Topic Map

Topic Map: Every Idea and Its Root

A tree of every topic and sub-topic raised in the conversation, with the established fact or field each one is rooted in. Indentation shows derivation — a child line is a specialization, application, or consequence of its parent.

Boolean truth tables (the seed of everything)
├─ OR, XOR as shaded 2x2 grids
├─ All 16 two-input operators, grouped by symmetry
│  └─ Every function's exact inverse (AND<->NAND, OR<->NOR, XOR<->XNOR, ...)
└─ CNOT as a classical reversible gate
   ├─ State numbering (State 1-4 <-> |00>,|01>,|10>,|11>) = standard ket convention
   ├─ CNOT as a 4x4 permutation matrix
   └─ The jump to real quantum behavior
      ├─ Amplitude vectors [a,b,c,d], |a|^2+...=1
      └─ Superposition input -> Bell state -> entanglement (provably non-factorable)

Recursive halving optimization (independently derived, then named)
├─ AND_n, OR_n, XOR_n recursive doubling rules
├─ XOR's doubling rule = the Thue-Morse sequence's generating rule
├─ Formal name: Shannon cofactor expansion
│  ├─ f0 XOR f1 = the Boolean derivative (sensitivity to that variable)
│  ├─ f0 OR f1  = existential quantification / smoothing
│  └─ f0 AND f1 = universal quantification / consensus
└─ Real-world use: Binary Decision Diagrams (BDDs), variable-ordering algorithms

The XYZ+T parameter space (proposed, then corrected three times)
├─ X = n, Y = redundancy, Z = output value, T = recombination operator
├─ Correction 1: T is a 3-valued dial, not a signed axis
├─ Correction 2: T's irreversibility != physical time's arrow (combinatorial vs. thermodynamic)
├─ Correction 3 / resolution: T = Shannon/decision-tree depth = real algorithmic time
├─ Row-split vs column-split = variable ordering (a real, reversible BDD concept)
└─ n=3 closure: 3 reversible variable-axes + 1 irreversible depth-axis = 3D+time SHAPE
   └─ Explicitly flagged as a coincidence of choosing n=3, not a physics discovery

Fourier / Walsh-Hadamard analysis of boolean functions (the real "force vector")
├─ c0 (bias), c1, c2, c12 (interaction) computed directly for AND/OR/XOR
├─ Real physics correspondence: the Ising model (fields + pairwise couplings)
├─ Fast Walsh-Hadamard Transform (FWHT): O(n*2^n) vs. naive O(4^n)
├─ Concentration of measure: real functions cluster near the origin (provable)
│  └─ "Atom" visual (dense core/diffuse cloud) explained by this, not nuclear physics
├─ "Strings/loops" visual explained by discrete-lattice geometry (crystallography zone axes)
│  └─ String-theory connection = pattern-matching, EXCEPT: both share the real QM formalism
└─ Dark matter / restricted N-body analogy (judged a genuinely good fit)
   └─ Precise name: restricted N-body problem, not literally the three-body problem

Bit-hacking / "Doom hacks" (representation instead of computation)
├─ Lookup tables, XOR swap, popcount, bitboards
├─ Fast inverse square root (0x5f3759df)
├─ CORDIC (rotation/trig via shift+add only, no float unit)
├─ Integer angles / Binary Angle Measurement (Doom's real technique)
├─ Float-to-int "magic number" rounding trick
└─ The provable ceiling: Shannon's 1949 counting argument
   ├─ Almost all boolean functions have no compact formula (proof by counting)
   └─ Separately: 2^(2^n) enumeration wall is physical (storage), not algorithmic

Composing operators (closing the gap Shannon's argument opened)
├─ NAND is functionally complete (reaches every function, given enough gates)
├─ Circuit minimization = the real field (Quine-McCluskey, ESPRESSO)
├─ Bitslicing = real parallel speedup (one gate chain, many bits at once)
└─ Full correspondence to real quantum-circuit-simulation research
   ├─ Table-as-vector <-> quantum state vector
   ├─ Shannon split <-> QMDD/decision-diagram node split
   ├─ Redundancy <-> entanglement / compressibility
   ├─ Closed-form evaluator <-> stabilizer formalism (Gottesman-Knill theorem)
   ├─ NAND completeness <-> universal quantum gate sets
   ├─ Circuit minimization <-> Solovay-Kitaev theorem
   └─ Two different "walls" distinguished: 2^n (one state's cost) vs 2^(2^n) (search space)

The meta-thread (the collaboration examining itself)
├─ Pattern named: validate shape -> find real structure -> correct the exact gap
├─ Re-named on request: "imagination," not a test
└─ Resolution: imagination and correction aren't in tension; one is what makes the other durable

The builds (every idea made runnable — see docs/08 and /source for detail)
├─ xyzt-lattice.jsx        - first 3D view, n=2, Walsh coefficients vs. random cloud
├─ boolean-solver.jsx      - first live solver, n=2 (FWHT-precursor + Shannon + NAND-BFS)
├─ xyzt-lattice-n3.jsx     - capstone 3D view, n=3, X/Y/Z=per-variable influence, T=relevance
├─ ultimate-solver.jsx     - solver generalized to n=3, real FWHT
├─ quantum-composer.jsx    - real 3-qubit gate composer (X,H,Z,S,CNOT,Toffoli), entanglement check
└─ unified-app.jsx         - everything merged: Solver + 3D View + Quantum + Trajectories,
                             one shared n=0..3 selector, generalized composer, N-body panel