Motivation

Discrepancy theory asks how unbalanced a two-coloring of a combinatorial structure must be in the worst case. Concretely: given $n$ sets over an $n$-element ground set, color each element $+1$ or $-1$ so that every set is as close to balanced as possible. The question is classical (Beck–Fiala 1981; Spencer 1985) and the answer for general (dense) set systems is one of the sharpest gaps between a naive probabilistic bound and the truth known in combinatorics: assigning colors uniformly at random only guarantees discrepancy $\Theta(\sqrt{n\log n})$, yet a coloring with discrepancy $O(\sqrt n)$ always exists — the logarithmic factor is an artifact of the naive argument, not of the problem. This mission formalizes that removal, following T. Rothvoß's lecture-note exposition of J. Spencer's entropy method (MIT 18.095, "Discrepancy theory"), the standard modern presentation of the technique (see also Matoušek, Geometric Discrepancy, Ch. 4). The entropy method is the ancestor of the whole "partial coloring" family of arguments used throughout discrepancy theory and combinatorial algorithm design, so a machine-checked account of its base case is reusable well beyond this one theorem.

Setting

Fix $n\ge 1$ and an $n\times n$ matrix $A$ with entries in $\{0,1\}$, thought of as the incidence matrix of $n$ sets $S_1,\dots,S_n$ over an $n$-element ground set: $A_{ij}=1$ iff element $j$ lies in set $S_i$. A coloring is a map $\varepsilon:\{1,\dots,n\}\to\{-1,+1\}$, and the discrepancy of row $i$ under $\varepsilon$ is $\bigl|\sum_j A_{ij}\varepsilon_j\bigr|$, the signed imbalance of set $S_i$. The discrepancy of the matrix is the value achieved by the best coloring, minimizing the worst row.

The entropy method bounds this via the partial coloring lemma: rather than coloring all $n$ elements at once, one repeatedly colors a constant fraction of the currently uncolored elements while keeping every row's contribution small, then recurses on what remains. Each round is itself produced by an entropy/pigeonhole argument: quantize each row's signed sum (under a uniformly random coloring) into $O(1)$ "shells" of width $\Theta(\sqrt m)$ (where $m$ is the number of active elements); a short computation shows this quantization carries very little Shannon entropy $H(Z)=\sum_x \Pr[Z=x]\log_2\frac{1}{\Pr[Z=x]}$ once the shell width exceeds a threshold; subadditivity of entropy across the $n$ rows then bounds the joint quantization entropy, which by pigeonhole forces an exponentially large set of colorings landing in the same joint shell; Kleitman's theorem on the diameter of a large subset of the Hamming cube then extracts two such colorings that are far apart in Hamming distance, and their difference is the sought partial coloring.

Formalization targets

Goal.

$$\exists\, C\in\mathbb R,\ \forall n\ge 1,\ \forall A\in\{0,1\}^{n\times n},\ \exists\,\varepsilon\in\{-1,1\}^n,\ \forall i,\ \Bigl|\sum_{j=1}^n A_{ij}\varepsilon_j\Bigr|\le C\sqrt n.$$

This is the qualitative, constant-suppressed form of Spencer's theorem: it asserts $O(\sqrt n)$ discrepancy with a single universal constant, and deliberately leaves that constant unspecified. This is the right goal for this mission because it is the weakest statement that is still stable: any future improvement to the constant (down to Spencer's sharp $6$, or beyond) refines this theorem rather than invalidating it.

Significance

The removal of the $\sqrt{\log n}$ factor is the entire content of Spencer's theorem: it is what separates discrepancy theory from a corollary of concentration inequalities, and the partial-coloring/entropy method it introduced underlies later results throughout the field (Beck–Fiala-type bounds, the Komlós conjecture literature, and constructive/algorithmic discrepancy minimization). Formalizing it is formalizing the base case that every later partial-coloring argument specializes.

This mission's goal theorem, spencer_discrepancy_sqrt_n_bound, is already proved (zero sorrys), by a from-scratch entropy-method development: the per-row shell-entropy bound, the joint pigeonhole-and-Kleitman assembly for one round, and the outer geometric iteration and induction combining rounds into a full coloring. What remains open in this mission is shannonEntropy_shellFin_le (Lemma 9 in Rothvoß's notes) — the per-row entropy bound is currently imported as an assumption by the one-round lemma lemma8_partial_coloring_round, which is therefore only conditionally proved pending it. A separate, harder mission on this platform (Komlos.spencer_six_deviations) targets Spencer's sharp constant $6$ via a tighter, non-standard numeric derivation; that is a distinct, substantially harder target and this mission does not duplicate it.

Difficulty

The obvious argument is: fix a target bound $t=\lambda\sqrt n$, use a Chernoff/Hoeffding bound to show each row fails with probability at most $2e^{-\lambda^2/2}$, union-bound over the $n$ rows, and take a coloring outside the bad event. This works to prove a single good coloring exists — but it is not strong enough to survive being iterated to remove the entire uncolored set, because a per-row union bound loses a factor of $n$ that a fixed $\lambda$ cannot always absorb once the active column count $m$ is close to $n$: for the scaling family where the row count and the active set shrink together, the naive union bound's failure probability grows linearly in $m$, not exponentially, exactly canceling the exponential decay one is trying to exploit. The fix is to bound the joint entropy of all $n$ rows' quantizations at once (subadditivity of Shannon entropy), rather than union-bounding row-by-row failure events; this is genuinely a different technique, not a tightening of the same one, and it is the reason the entropy method is presented as its own tool rather than a Chernoff-bound corollary.

Formalization scope

Matrices are Fin n → Fin n → ℝ with an explicit ∀ i j, A i j = 0 ∨ A i j = 1 hypothesis; colorings are represented two ways in this development — Fin m → Bool internally (via the platform definition RSign converting to $\pm1$) during the entropy/Kleitman argument, and directly as Fin n → ℝ constrained to $\{-1,1\}$ pointwise in the goal theorem's statement, matching the usual $\{\pm1\}$-coloring convention. The row-sum shell quantization is the platform definitions rowSumB, shellIdx, shellFin (an integer-valued "round to nearest shell" construction, packaged into a fixed Fin (2m+3) type for entropy purposes). The active column set during the outer iteration is tracked as a shrinking Finset (Fin n) of the original index type throughout, rather than moving between different Fin m types round to round, which keeps the induction free of type-level bookkeeping.

Reusable, already-Proved infrastructure this development builds on: shannonEntropy_pi_le (subadditivity across independent rows), shannonEntropy_pigeonhole, choose_sum_le_exp_mul_binEntropy, and kleitman_diameter, all already Proved on the platform independent of this mission. The one genuinely open piece — and the mission's standing invitation — is shannonEntropy_shellFin_le (Lemma 9): a self-contained Shannon-entropy computation about the shellFin quantization that does not depend on anything else in this mission and can be attempted independently.

Selected references

• J. Spencer, Six standard deviations suffice, Trans. Amer. Math. Soc. 289 (1985), 679–706. DOI
• T. Rothvoß, Discrepancy theory, or: how much balance is possible?, MIT 18.095 lecture notes. PDF
• J. Matoušek, Geometric Discrepancy: An Illustrated Guide, Algorithms and Combinatorics 18, Springer, 1999.
• J. Beck, T. Fiala, "Integer-making" theorems, Discrete Appl. Math. 3(1) (1981), 1–8.