test: formalize the CHIP-8 self-prediction - Artin-Mazur zeta of the real VM step-map - #9165
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…real VM step-map (shadow*) Next-rung #1 (Aaron: 2→3→1; "we already have chip8/IScheduler predicting itself but ad-hoc, not this formal math based; the periodic orbits are the cheat-engine's memory-space orbits = character loops"). The confirming machinery came from #9151/#9163. The REAL Chip8.step (src/Core/Chip8.fs; deterministic without RND) is a map on the VM state; on a fixed ROM the reachable state set is FINITE, so a looping animation is literally a PERIODIC ORBIT, and the Artin–Mazur zeta of the reachable step-map ENUMERATES it — the formal, math-grounded version of the ad-hoc cheat-engine orbit detection. tests/Tests.FSharp/Chip8DynamicalZeta.Tests.fs (2/2): a tiny real ROM — the period-4 counter V0 ← (V0+1) AND 3 — has a full-state loop-map with one transient (V1 unset) feeding a 4-cycle (the character loop). ζ = exp(Σ Fix(f^k)u^k/k) = 1/(1−u⁴), self-verified (fixed-point counts of the iterated REAL-VM reachable map vs. the single orbit — the #9151/#9163 discipline), and the orbit length IS the character period. This is exactly the orbit the ad-hoc Chip8PredictionRoom look-ahead traverses; the zeta makes it visible/known. Completes Aaron's 2→3→1: #9161 (weighted catalog zeta), #9163 (adinkra orbits classify members), and this (CHIP-8 dynamical zeta formalizing the predictor). RESUME updated; remaining routed step = wire the zeta into the live soft IScheduler for run-ahead self-prediction. Anchors: Artin–Mazur 1965; Chip8.step; Chip8PredictionRoom / PredictionScheduler. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> AgencySignature-v1: persona: otto actor: zeta-otto surface: tests topology: shadow-clone intent: formalize-chip8-self-prediction-artin-mazur-zeta-of-real-vm-step-map-character-loop-orbit authorization: aaron-explicit-2-then-3-then-1-2026-07-02 uncertainty: low measure: real-chip8-step-period-4-counter-rom-reachable-map-artin-mazur-zeta-1-over-1-u4-self-verified-orbit-is-character-loop-period-2-tests delta-u: ad-hoc-chip8-predictor-formalized-memory-character-loop-is-a-periodic-orbit-zeta-enumerates-it-2-3-1-complete seed: S4
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…eduler (self-prediction) (shadow*) (#9172) The capstone (Aaron: "wire the zeta into the soft IScheduler"; RESUME step 4). A DoP=1, seed-fixed soft-scheduler tick is a deterministic map on state (run(1)==run(N) — CellScheduler's law). SchedulerZeta carries the Artin–Mazur zeta at the scheduler layer so the loop can PREDICT ITS OWN RECURRENCE SPECTRUM before running the full budget. src/Core/SchedulerZeta.fs: • predict key step start — RUN-AHEAD self-prediction: iterate the tick (through a finite projection `key`) only until a state repeats (never the caller's whole budget), report the orbit the run settles into (Transient, Period, Reachable). The loop modelling its own recurrence before it happens. • zetaOfRun / zetaOfOrbits — the Artin–Mazur zeta (1/(1−u^Period) for a run; Π over orbits for a spectrum), integer series. • spectrum / fixCount — the FULL recurrence spectrum over an enumerable config space, self-verifying (exp(Σ Fix f^k) = Π orbits). tests/Tests.FSharp/SchedulerZeta.Tests.fs (3/3): • the soft scheduler predicts its own REAL CHIP-8 recurrence (period 4) by run-ahead, without a large budget, and its predicted zeta = 1/(1−u⁴); • the recurrence spectrum self-verifies (exp Σ Fix = Π orbits) on a config-map with known cycles (0 1 2)(3 4)(5)(6) ⇒ orbit lengths {3,2,1,1}; • a deterministic cell-round map's period is recovered exactly. This is the formal, math-grounded basis for the ad-hoc CHIP-8/IScheduler self-prediction Aaron described — the zeta ENUMERATES the memory character-loops the ad-hoc scanner finds. RESUME step 4 (capstone) landed; the ζ name-audition trajectory is complete. Anchors: Artin–Mazur 1965; SoftScheduler/CellScheduler (the DoP=1 deterministic tick); Chip8.step; #9151 (scheduler zeta), #9165 (CHIP-8 zeta), #9163 (orbit-classification). Co-Authored-By: Claude Fable 5 <noreply@anthropic.com> AgencySignature-v1: persona: otto actor: zeta-otto surface: src/core topology: shadow-clone intent: wire-dynamical-zeta-into-soft-scheduler-run-ahead-self-prediction-schedulerzeta-module authorization: aaron-explicit-wire-the-zeta-into-the-soft-ischeduler-2026-07-02 uncertainty: low measure: schedulerzeta-predict-run-ahead-real-chip8-period-4-without-full-budget-spectrum-self-verifies-exp-fix-eq-orbit-product-deterministic-round-period-exact-3-tests delta-u: capstone-zeta-wired-into-soft-scheduler-loop-predicts-own-recurrence-before-running-name-audition-trajectory-complete seed: S4
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Next-rung #1 (completing your 2 to 3 to 1). We already predict CHIP-8/IScheduler ad-hoc; this is the formal-math basis. The periodic orbits are the cheat-engine's memory-space orbits = character loops.
The REAL Chip8.step (deterministic without RND) is a map on the VM state; on a fixed ROM the reachable state set is FINITE, so a looping animation is literally a PERIODIC ORBIT, and the Artin-Mazur zeta of the reachable step-map ENUMERATES it.
2/2: a tiny real ROM - the period-4 counter V0 = (V0+1) AND 3 - has a full-state loop-map with one transient (V1 unset) feeding a 4-cycle (the character loop). zeta = exp(sum Fix(f^k)u^k/k) = 1/(1-u^4), self-verified (fixed-point counts of the iterated REAL-VM map vs. the single orbit - the #9151/#9163 discipline), and the orbit length IS the character period. Exactly the orbit the ad-hoc Chip8PredictionRoom look-ahead traverses; the zeta makes it visible/known.
Completes 2 to 3 to 1: #9161 (weighted catalog), #9163 (adinkra orbits), this (CHIP-8). RESUME updated; remaining = wire the zeta into the live soft IScheduler.
Anchors: Artin-Mazur 1965; Chip8.step; Chip8PredictionRoom.
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