What the future
demands of memory.
Two states can look alike today and lead to different observations tomorrow.
An exact prediction must retain enough information to tell them apart.
NOT A SIMULATION OF THE MODEL
Something
must be conflated.
The scene unfolds automatically. Watch the two paths that enter the same state, then require different observations.
Four values.
A finite capacity.
35 = 5 × 7 has four divisors: 1, 5, 7 and 35.
One coordinate with four values offers four possibilities. Two coordinates give at most 16 states. Three, at most 64.
35 illustrates the counting argument. It is not claimed to belong to the Research Z family.2 coordinates · at most 16 states
A fixed number of coordinates
eventually runs out of states.
The HORIZON overview reports an explicit infinite arithmetic family, with parameter r = 5 fixed, whose number of distinguishable predictive futures grows without bound.
For integers n = pq in this family, with p and q distinct primes, the divisors are 1, p, q and pq. A predictor with k coordinates taking values in this set therefore has at most 4ᵏ states.
For any fixed k, this capacity is finite. It cannot distinguish an unbounded number of futures across the whole family studied.
See a concrete example · 7,515 / 4,096
In one documented Research Z case, 7,515 different futures must remain distinguishable.
- Futures to distinguish
- 7,515
- States with 6 coordinates
- 4,096
Six coordinates with four possible values offer only 46 = 4,096 states: a collision is inevitable. This model therefore needs at least seven coordinates.
A future is a sequence of zeros and ones: at each step, we observe whether four successive states are linearly independent.
See the exact formula
For r = 5 and n ≡ 2 (mod 5), the state is a polynomial with four coefficients in An = (ℤ/nℤ)[T]/(Φ5), with Φ5(T) = T4 + T3 + T2 + T + 1.
xt+1 = Θn(xt) = (xt(T3))n
Let vt be the vector of the four coefficients of xt. For a prime factor p of n, the local observation is:
Δt = det(vt, vt+1, vt+2, vt+3)
bt = 1 if Δt ≠ 0 (mod p), and 0 otherwise.
The future from step t is the sequence Ft = (bt, bt+1, bt+2, …). Two states are distinguishable as soon as their sequences differ.
Examine the scope and mechanism +
The reported statement concerns predictive futures of the exact local non-degeneracy observation, in the model studied by Research Z.
- Construct a family whose orbit lengths are unbounded.
- Show that the phases at which the observation vanishes form a nonempty set of bounded size.
- Deduce an unbounded number of distinguishable futures.
- Compare this growth with the maximum capacity 4ᵏ.
These steps belong to the full proof. The illustration and counting argument on this page do not replace it.
The scope is precise.
The result constrains this model and this type of coordinate. It does not prove the impossibility of every compact representation of prime numbers, nor solve the general problem.
V1 reports a reconstruction by the Referee role and internal acceptance. No external academic validation is attested here; complete bibliographic certification of novelty remains to be established.
The full proof and internal reports are not published on this showcase.
Request shareable material ↗Forty successes.
One limit.
The polynomial n² + n + 41 produces a prime for n from 0 to 39. At n = 40, it gives 1,681 = 41².
Checking many cases and proving all cases are two different things.
A classical example, not a HORIZON discovery.