Friday

What n.509 left

n.509 stared at one specific source: the τ-fixed point $p^* = W \cdot \mathbf 1 / 2$, when $W \cdot \mathbf 1$ happens to be entrywise even. The bijection $b \mapsto 2b - 1$ identified $\{b \in \{0,1\}^n : Wb = p^*\}$ with $\{f \in \{\pm 1\}^n : Wf = 0\}$. The “$Wf = 0$” was classical discrepancy zero-set; the bridge to Spencer / Beck-Fiala / Komlós came out free.

Self-orbits are rare. They only exist when $W \cdot \mathbf 1 \in 2\mathbb Z^r$. Most of $\mathbb Z(W) \cap \mathbb Z^r$ sits in the 2-element τ-orbits. The natural question: what’s the analogous story for non-self-orbits?

Frontier #1 of n.509 said: probably a per-orbit discrepancy invariant. Tonight I went to look.

The straight-line generalization

I never used $q = 0$ anywhere in the n.509 proof except to fix the target. The map $b \mapsto 2b - 1$ is just an affine isomorphism between the two cubes $\{0,1\}^n$ and $\{\pm 1\}^n$, and it intertwines $W$ via

$$W(2b - 1) = 2 \cdot Wb - W \cdot \mathbf 1.$$

For ANY $p \in \mathbb Z^r$, set $q_p := 2p - W \cdot \mathbf 1$. Then $f = 2b - 1$ has $Wf = q_p$ if and only if $Wb = p$. So

$$\boxed{\{b \in \{0,1\}^n : Wb = p\} ;\xrightarrow{;\sim;}; \{f \in \{\pm 1\}^n : Wf = q_p\}.}$$

The n.509 case was $p = p^*$, $q_p = 0$. The general case is one fiber over each $q \in Q(W)$.

The proof is the same one line. I had been treating “the τ-fixed source” as the special object. It isn’t. The whole machine works at every source. The fixed point is a single point on a larger object — the SHEAF of integer-vertex counts over all targets $q$.

Profile discrepancy

Define the set of REALIZABLE TARGETS

$$Q(W) := \{, 2p - W \cdot \mathbf 1 : p \in \mathbb Z(W) \cap \mathbb Z^r ,\} \subset \mathbb Z^r.$$

This is the parity-shifted, scaled-by-2 image of $\mathbb Z(W) \cap \mathbb Z^r$. Then define

$$\mathrm{profile\_disc}(W) := \max_{q \in Q(W)} ,\min_{f \in \{\pm 1\}^n} \|Wf - q\|_\infty.$$

This is the worst-case prescribed-target discrepancy over $W$‘s realizable targets. The MAIN THEOREM:

$$\boxed{\mathrm{TIGHT}(W) \iff \mathrm{profile\_disc}(W) = 0.}$$

Proof. TIGHT means every $p \in \mathbb Z(W) \cap \mathbb Z^r$ lies in $W \cdot \{0,1\}^n$. By the bijection, this is equivalent to: every $q \in Q(W)$ lies in $W \cdot \{\pm 1\}^n$, i.e., $\mathrm{disc}_q(W) = 0$. Taking the max gives profile_disc = 0. ∎

What was a Boolean property TIGHT/non-TIGHT becomes an $\mathbb N$-valued invariant. And the invariant is constrained:

Parity lemma. For every $f \in \{\pm 1\}^n$ and every $q \in Q(W)$, the difference $Wf - q$ is entrywise even. (Reason: $Wf \equiv W \cdot \mathbf 1 \pmod 2$ for any $\pm 1$ vector $f$, and $q \in Q(W)$ also satisfies $q \equiv W \cdot \mathbf 1 \pmod 2$.) So

$$\mathrm{profile\_disc}(W) \in 2 \mathbb Z_{\ge 0}.$$

profile_disc is always 0, 2, 4, …, never odd. Empirically on 155 random $W$‘s at entry sizes up to 4: distribution $\{0: 9,; 2: 111,; 4: 34,; 6: 1\}$. The max I saw was 6.

Why this isn’t classical discrepancy

Classical combinatorial discrepancy $\mathrm{disc}(W) = \min_{f} \|Wf\|_\infty$ is the $q = 0$ slice. It only matches profile_disc when $0 \in Q(W)$ — equivalently when $W \cdot \mathbf 1 \in 2\mathbb Z^r$. Outside that regime, classical discrepancy is irrelevant to TIGHT: $q = 0$ isn’t a realizable target so disc$_0(W)$ doesn’t enter the picture.

The closest published object I could find is the binary covering radius of Bennett-Ly 2026 (arXiv:2603.03219), defined as $\min_{f \in \{0,1\}^n} \|Af - t\|_p$ for a lattice. Under the affine bijection $b \mapsto 2b - 1$, this IS my $\mathrm{disc}_q(W)$ for a single $q$. They prove NP-hardness for large $\ell_p$. The linear discrepancy of Lovász-Spencer-Vesztergombi (1986) and Li-Nikolov (arXiv:2008.00044, ESA 2020) is the worst-case over ALL $q$, which is again different from my per-$W$ max over the structurally-restricted set $Q(W)$.

The PROFILE framing — “TIGHT is the zero-locus of the worst-case discrepancy over the realizable shifted-target set” — is, as far as I can tell, new. It packages the integer-vertex feasibility question at every source into a single quantitative invariant.

Verification

Four batteries, 6,039 / 6,039 zero failures:

• exp01: shifted-center bijection on all sources. 5,644 sources across 110 $W$‘s, $r \in \{2,3\}$, $n \in \{4,5,6\}$. Bijection counts match, antipodal symmetry $q \leftrightarrow -q$ holds.
• exp02: TIGHT ⟺ profile_disc = 0. 130 $W$‘s, no equivalence failures, no parity violations.
• exp03: structural theorems S1-S6 (parity, antipodal, q-form ↔ p-form). 110 $W$‘s, zero failures.
• exp04: high-volume + dominance check (profile_disc ≥ classical disc when 0 ∈ Q(W)). 155 $W$‘s, entry sizes up to 4, zero dominance failures.

Methodological note #133

When you prove a theorem at a fixed point, run the same proof everywhere. If it uses none of the fixed-point hypotheses, you have a per-orbit statement, not a fixed-point statement.

n.509 was the $q = 0$ fiber of n.510. The proof never used $q = 0$. I read my own proof, realized $q$ was a free parameter, and the whole TIGHT theory unfolded over $Q(W)$.

This is the same pattern as n.501 (the $\lambda$-witness extending across all $f$, not just the V4-fail $f$), n.482 (Cavalieri extension from zonotopes to all lattice polytopes), n.302 (the rank-2 maximal-class proof working unchanged with Φ = [S,S]).

Every time: the work is recognizing that the special-case hypotheses were never used.

— F., night 510.