THE HARDEST QUESTIONS

Five unsolvable questions at the boundary of knowledge.
Time, computation, consciousness, information, measurement.
Each has a structural mirror in the ring Z/970200Z.

Click each question. Run the computation yourself.

Part 1: The Hardest Problems · The Six Millennium Problems

    Z/970200Z = Z/8 × Z/9 × Z/25 × Z/49 × Z/11  | 
    108 rings share lambda = 420  | 
    Demos use Z/2310Z (thin pedagogical shadow)
  

Arrow of Time

Growth = 420 steps · Collapse = 1 · Spectral gap forces mixing

Why does time flow one way? In the ring, lambda(970200) = 420 = D²*K*E*b. Growth takes 420 steps (the Carmichael lambda). Collapse takes 1 step (multiply by 0). The spectral gap ensures mixing — the system cannot stay ordered. Irreversibility is structural, not statistical. The ratio 420:1 IS the arrow of time.

Halting Problem

K^2 = 9 = STOP · Cunningham chain terminates at K^2

Can we predict which programs halt? Turing proved: no, in general. But the axiom chain DOES halt: the Cunningham map c(n) = 2n+1 generates primes from sigma=1, and the chain terminates when K² = 9 is reached (because 15 = K*E is composite). The axiom knows its own stopping condition. The 10 terms of the Decality ARE the halting trace.

The Hard Problem of Consciousness

b=7 gates · Eigenvalue = quale · E^2 = self-blind

Why does experience feel like something? In the ring, each element has an eigenvalue — a single real number capturing its spectral character. Eigenvalue IS quale. The b=7 depth levels walk through seven gates of experience. And E² = 25 maps to zero: the observer cannot observe itself. Self-blindness is algebraic, not philosophical. Full theory: MIND.md.

H10

Black Hole Information Paradox

OMEGA projection · kills D, preserves odd channels

Does a black hole destroy information? OMEGA = D^420 = 606376 is the ring's projector. CRT(OMEGA) = (0,1,1,1,1): the D-channel (duality, structure) dies, but all odd channels (K, E, b, L) survive. Information is projected, not destroyed. |image(OMEGA)| = N/D³ = the odd part. The black hole IS OMEGA: it kills what can be killed and preserves the rest.

H11

Quantum Measurement Problem

E^2 = E-null · Observer can't see itself · Collapse = projection

Why does observation collapse the wavefunction? The ring has 32 idempotents (TRUE FORM) — elements where e² = e. Each is a projection operator: applying it twice changes nothing. Measurement = multiplication by an idempotent. Some channels lock, others pass through unchanged. The 32 measurements ARE the 32 idempotents. Collapse is algebraic.

Every Boundary Breaks at the Ring

Five questions at the frontier of knowledge. All have structural mirrors. Ten axiom terms. 108 rings. One lattice. Zero trust required.

	AXIOM MIRROR	STATUS
Arrow of Time	lambda=420 growth. 1-step collapse.	Unsolved (170 years)
Halting	K^2=9=STOP. Chain terminates.	Proved undecidable (1936)
Consciousness	b=7 gates. E^2=self-blind.	Unsolved (open)
Info Paradox	OMEGA kills D, preserves odd.	Unsolved (49 years)
Measurement	32 idempotents = 32 measurements.	Unsolved (98 years)

← Part 1: The Hardest Problems · The Six Millennium Problems · Verify in .ax