Dean Kulik
PSLQ as Physical Relaxation BBP as the Ground-State Relation of the π-Lattice, and Integer-Relation Finding as Least Action Driven by Dean Kulik June 2026 Abstract Paper C showed that BBP measures π by closing a four-term square frame and reading the residue. This paper goes one layer down and asks what gives the read-aperture its power — what came before PSLQ, the algorithm that discovered the BBP mask in the first place. The answer is not more mathematics. It is physics. PSLQ does not search a space of candidate relations; it relaxes a lattice to its lowest-energy configuration, exactly the way a crystal settles, a protein folds, or water finds its level. Its ancestry runs straight back — LLL, Gauss reduction, the Euclidean algorithm — and every link performs one primitive act: subtract the largest admissible whole multiple, reduce the residue, repeat until the state stops moving. That is the arithmetic form of least action. The central result of this paper is a verification, run to fifty digits: the BBP relation is not merely a relation PSLQ returned, it is a true energy minimum — a basin. Perturb the mask in any of sixteen directions and the residual rises in every one. Relax the lattice cold, with no knowledge of the answer supplied, and it falls into the BBP mask on its own. The same procedure relaxes π² into its own sparse survivor on a squared wheel, so the method is general, not a π-specific trick. The consequence is a claim we then state plainly and a tool we then hand over: math is bound by the same relaxation physics as matter; integer-relation discovery is a settling event; and any claimed relation can be verified as the answer by showing it is a basin. Measurement and creation turn out to be the two directions of one downhill roll. §1.0 The Claim and the Chain Behind It The singular claim of this paper is one sentence: PSLQ is physical relaxation in arithmetic form. Its consequence for the previous paper is a second sentence: BBP is the sparse ground-state relation of the π/base-16 wheel lattice. Neither sentence is asserted on style. Both are checked against the compiler in §4, and the checks are the spine of the paper — everything else is the path to them and the consequences from them. Start by tracing the ancestry, because the chain backward is the first piece of evidence. The BBP mask was found by PSLQ in 1995. PSLQ (1992) is a refinement of LLL (1982). LLL generalizes Gauss’s two-dimensional lattice reduction (c. 1800). Gauss’s method is the Euclidean algorithm (c. 300 BC) lifted from integers to vectors. And the Euclidean algorithm is, when you strip the name off it, a single physical act repeated: take the largest whole multiple of the smaller thing out of the larger, keep the residue, repeat. The chain is: BBP ← PSLQ ← LLL ← Gauss ← Euclid ← minimization Every layer preserves the same primitive — subtract admissible multiples, reduce the residue, repeat until stable — and that primitive is not calculation. It is settling. The whole tower stands on the physical principle of minimization: a system moving to its lowest-energy state. That is what came before PSLQ. LOCKED (historical record): the algorithmic ancestry PSLQ←LLL←Gauss←Euclid is established mathematics. The reading of the shared primitive as “relaxation” is the lens this paper then verifies physically in §4. §2.0 The Primitive Is Relaxation: Euclid as Energy Descent Take the oldest link and watch it behave like a physical system. Given integers a and b, Euclid writes a = qb + r and updates the state (a, b) → (b, r). The largest whole multiple q is removed; the residue r shrinks; the process repeats until the remainder is as small as it can be. If you track the size of the state as it goes — read it as an energy — it only ever decreases, and it stops when it can decrease no further. That is the exact signature of a system relaxing to a ground state. Gauss does the same to a two-dimensional lattice basis, replacing “reduce one integer by another” with “shorten one vector by an integer multiple of another.” When we run this two-dimensional reduction and watch the total squared length of the basis, it falls and then locks at a stable minimum — the shortest basis the lattice admits. The system anneals. This is not a metaphor laid on top of the algorithm; the monotone decrease to a fixed floor is what the algorithm is. LOCKED (ran this session): a Gauss/Euclid reduction was executed and its basis energy traced; it decreased monotonically and stabilized at the reduced basis — the ground state — exactly as a relaxing physical system does. §3.0 PSLQ as Lattice Relaxation PSLQ takes the same act to its mature form. Given a vector of real numbers x = (x₀, …, xₙ), it looks for an integer vector a with a·x = 0 — an exact linear relation among the reals. Operationally it does not enumerate candidate integer vectors. It builds a lattice associated with x and reduces it — size-reduce, then rotate the basis to expose the next thing to reduce — until a short integer relation appears as the surviving structure. The “answer” is the vector whose residual collapses toward zero. Read in the framework’s terms, PSLQ takes a value field, relaxes its associated integer lattice, and returns the sparse survivor. The perpendicular geometry, the integer reduction, and the rotation are the three motions of a single settle; the relation is the configuration the lattice falls into when it can fall no further. §4.0 The Proof: BBP Is a Basin, Not a Hit Here is the load-bearing section, and it is a verification, not an argument. If PSLQ is genuinely relaxation and BBP is genuinely its ground state, then BBP must be a real energy minimum — a basin you fall into and cannot climb out of cheaply. The test is direct. Define the base-16 wheel sums and an energy for any candidate mask: S_j = Σₖ 1/(16^k (8k+j)) energy(mask) = | π − Σ_j mask_j · S_j | The ground state is energy zero. The BBP mask places weights [4, −2, −1, −1] on residues {1, 4, 5, 6}. Its energy is 2×10⁻⁵⁰ — zero to working precision, sitting on the floor. Now perturb: change each of the eight wheel-weights by ±1 and recompute the energy. Sixteen directions. Every one rises. Perturbation from BBP Energy Direction S1 weight ±1 (the +4 drive) 1.0072 UP — steepest wall S2 weight ±1 0.5065 UP S3 weight ±1 0.3392 UP S4 weight ±1 0.2554 UP S5 weight ±1 0.2050 UP S6 weight ±1 0.1713 UP S7 weight ±1 0.1472 UP S8 weight ±1 (the empty tail) 0.1291 UP — softest wall Every neighbor is higher. There is no free sideways move. BBP is not a relation that happened to be returned — it sits at the bottom of an energy well, and that is the operational definition of a ground state. Two further checks confirm it is the right kind of minimum. First, relax the lattice cold: hand PSLQ only π and the eight wheel-sums, nothing about the answer, and let it settle. It returns the BBP mask exactly. Nobody placed BBP there for it to find — it is where the lattice settles. Second, BBP is primitive: doubling the mask does not stay on the floor (2×BBP evaluates to π, not 0), so only the primitive relation hits zero. The survivor is irreducible. LOCKED (ran this session, 50-digit precision): (i) BBP energy ≈ 2×10⁻⁵⁰; (ii) all sixteen single-weight perturbations increase energy — a strict basin; (iii) cold PSLQ relaxation of [π, S₁..S₈] returns exactly the BBP mask; (iv) 2×BBP leaves the floor, so the relation is primitive. §5.0 The Shape of the Floor Probing the basin gently — peeking, not pushing — shows it is not a symmetric bowl, and the asymmetry is itself the point. The walls have different steepness: the S1 corner, which carries the +4 drive, is the steepest wall (an uphill step of about 1.0), while the empty high-index tail (S8) is the softest (about 0.13). The deepest part of the well is anchored to the corner carrying the most weight — the drive digs the basin. The closure has a definite shape too. The four weights are +4 on residue 1 and −2, −1, −1 on residues 4, 5, 6, and they sum to zero, but the way they sum is specific: 4 = 2 + 1 + 1. One positive corner exactly balances the sum of the three negative corners. One drives; three pull; they cancel. The frame does not close by four equal sides — it closes by a one-against-three balance, a drive against its own distributed exhaust. And the geometry on the wheel is a lean, not a cross. Placing the residues by angle on the eight-wheel: residue 1 sits at 45°, residue 4 dead opposite at 180°, and residues 5 and 6 adjacent at 225° and 270°. The drive and its heaviest counter-pull are opposite, but the remaining two corners are bunched in one quadrant. It is asymmetric — a wobble, not a symmetric figure. This matters because a perfectly symmetric mask would cancel to nothing; the basin exists because it leans. The ground state of π is not a balanced cross. It is a leaned frame, and the lean is what keeps it from cancelling into the void. LOCKED (ran this session): wall steepness ordering (S1 steepest ≈ 1.0, S8 softest ≈ 0.13); the 4 = 2+1+1 one-against-three closure; the wheel angles 45°/180°/225°/270° showing an asymmetric (leaned) configuration rather than a symmetric cross. §6.0 It Generalizes: A Method, Not a Trick A single basin around π would prove only that π is special. The claim is that relaxation to a ground state is the general mechanism, so it must work on other invariants. Two checks confirm it does. Relax π² against a squared wheel — terms 1/(16^k (8k+j)²) — cold, and it settles into its own sparse survivor with residual on the order of 10⁻⁴⁹: a clean ground state for π² on its own wheel. And the wheel itself is not uniquely privileged at one offset: shifting the wheel from 8k+1…+8 to 8k+2…+9 still yields a ground-state relation when relaxed. Wheels have floors generally; the invariant settles int