Author notes — full detail, auditor-facing

Earlier framework formulations described the dimensional cascade as a single-peak progression — one monotonic curve from dim 1 through dim N, with framework parameters varying smoothly along it. This entry documents the *revision* to a cycle-restart model and its eventual subsumption into the Tribonacci-cycle framework.

The earlier model

The original framework treated dimensions as a single sequence with parameters (framerate, c-value, spiral ratio) varying monotonically. The implicit assumption: dim 4 is just "one step past dim 3" with the same machinery extended.

This was *partially* correct (parameters do vary along the sequence) but *missed the cycle structure*: dimensions don't form one monotonic ladder — they form cycles of three (seed → flat → volumetric) with the meta-pattern *restarting* at each cycle boundary.

The cycle-restart insight

The framework's discovery:

• Cycle 1 = dims 1, 2, 3. Pattern: seed (1D, linear) → flat (2D,

planar) → volumetric (3D, solid).

• Cycle 2 = dims 4, 5, 6. Pattern repeats: seed (4D, 4D-linear

analog) → flat (5D, 5D-planar analog) → volumetric (6D, 6D-solid analog).

• Cycle 3 = dims 7, 8, 9. Pattern repeats again.

Within each cycle, parameters scale by π relative to the previous cycle's analogous slot. The 2D→3D boundary energy and the 5D→6D boundary energy differ by a factor close to π (predicted ~3.14).

Why the single-peak model was wrong

The single-peak model didn't predict the cycle structure. It predicted *smooth* parameter variation across the entire sequence — so framework parameters at dim 4 should look like extrapolations of dim 3.

But the cipher's empirical results (and the magic-numbers derivation, the c-ladder findings) showed that dim 4 parameters are *not* simple extrapolations of dim 3. They're *analogs* at the next cycle level. The single-peak model was missing the cycle restart.

How this evolved into the Tribonacci framework

The cycle-restart model was the framework's intermediate step between the single-peak model and the full Tribonacci cycle-specific framework.

• Single-peak → predicts smooth monotonic variation. Wrong.
• Cycle-restart → predicts the same machinery restarts at each

cycle boundary, with parameters scaled by π. Closer but incomplete.

• Tribonacci → predicts each cycle has its *own* recurrence engine

(Fibonacci for cycle 1, Tribonacci for cycle 2, Pentanacci for cycle 3). The cycle structure is real and the cycle *content* is different per cycle.

The cycle-restart model identified the structure; the Tribonacci framework identified the per-cycle mechanism.

Why this entry is preserved as superseded

The cycle-restart model is no longer canonical — the Tribonacci framework supersedes it. But: 1. The cycle structure itself (3-dimension cycles, meta-pattern restart) is correct and preserved in the Tribonacci framework. 2. The π scaling between analogous slots in adjacent cycles is approximately correct, though the Tribonacci framework gives a sharper derivation. 3. The framework's intellectual-honesty discipline preserves intermediate-step documents as the audit trail. This entry is that audit trail for the single-peak → cycle-restart → Tribonacci progression.

What this entry retains as solid

• The cycle structure (three dimensions per cycle).
• The meta-pattern restart at cycle boundaries.
• The approximate π scaling between adjacent cycle slots.
• The recognition that the framework's machinery applies

cycle-by-cycle, not monotonically.

What this entry retracts

• The claim that the same recurrence engine governs all cycles.
• The implicit equivalence of "dim 4" and "dim 3 + 1." Dim 4 is

the seed of cycle 2, structurally analogous to dim 1 (the seed of cycle 1), not a continuation of dim 3.