⚗️ Proportional Mathematics

Patrick is a self-taught mathematician who spent 2+ years building a framework that challenges how we think about numbers. Not as abstract quantities, but as constructed objects with history. Not as points on a line, but as fill-states in vessels.

This site walks through his core mathematical ideas — from first axioms to the power spectrum tables where e keeps appearing uninvited — and connects them to ideas in non-commutative algebra, fractal geometry, and theoretical physics.

Built to help Myk (and anyone else) understand Patrick's work in his own language, from first principles.

Standard math treats numbers as dimensionless points. Patrick treats them as states of a container. A number isn't "3" floating in the void — it's "3 out of some capacity n."

This isn't just notation. It changes what operations mean.

Why multiplication first?

Euclid privileges addition because geometry measures by piling lengths. Patrick privileges multiplication and scaling because construction is proportional. When you build something — a house, a crystal, a number — you're scaling and composing, not stacking unit blocks.

Additive descriptions are derivative projections of multiplicative constructions. "3 + 2 = 5" is a shadow of the proportional relationships that created those quantities.

In recursive proportions, Patrick uses a +1 that does something unusual. It doesn't add a unit of quantity. It records the act of construction itself.

Watch how the +1 accumulates through recursive construction:

f = the current fill-state (what's in the vessel now). a, b = scaling factors. n = vessel capacity. Each level takes the current fill, scales it, and adds a +1 — the trace of that construction step. The expression grows as: f/n → (af+1)/n → (a(bf+1)+1)/n = (abf+a+1)/n. The +a+1 isn't a sum of values — it's a history of what was built and in what order.

Standard math says √2 is irrational — it can't be expressed as a ratio of integers. Patrick reframes this:

Drag the sliders to change the grid resolution and target value. The grid can only express values that land on its marks — everything else falls between, with a gap.

Think of a grid. Your reference frame has resolution n. You can express any position that lands on a grid line perfectly. But √2 doesn't land on any grid line in an integer grid. It's between the marks.

Patrick's claim: this isn't a contradiction. It's a boundary marker. Useful to draw boundaries, but requiring derivation or reinterpretation. The value is accurate but has lost resolution within its reference frame.

Irrationals aren't mysterious — they're what happens when you ask a question below your measurement scale.

This is a strong claim: 3D is a hard ceiling. What mathematics calls "4D" or "5D" are faceted objects that exist within three-dimensional space. The fourth "dimension" in a tesseract is a projection artifact, not a physical direction.

Applied to algebra: x⁴ isn't "four-dimensional" — it's a scaled 3D object. The exponent is a construction factor, not a spatial axis. This connects to his insistence that numbers refer to real objects — real things can only exist in 3D, so mathematical constructions that claim more dimensions are being read incorrectly.

Patrick's framework for reading numbers as things that happened:

((km)f ±1) ≈ (ab ±1)

A product of factors combined with a fill-capacity relation, offset by a single proportional increment, can be interpreted as a power construction offset by that increment.

Reading the Events

This is where it gets really interesting. Take any number n. Form the triple (n−1, n, n+1). Compute all nine possible powers between them:

The Crossover at e

As you slide n, the ordering of the nine values changes. Some powers swap rank. Different pairs commute (go orange) at different values of n — for instance, 2^4 = 4^2 exactly at n=3, and 2.3^3.3 ≈ 3.3^2.3 near n=3.3.

The crossover point for a^b = b^a is governed by x^(1/x), which peaks at x = e. This peak is why e keeps appearing: it's where the function that determines whether two numbers can commute under exponentiation reaches its maximum. The number e is the symmetry point of exponentiation — not because the table is most orange there, but because it's the fulcrum around which all commutative pairs organize.

Key observations:

• At n=3: 2^4 = 4^2 = 16 — exact commutation! The only integer solution to x^y = y^x (besides x=y).
• At n≈3.3: 2.3^3.3 ≈ 3.3^2.3 — the (a,b) pair nearly commutes.
• At n=e: no single pair is especially close, but e is where x^(1/x) peaks — the theoretical maximum of "commutability." All pairs that DO commute at other n values owe their existence to this peak.
• As n grows larger, ALL pairs diverge further — exponentiation becomes increasingly non-commutative at scale.

Take the power spectrum and tick it forward by multiplying each value by an invariant I. Each tick shifts the whole spectrum, creating columns of evolution.

With I = 2: the clock ticks in octaves. Every value is a power of 2. Pure binary — no new information, just translation along the powers-of-2 lattice.

With I = e: the clock is quasiperiodic. The tick marks never land on the same spot twice. In log space, each tick adds ln(I), so the columns are evenly spaced. The ordered spectrum gives you the offsets within each period. It's a logarithmic ruler scrolling past a fixed window.

The e version is richer because e is transcendental — the orbit never closes. Same structure, infinite density of information per tick.

Start with seed 1. Apply f(x) = 8x + 1 repeatedly:

Now decompose each term. The recurrence A(n) = 8·A(n−1) + 1 splits into:

• B(n) = 8^n (the "skeleton" — pure scaling without memory)
• C(n) = (8^n − 1)/7 (the "structure" — accumulated +1 terms)
• A(n) = B(n) + C(n)

The Sierpiński Connection

C(n) = (8^n − 1)/7 is exactly the number of holes in a Sierpiński carpet at depth n. The carpet rule is: take a 3×3 grid, remove the center, recurse into the remaining 8 squares. At each level, you create 8× the previous structure and add new holes.

This isn't an analogy. It's an exact geometric identity. The structure sequence is the Sierpiński carpet's hole count.

The Fine Structure Constant

At depth 2: B(2) = 64, C(2) = 9, A(2) = 73. At depth 2 summed: B = 8 + 64 = 72, C = 1 + 9 = 10. Cumulative: 128 and 9.

The inverse fine structure constant α⁻¹ ≈ 137.036. And 128 + 9 = 137.

Brookcub's thesis: 128 = 2⁷ is the "skeleton" (pure geometric scaling via 8-fold branching in 3 spatial dimensions with binary degrees of freedom), and the 9 is the "flesh" — the accumulated structure/matter content. The +1 at each level is the irreducible cost of existence at that scale.

Note: α⁻¹(M_Z) ≈ 127.9, not exactly 128. The framework is suggestive but approximate. Patrick's position: these are human-made descriptions of a universe that just exists. The near-match is interesting but not a claim about physics — it's a property of the math itself.

The deepest result in Patrick's framework: the +1 that appears at every level of the 8n+1 recurrence is not an arbitrary constant. It is the scale divided by itself — the self-ratio of each level.

At each step of the recursion, the current scale value d divides by itself (d/d = 1), generating identity from within. The +1 is what a scale level deposits by the act of existing.

The Construction

Set invariant a = 8, seed b = 1/8. The recursive formula at each level uses the scale d, which grows as powers of 8:

The orange terms are each scale dividing by itself. 16/16, 128/128, 1024/1024 — different magnitudes, same self-reference. The +1 is not the number one. It is the act of self-recognition.

The Nested Form

Written as a nested expression, the construction history becomes visible:

Read from inside out: start with the seed, wrap in scale, deposit identity, repeat. Each layer is one scale level having existed and recognized itself. The expression is the construction history.

The Proportional Generator

Patrick's proportional tables with invariant 8 and seed 1/8, using recursive accumulation, generate all three of Brooklyn's sequences simultaneously:

Three sequences that Brooklyn derived independently from physics, fractal geometry, and number theory — all generated by one proportional framework. Patrick built the machine; Brooklyn found the output.

The Vessel and the Fill

Patrick's multiplicative tables (without recursive +1) give: 17, 136, 1088, 8704... — the clean skeleton.

Brooklyn's affine recurrence (with recursive +1) gives: 17, 137, 1097, 8777... — the physics sequence.

The difference between them: 0, 1, 9, 73, 585 — the Sierpinski carpet holes.

Patrick computes the vessel (the clean multiplicative structure). Brooklyn computes the vessel + fill (structure plus accumulated deposits). The Sierpinski carpet is the fill — the record of what each scale level deposited by existing.

The same self-ratio mechanism works at any invariant. When you set invariant = 3 (the Sierpinski triangle's branching factor), something remarkable happens: pi and e emerge as complementary operations on the same structure.

The pi Row: Seed 1/3, Operation +1

Sequence: 7, 22, 67, 202... — and 22/7 ≈ 3.14286 ≈ pi, the most famous rational approximation.

The e Row: Seed 4/3, Operation -1

Sequence: 8, 23, 68, 203... — and 8/3 ≈ 2.667, a known convergent of e.

Key distinction: The -1 here comes from n/n (the value's self-ratio), not d/d (the scale's self-ratio). The +1 in the pi row is the scale recognizing itself; the -1 in the e row is the value recognizing itself. They are dual, not identical.

The Duality

The seeds differ by exactly 1 in the numerator: 1/3 vs 4/3. The operations are mirrors: +1 vs -1. But they are not the same operation with a sign flip.

+1 = d/d: the vessel (scale) recognizes itself → deposit → pi

-1 = n/n: the fill (value) recognizes itself → withdrawal → e

pi and e are self-recognition of two different aspects of the same system: the container vs the contents. The asymmetry between them is the asymmetry between vessel and fill.

The Dimensional Pattern

The invariant k connects to the spatial dimension of the corresponding Sierpinski fractal:

The recipe: pick spatial dimension → set branching factor → seed with 1/k → recurse with self-ratio → read off the physics. The self-ratio d/d = 1 is universal across all dimensions.

The visualization above plots |a^b − b^a| for pairs (a,b). The zero contour (where exponentiation commutes) follows the curve x^(1/x) = y^(1/y), which peaks at e.

🌊 See the full Wet Math framework →

The synthesis: Both are measuring where non-commutativity has structure. Not the commutative regions (boring, predictable — Class I/II). Not the fully non-commutative regions (chaotic, random — Class III). The edge — where operations almost commute but don't quite, and the gap has pattern.

Patrick's +1 construction history is the trace of non-commutativity: doing a then b leaves a different +1 history than b then a. The Groovy Commutator measures the same thing for CA evolution. Same ghost, different staircase.

The Squaring the Circle (Almost)

Draw diagonals from the center of a circle to the corners of a square — each diagonal is 5r/4 long, making the corner-to-corner X span 2.5r.

Square area = (5/4 × 2r)² / 4 = (25/8)r² = 3.125r²

Circle area = πr² ≈ 3.14159r²

Difference: 0.53%. Invisible at drawing scale. A magic trick from the relationship between hypotenuse and radius — they're nearly exactly 2.5 apart.

1. The +1 history and the commutator: If (abf+a+1)/n ≠ (abf+b+1)/n is the "construction commutator," can we define a formal operator [a,b]_c = (a+1) − (b+1) = a − b and study its properties? Does this commutator have eigenvalues? Fixed points?
2. The ⅔ rounding threshold: In the McGuffin game system, Patrick uses 0.667 instead of 0.5 as the rounding boundary. Is there a proportional-axiom derivation of why 2/3 is the "right" threshold? It feels logarithmic rather than arithmetic.
3. Irrationals as fractal boundaries: If √2 is "below resolution," and the Sierpiński carpet is the geometry of the 8n+1 sequence... are irrationals the boundaries between carpet levels? Does "resolution" correspond to fractal depth?
4. The e-crossover in higher constructions: The power spectrum crossover happens at e for pairs. What happens for triples? For the full tetrahedron? Does the "ghost of e" generalize to higher-arity power structures?
5. The shell decomposition: (i+1)^(i+1) = i^i + i^(i+1) + (i+1)^i + remainder, and remainder/(i(i+1)) gives clean integers. Why? What's the algebraic identity underlying this? Is this related to the binomial theorem applied to exponentiation?
6. Connection to E₈: The σ₃ divisor sums connecting the 8n+1 structure sequence to Eisenstein series E₄ (which governs E₈ lattice theta functions) — is this a deep connection or a coincidence of modular arithmetic? The fact that three unrelated areas (Sierpiński holes, octal repunits, modular form coefficients) agree on the same numbers feels significant.