QH4 Pattern Matching
★ Open Conjecture
O(1) structural classification of ring positions by pattern type — the conjecture to derive — all priors established in Chapter 15 · Research Seed
This chapter is a research seed. The prior results (Chapters 13–15, including all 9 theorems of the Pattern Mining Machine) are established and locked. The Pattern Matching conjecture stated here is not yet derived. This chapter records the conjecture precisely so future work begins from known ground, not guesswork.
From WHERE to WHAT
Chapter 15 established the O(1) inverse formula: given position p, compute its Λ(ρ,γ,σ,θ) address in constant time. This tells us where position p is in the ring. The Pattern Matching conjecture asks a different question: what is position p — its structural role, its relationship to neighbours, and how to recognize a known pattern without searching.
Given a set of positions S ⊆ Z₂₅₆, classify S by its structural signature — without iterating through bins. The answer is O(1) per position, derived purely from (γ, σ, θ) arithmetic.
Do Not Re-Derive — Reference Only
All of the following are proven and locked in Chapters 13–15. They are the starting point for the conjecture, not the subject of it.
| Prior | Content | Status |
|---|---|---|
| A1 — Ring constants | 256=4⁴, 4 vacuums, 252 active, 36 gates × 7 steps, 9 gates/epoch | Locked |
| A2 — Forward formula | pos(γ,σ) = (7×(7(γ−1)+σ+⌊(γ−1)/9⌋)) mod 256 | 252/252 |
| A3 — Inverse formula | locate(p): m=(p×183)%256, adj correction, γ=div, σ=mod — O(1) | 252/252 |
| A4 — Quadrant midpoints | {32,96,160,224} — slot σ=4, equidistant at spacing 64 | Locked |
| T3 — Anchor condition | p is anchor ⟺ p mod 32 = 0 — O(1), no locate() required | Proven |
| T4 — Vacuum adjacency | p mod 64 ∈ {1,63} — O(1), no locate() required | Proven |
| J2 — Pattern classifier | 6-type classification in O(n) for sets of positions | Proven |
Structural Classification from Arithmetic
The six pattern types defined in J2 (Ch15) classify any set S by iterating locate() over each element. The Pattern Matching conjecture claims something stronger: given positions p₁, p₂, …, pₙ, their pattern type is determined O(1) from (γ, σ, θ) arithmetic — without calling locate() at all for individual comparisons.
| Pattern Name | Definition |
|---|---|
| Bin cluster | All positions share the same gate γ — differ by multiples of 7 |
| Step column | All positions share the same step σ across different gates |
| Epoch stripe | All positions within one epoch (quarter of ring) |
| Vacuum-adjacent | Position is first or last slot of an epoch (1 step from vacuum) |
| Midpoint | Position is slot σ=4 of its gate (centre of gate window) |
| Cross-epoch | Set of positions spanning multiple epochs |
Five Questions to Open the Derivation
| # | Question |
|---|---|
| Q1 | What is the algebraic condition for two positions to be in the same gate? Is it (p₁×183 − p₂×183) mod 7 = 0 after vacuum correction? |
| Q2 | What is the condition for same step across gates? Does pᵢ mod 7 = constant hold directly? |
| Q3 | Can we detect a vacuum-adjacent position directly from p without locate()? (T4 does this — but can we also classify the pattern type of a set without locate()?) |
| Q4 | Does the distance between two positions in (γ,σ) space encode physical meaning in MPRC? |
| Q5 | What is the pattern signature of the 4 quadrant midpoints {32,96,160,224}? |
Q1 and Q2 are the algebraic core. If the same-gate and same-step conditions can be read directly from p (without going through locate()), the full classifier becomes a sequence of modular arithmetic tests on raw position values — no address computation needed per comparison.
Physical Meaning of Pattern Type
In MPRC, the ring position p encodes a physical state: φ (position on the oloid), u-spinor direction, and quadrant polarity. Pattern types are not abstract classifications — they correspond to physical groupings:
| Pattern Type | MPRC Interpretation |
|---|---|
| Gate cluster | Same coherent u-axis window — same angular momentum range |
| Step column | Same phase position within each window — same φ modular offset |
| Epoch stripe | Same quadrant — same polarity domain (Q⁰..Q³) |
| Vacuum-adjacent | Boundary state — 1 step from a frame crossing at {0,64,128,192} |
| Midpoint | σ=4 structural node — centre of coherent window |
Pattern matching in MPRC context means recognizing which physical state a position belongs to without scanning the ring — directly from the position value. This enables O(1) state identification across any modality.