IPS_{Lin'} in Algebraic Proof Systems

• IPS_{Lin'} is a fragment of the Ideal Proof System defined by requiring linearity solely in non-Boolean axiom slots, highlighting key complexity constraints.
• The system employs a structured polynomial decomposition that, when represented as sums of ROABPs, leads to exponential width lower bounds in refutations.
• Extensions to multilinear variants and adaptations for positive characteristic fields underscore the model's broad significance in algebraic proof complexity.

Searching arXiv for the specific IPS_{Lin'} lower-bounds paper and closely related IPS background papers. First, finding the main 2025 paper on IPS_{Lin'}. Searching for Grochow–Pitassi's Ideal Proof System paper. Searching for FSTW21 / the functional lower-bound reference mentioned in the provided material. Searching for the STOC 2024 Hakoniemi–Limaye–Tzameret paper and related positive-characteristic references. IPS$_{\mathrm{Lin}'}$ is a fragment of the Ideal Proof System (IPS) in algebraic proof complexity in which the refutation polynomial is required to be linear only in the non-Boolean axiom slots. In the formulation studied by Chatterjee, Ghosal, Mukhopadhyay, and Sinhababu, IPS$_{\mathrm{Lin}'}$ is analyzed through lower bounds against restricted algebraic models, most notably sums of ROABPs, yielding an exponential-size lower bound for refutations of a variant of the subset-sum axiom polynomial over fields of characteristic zero, with corresponding extensions to positive characteristic after suitable modification of the target polynomial (Chatterjee et al., 13 Jul 2025).

1. Placement within the Ideal Proof System

The full IPS of Grochow and Pitassi refutes the unsatisfiability of a polynomial system

$$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$

by an algebraic circuit $P(X,Y,Z)$ satisfying

$P(X,0,0)=0,$

and

$$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$

(Chatterjee et al., 13 Jul 2025). Here $Y=(y_{1},\dots,y_{m})$ are fresh axiom-slot variables for the non-Boolean axioms, and $Z=(z_{1},\dots,z_{n})$ are slots for the Boolean axioms.

Within this framework, IPS$_{\mathrm{Lin}}$ imposes linearity in all axiom slots: $$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$ IPS$_{\mathrm{Lin}'}$0 weakens this by requiring linearity only in the non-Boolean axiom slots: $_{\mathrm{Lin}'}$1 (Chatterjee et al., 13 Jul 2025).

A common source of confusion is to treat IPS$_{\mathrm{Lin}'}$2 as identical to IPS$_{\mathrm{Lin}'}$3. It is not: the primed system drops the linearity requirement on the Boolean-slot variables $_{\mathrm{Lin}'}$4, and this distinction is central to the lower-bound regime established in the 2025 paper.

2. Formal structure of IPS$_{\mathrm{Lin}'}$5

Because the $_{\mathrm{Lin}'}$6-variables occur only linearly, an IPS$_{\mathrm{Lin}'}$7 proof can be written in the form

$_{\mathrm{Lin}'}$8

with the condition that substituting $_{\mathrm{Lin}'}$9 and $$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$0 yields the constant polynomial $$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$1 (Chatterjee et al., 13 Jul 2025). The size of the proof is the size of the smallest algebraic circuit computing $$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$2.

The same work also isolates a multilinear variant. A proof $$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$3 in IPS$$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$4 is in %%%%75$_{\mathrm{Lin}}$75%%%%6 if, in addition,

$$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$7

is multilinear in $$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$8 (Chatterjee et al., 13 Jul 2025). This auxiliary restriction is used for a separate lower-bound theorem: the paper proves a nearly quadratic-size formula lower bound for $$\{ f_{1}(X)=0,\dots,f_{m}(X)=0,\ x_{1}^{2}-x_{1}=0,\dots,x_{n}^{2}-x_{n}=0\}$$9-IPS$P(X,Y,Z)$0 on multilinear refutation over the Boolean hypercube of a variant of the subset-sum axiom polynomial, and also obtains a nearly matching qualitative statement for a constant degree target polynomial (Chatterjee et al., 13 Jul 2025).

This division between IPS$P(X,Y,Z)$1 and $P(X,Y,Z)$2-IPS$P(X,Y,Z)$3 is methodologically important. The former is studied against sums of ROABPs, while the latter is studied against formulas.

3. Hard instance: the subset-sum style axiom polynomial

The lower bound for IPS$P(X,Y,Z)$4 is built on a quadratic analogue of the usual linear subset-sum polynomial. The variables are

$P(X,Y,Z)$5

and the scalar is fixed as

$P(X,Y,Z)$6

The axiom polynomial is

$P(X,Y,Z)$7

(Chatterjee et al., 13 Jul 2025).

Over the Boolean hypercube $P(X,Y,Z)$8 and $P(X,Y,Z)$9, one has

$P(X,0,0)=0,$0

so after shifting by $P(X,0,0)=0,$1 the polynomial is never zero. Consequently,

$P(X,0,0)=0,$2

is unsatisfiable on the Boolean cube, and the theorem is stated for the fully Booleanized system

$P(X,0,0)=0,$3

(Chatterjee et al., 13 Jul 2025).

The choice of this polynomial is not incidental. The paper uses it as the canonical hard instance for translating a proof lower bound into a functional lower bound for the Boolean-cube inverse $P(X,0,0)=0,$4, modulo the Boolean axioms.

4. Main lower bound against sums of ROABPs

The principal theorem for IPS$P(X,0,0)=0,$5 is formulated over fields of characteristic zero. Let $P(X,0,0)=0,$6 be any such field, let $P(X,0,0)=0,$7 be the polynomial above, and consider IPS$P(X,0,0)=0,$8 refutations of

$P(X,0,0)=0,$9

Then, for some absolute $$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$0, any IPS$$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$1 refutation realized as a sum of ROABPs must have total width at least

$$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$2

(Chatterjee et al., 13 Jul 2025).

The same theorem states more: if the refutation polynomial is written as

$$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$3

then the $$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$4's are computed by a sum of multilinear ROABPs, and the same lower bound applies (Chatterjee et al., 13 Jul 2025).

Two clarifications are essential. First, the theorem is not a lower bound for full IPS. It is a lower bound for the fragment IPS$$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$5 under the additional representation restriction that the proof be realized as a sum of ROABPs. Second, the quantitative conclusion is stated as an exponential lower bound on total width, not merely on the number of summands.

This result extends to fields of positive characteristic when the target polynomial is suitably modified, and the modification is described as being inspired by recent results of Hakoniemi, Limaye, and Tzameret, and of Behera, Limaye, Ramanathan, and Srinivasan (Chatterjee et al., 13 Jul 2025).

5. Proof architecture

The proof strategy is organized around a functional-to-circuit reduction. In the form quoted in the paper, if $$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$6 is such that $$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$7 is unsatisfiable on $$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$8, and if $$P(X,f_{1}(X),\dots,f_{m}(X),x_{1}^{2}-x_{1},\dots,x_{n}^{2}-x_{n})=1$$9 is any class of $Y=(y_{1},\dots,y_{m})$0-polynomials closed under partial assignments, then the nonexistence of a polynomial $Y=(y_{1},\dots,y_{m})$1 satisfying

$Y=(y_{1},\dots,y_{m})$2

implies that no IPS$Y=(y_{1},\dots,y_{m})$3 or IPS$Y=(y_{1},\dots,y_{m})$4 proof of unsatisfiability lies in $Y=(y_{1},\dots,y_{m})$5 (Chatterjee et al., 13 Jul 2025). Concretely, an IPS$Y=(y_{1},\dots,y_{m})$6 refutation yields a polynomial

$Y=(y_{1},\dots,y_{m})$7

which is multilinear and agrees with $Y=(y_{1},\dots,y_{m})$8 on the Boolean cube.

The next step is multilinearization. The paper states that any ROABP can be efficiently turned into a multilinear ROABP computing the Boolean-cube restriction of $Y=(y_{1},\dots,y_{m})$9, so the lower bound may be proved against multilinear ROABPs without loss in the relevant sense (Chatterjee et al., 13 Jul 2025).

The central hardness measure is the partial-derivative-matrix rank. In characteristic zero, the unique multilinear polynomial $Z=(z_{1},\dots,z_{n})$0 satisfying

$Z=(z_{1},\dots,z_{n})$1

has full partial-derivative-matrix rank

$Z=(z_{1},\dots,z_{n})$2

on every balanced partition of $Z=(z_{1},\dots,z_{n})$3 into $Z=(z_{1},\dots,z_{n})$4 variables. The paper states this as

$Z=(z_{1},\dots,z_{n})$5

for every balanced partition $Z=(z_{1},\dots,z_{n})$6 of $Z=(z_{1},\dots,z_{n})$7 (Chatterjee et al., 13 Jul 2025).

This is contrasted with a low-rank phenomenon for sums of ROABPs. Adapting the lower-bound method of Chatterjee, Kush, Saraf, and Shpilka, the paper states that if one samples a balanced partition $Z=(z_{1},\dots,z_{n})$8 at random, then any sum of $Z=(z_{1},\dots,z_{n})$9 multilinear ROABPs of maximum width $_{\mathrm{Lin}}$0 satisfies

$_{\mathrm{Lin}}$1

with overwhelming probability, for parameters $_{\mathrm{Lin}}$2, provided $_{\mathrm{Lin}}$3 remains small (Chatterjee et al., 13 Jul 2025).

Combining the full-rank property of the target function with the random-partition low-rank upper bound yields a contradiction unless the total width is exponential. The argument is summarized in the paper as forcing

$_{\mathrm{Lin}}$4

and hence $_{\mathrm{Lin}}$5, from which the stated $_{\mathrm{Lin}}$6 lower bound follows (Chatterjee et al., 13 Jul 2025).

6. Characteristic dependence, related fragments, and interpretive cautions

The characteristic-zero theorem and the positive-characteristic extensions are described as almost identical except for the ingredient establishing the high-rank property of the target multilinear inverse. In positive characteristic $_{\mathrm{Lin}}$7, one route uses a field extension $_{\mathrm{Lin}}$8, choosing $_{\mathrm{Lin}}$9 in a sufficiently large extension so as to preserve the full-rank behavior of the corresponding inverse $$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$0 over $$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$1. A second route, attributed in the paper to Hakoniemi, Limaye, and Tzameret for $$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$2, uses an alternate vector-invariant polynomial of fourth degree in $$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$3, again with full rank on every balanced partition (Chatterjee et al., 13 Jul 2025).

The paper’s contributions therefore separate into two lower-bound directions. One concerns $$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$4-IPS$$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$5, where the proof model is formula size and the lower bound is nearly quadratic. The other concerns IPS$$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$6, where the proof model is a sum of ROABPs and the lower bound is exponential in total width (Chatterjee et al., 13 Jul 2025). These are distinct statements about distinct fragments and proof representations.

A further interpretive caution is that the results do not amount to a general lower bound for arbitrary algebraic proofs of the same contradiction. They isolate a specific linear-in-the-non-Boolean-slots fragment and a specific circuit class. What they do show is that, for the subset-sum style axiom polynomial used in the paper, this fragment already exhibits strong proof-complexity barriers under natural algebraic restrictions. This suggests that linearity in only the non-Boolean axiom slots remains a nontrivial constraint, even though it is weaker than the requirement defining IPS$$\deg_{y_i} P \le 1 \quad\text{and}\quad \deg_{z_j} P \le 1.$$7 (Chatterjee et al., 13 Jul 2025).