406295556649228900462157733493τ3 111794210406295556649228900462157733493\,τ^3 (A.10) +23105776975281816108275814441284422085171521τ2 +23105776975281816108275814441284422085171521\,τ^2 −244157339715898821440243649673959463071543521τ -244157339715898821440243649673959463071543521\,τ +208410060660460340386576638889814578828638507. +208410060660460340386576638889814578828638507. An exact Sturm count gives one root of pstabp_ stab in (0,1)(0,1), and rational endpoint evaluation gives the bracket in Theorem 3.4. Appendix B Period-23 Certificate This appendix records the reduced branch maps, the return-map construction, the proofs, and the invariant-ellipsoid data for the exact m=3m=3 period-2323 certificate stated in Section 4.1 (Proposition 4.1). It is a fixed-QP initialization result and does not assert robustness to perturbations of the problem data. B.1 Branch Map Record the state immediately after the z- and multiplier updates. With β=1β=1, set t=z+λ,v=(y,t)∈ℝ6,St=[ 03×3I3],t=z+λ, v=(y,t) ^6, S_t=[\,0_3× 3\ I_3\,], so that t=Stvt=S_tv. The projection identity gives z=[t]+z=[t]_+ and λ=[t]−λ=[t]_-; thus v determines (z,λ)(z,λ) and the next ADMM step. Each strict projection branch is determined by σ(t):=(sgnt1,sgnt2,sgnt3)∈−,+3.σ(t):=(sgnt_1,sgnt_2,sgnt_3)∈\-,+\^3. On a fixed branch the projection is linear, and eliminating the x- and z-updates gives Φσ(v)=Rσv+rσ,Rσ∈ℚ6×6,rσ∈ℚ6. _σ(v)=R_σv+r_σ, R_σ ^6× 6, r_σ ^6. For quadratic objectives F(x)=12x⊤Qxx+c1⊤x,G(y)=12y⊤Qyy+c2⊤y,F(x)= 12x Q_xx+c_1 x, G(y)= 12y Q_yy+c_2 y, set Kx=(Qx+A⊤A)−1,Ky=(Qy+B⊤B)−1,Hx=AKxA⊤,Hy=BKyB⊤,K_x=(Q_x+A A)^-1, K_y=(Q_y+B B)^-1, H_x=AK_xA , H_y=BK_yB , Dσ=diag(σi=+),Jσ=2Dσ−I3,Eσ=I3−Dσ,D_σ=diag\! (1_\ _i=+\ ), J_σ=2D_σ-I_3, E_σ=I_3-D_σ, ξ=Kx(A⊤b−c1),η=Ky(B⊤b−c2−B⊤Aξ).ξ=K_x(A b-c_1), η=K_y(B b-c_2-B Aξ). Direct substitution in the x-, y-, and projection–multiplier updates gives Rσ=[KyB⊤HxBKyB⊤(Hx−I3)Jσ(I3−Hy)HxBEσ+((I3−Hy)Hx+Hy)Jσ],rσ=[ηb−Aξ−Bη].R_σ= bmatrixK_yB H_xB&K_yB (H_x-I_3)J_σ\\ (I_3-H_y)H_xB&E_σ+ ((I_3-H_y)H_x+H_y )J_σ bmatrix, r_σ= bmatrixη\\ b-Aξ-Bη bmatrix. (B.1) If the QP data are rational, then every entry of RσR_σ and rσr_σ is rational. B.2 Exact Data and Certificate Apply the reduced map above to the rational instance (4.1) stated in Section 4.1. The resulting strict projection-sign word is 23W_23 in (4.2). Take ω=23=(σ0,…,σ22)ω=W_23=( _0,…, _22). Write the composition of its first j branch maps as Φω(j)(v) _ω^(j)(v) =Ljv+dj, =L_jv+d_j, L0 L_0 =I6,d0=0, =I_6, d_0=0, Lj+1 L_j+1 =RσjLj, =R_ _jL_j, dj+1 d_j+1 =Rσjdj+rσj. =R_ _jd_j+r_ _j. The full 2323-step return map is therefore v23(ℓ+1) v^23( +1) =Φper(v23ℓ), = _ per (v^23 ), (B.2) Φper(v) _ per(v) :=Φω(23)(v)=Mperv+cper, := _ω^(23)(v)=M_ perv+c_ per, Mper M_ per :=L23=Rσ22⋯Rσ0,cper:=d23. :=L_23=R_ _22·s R_ _0, c_ per:=d_23. Exact elimination shows that I6−MperI_6-M_ per is nonsingular, so define the phase-zero state by v^0:=(I6−Mper)−1cper∈ℚ6,v^j:=Ljv^0+dj(j=0,…,22). v^0:=(I_6-M_ per)^-1c_ per ^6, v^j:=L_j v^0+d_j (j=0,…,22). The complete exact rational initialization is recovered by t^0=Stv^0,z^0=[t^0]+,λ^0=[t^0]−, t^0=S_t v^0, z^0=[ t^0]_+, λ^0=[ t^0]_-, K^x:=(Q^x+A^⊤A^)−1,x^0=K^x[A^⊤(b^−B^y^22−z^22+λ^22)−c^1]. K_x:=( Q_x+ A A)^-1, x^0= K_x\! [ A ( b- B y^22- z^22+ λ^22)- c_1 ]. These equations define the exact rational initialization; the machine certificate stores its expanded numerators and denominators. Proof of Proposition 4.1. For ω=23ω=W_23, the recursion above gives the exact rational return map (Mper,cper)=(L23,d23).(M_ per,c_ per)=(L_23,d_23). Exact rational elimination solves the fixed-point system for v^0 v^0. Exact replay reconstructs every phase state and verifies the pattern, the strict margin, Φper(v^0)=v^0 _ per( v^0)= v^0, and that the 2323 phase states are pairwise distinct. Hence the sequence has minimal period 2323. Positive definiteness of Q^x Q_x and Q^y Q_y gives a unique primal solution; feasibility determines z, and nonsingularity of A A determines λ. Thus the KKT point is unique and is a fixed point of the single-valued ADMM map. It cannot belong to the nonconstant period-2323 sequence, which is therefore non-KKT. For nearby initializations, the matrix P in Appendix B.3 satisfies P−Mper⊤PMper≻0.P-M_ per PM_ per 0. It follows that the error decreases after each complete block of 2323 ADMM steps. In particular, every eigenvalue of MperM_ per lies strictly inside the unit disk. The strict sign margin guarantees that every point in ℰcertE_ cert continues to use the same 2323 affine branches. The return map sends this ellipsoid into itself, so induction gives the same projection pattern and phasewise convergence for all subsequent returns. The 2323 limiting phase points are distinct; hence the full sequence does not converge to a single point. ∎ Proof of Corollary 4.2. Set u=A^xu= Ax and w=B^yw= By. Nonsingularity of A A and B B makes the change of variables bijective and preserves the residual A^x+B^y+z−b^=u+w+z−b Ax+ By+z- b=u+w+z- b. The transformed objectives are Q~x=A^−⊤Q^xA^−1,Q~y=B^−⊤Q^yB^−1,c~1=A^−⊤c^1,c~2=B^−⊤c^2, Q_x= A^- Q_x A^-1, Q_y= B^- Q_y B^-1, c_1= A^- c_1, c_2= B^- c_2, which remain rational with Q~x,Q~y≻0 Q_x, Q_y 0. The x- and y-subproblems are invertible reparameterizations of the u- and w-subproblems, while the z- and multiplier updates are unchanged. Starting from u0=A^x0u^0= Ax^0 and w0=B^y0w^0= By^0, the two direct ADMM trajectories correspond step by step. ∎ B.3 Attracting Ellipsoid Use the partial-composition matrices LjL_j and the selector StS_t defined in Appendix B.1. Set P=12[2000−110210−220162−770023−33−1−2−7−316−131273−1314].P= 12 bmatrix2&0&0&0&-1&1\\ 0&2&1&0&-2&2\\ 0&1&6&2&-7&7\\ 0&0&2&3&-3&3\\ -1&-2&-7&-3&16&-13\\ 1&2&7&3&-13&14 bmatrix. Its leading principal minors are 1,1,114,258,33332,44132,1, 1, 114, 258, 33332, 44132, so P≻0P 0. Exact Sylvester tests also give P−Mper⊤PMper≻0.P-M_ per PM_ per 0. For the iith standard basis vector i∈ℝ3e_i ^3, define aji:=i⊤StLj∈ℚ1×6.a_ji:=e_i S_tL_j ^1× 6. Thus ajiea_jie is the perturbation of the iith projection input at phase j caused by the initial-state error e=v0−v^0e=v^0- v^0. Define the corresponding sign-preserving radius by r¯2:=min0≤j<23, 1≤i≤3aji≠0(t^ij)2ajiP−1aji⊤. r^2:= _ subarrayc0≤ j<23,\ 1≤ i≤ 3\\ a_ji≠ 0 subarray ( t_i^j)^2a_jiP^-1a_ji . The exact certificate gives r¯2>29/100000>1/4000 r^2>29/100000>1/4000; the minimum occurs at phase 1414 and coordinate t3t_3. The P-norm Cauchy–Schwarz inequality therefore preserves all 6969 projection signs for e⊤Pe<1/4000e Pe<1/4000. Since P−Mper⊤PMper≻0P-M_ per PM_ per 0, the return map leaves ℰcertE_ cert invariant. Induction gives the same branch word and phasewise convergence to the non-KKT sequence. The ellipsoid in (4.4) is sufficient but not claimed maximal. On the slice Δy=0 y=0, Δt3=0 t_3=0, 32(Δt1)2−3Δt1Δt2+8(Δt2)2<14000,Δt2=0⟹|Δt1|<16000. 32( t_1)^2-3 t_1 t_2+8( t_2)^2< 14000, t_2=0\ \ | t_1|< 1 6000. Every point in this interval is therefore a certified nonconvergent reduced initialization. B.4 Reproduction From the same GitHub certificate repository root, running python python/verify_period23_certificate.py verifies positive definiteness, nonsingularity, the unique KKT point, 2323-step closure, 2323 distinct phase states, strict projection consistency with margin >1/250>1/250, separation from the KKT point, and the exact Lyapunov inequality for the return matrix. The rational input, verifier, and generated certificate are provided in that repository. Appendix C AI-Assisted Research Records This appendix records route dossiers, workspace manifests, prompt classes, human interventions, promotion criteria, and the claim-to-artifact map. The accompanying GitHub repository https://github.com/ConanXu-math/identity-slack-admm-cycle-certificate stores certificates, verifiers, and retained research artifacts. Route dossiers live under provenance/routes/\codex-period66,kimi-period23\/. The discovery-to-verification workflow is summarized in Figure 4 (Section 4.2). Computer-assisted claims are supported by exact certificates and independently executable verifiers. Analytic claims are established by the proofs in the paper, with symbolic scripts used to check the stated algebraic and spectral predicates where applicable. Language-model transcripts carry no independent evidentiary weight. C.1 Prompt Records Archived prompts are classified into three provenance classes: • verbatim historical — retained exactly as issued; • reconstructed — rebuilt from retained transcripts or notes; • retrospective distilled — written after the fact to summarize a stage. No distilled prompt is asserted to be a historical input. Route dossiers and retained prompt artifacts live under provenance/routes/ in the GitHub repository; a consolidated prompt ledger may be archived alongside them. Representative entries (full texts in the GitHub route dossiers): P-K-01 (verbatim_historical), Kimi opening goal; P-C-01 (reconstructed), Codex brief; P-C-02 (retrospective_distilled), τ-only relaxation with explicit quantifier scopes. C.2 Human Interventions We distinguish human-specified, AI-proposed, and jointly refined steps. Human direction selected the mathematical question, configured the workspace and tools, set claim scopes, and audited literature; model outputs nominated representations, candidates, and certificate designs. Table 2 records the staged attribution for the Codex route. Attributions are based on the retained transcripts; where a representation emerged during the interaction and cannot be attributed to a single side, it is labeled jointly refined. Typical Codex gates include rejecting fixed-branch instability alone as a counterexample, requiring exact rational closure, and separating KKT-local from class-uniform claims in the τ-extension. The Kimi dossier records the blind opening and subsequent continue-search instructions under the frozen workspace rules. Route dossiers under provenance/routes/ retain the supporting artifacts. Table 2: Recorded stages of the Codex route, with the promotion condition that gated each stage. Stage Human input AI output Promotion condition Problem setup Identity-slack question; evidence requirements Proof and counterexample route proposals Correct literature boundary State-space reduction Request for a certifiable representation Signed reduced state s=(y,q)s=(y,q), q=z+λq=z+λ (jointly refined) Equivalence to raw ADMM Candidate screening Requirement that fixed-branch instability alone is insufficient Fixed-branch spectra; near-rotation heuristic Projection itinerary must be realizable Period search Requirement of a strict exact counterexample Rational data (2.3)–(2.5) and word (00)2(01)64(00)^2(01)^64 Exact rational closure and strict signs (G1G_1–G7G_7) Relaxation extension Frozen QP and initialization; vary only τ; quantifier gates Parameterized branch maps; candidate step regimes; certificate routes Quantifier audit and exact certificates C.3 Promotion Gates A candidate becomes a mathematical claim only after passing G1 G_1 :exact problem data, :\ exact problem data, G2 G_2 :KKT and subproblem validity, :\ KKT and subproblem validity, G3 G_3 :exact period closure, :\ exact period closure, G4 G_4 :strict branch admissibility, :\ strict branch admissibility, G5 G_5 :raw ADMM replay, :\ raw ADMM replay, G6 G_6 :minimal-period verification, :\ minimal-period verification, G7 G_7 :quantifier and literature audit. :\ quantifier and literature audit. Gates G1G_1–G6G_6 are mathematical or computational checks, carried out in exact arithmetic where applicable; G7G_7 is a human audit of quantifiers, novelty, and literature scope. Floating-point orbits and model-generated derivations nominate candidates only. For stability claims, G3G_3–G6G_6 are replaced by the relevant Lyapunov, Schur, Sturm, or interval predicates. In particular, KKT-local convergence, convergence from a specified initialization, fixed-problem guarantees, and class-uniform guarantees are treated as distinct promotion scopes rather than stylistic qualifications. C.4 Claim-to-Artifact Map Verification entry points in the GitHub repository (first exact replayable certificate as endpoint): Claim Verification command Theorem 2.1 python python/verify_certificate_pair.py Theorem 3.4 python python/certify_relaxed_multiplier_interval_theory.py Theorem 3.2 python python/verify_universal_step_obstruction.py --check Proposition 4.1 python python/verify_period23_certificate.py One-shot acceptance (66+23) python python/verify_all.py The computer-assisted claims listed in the table rest on these exact replays, independently of language-model transcripts.