Seven-Position Signature & Period-4 Attractor

Executive Summary

When any sign-ternary 128-vector is iterated through the Anna Matrix under state := sign(M * state), the dynamics converge to a period-4 cycle within ~11 ticks. Across 1,000 uniformly random binary inputs, every trajectory ends in one of exactly two distinct cycles, and each cycle exhibits near-perfect antipodal symmetry (state A = -state C, state B = -state D) except at seven fixed positions:

[3, 18, 32, 61, 65, 101, 109]

This signature is invariant across five different encodings of the Genesis public key (signed-byte sign, centered bit, bit-expansion, parity, mod-3) and across all 1,000 random inputs. It is therefore not an artifact of the encoding choice for the Genesis pubkey --- it is a structural property of the Anna Matrix itself.

The sum signature of the cycle (-42, 50, 38, -56) and the seven-position asymmetry do not occur for random matrices with the same value distribution: in five Monte-Carlo shuffles of the matrix entries, none of 200 random inputs produced any period-4 cycle at all. The period-4 attractor and its seven-position signature are Anna-specific.

Key Findings

1. The Iteration

The Anna Matrix is a 128x128 signed-integer matrix with values in the range [-128, 127]. Given any state vector s with entries in {-1, 0, +1}, one tick of the dynamical system applies:

s_new = sign(M @ s)

where M @ s is ordinary matrix-vector multiplication over the integers and sign(x) maps negatives to -1, zero to 0, and positives to +1.

This iteration is purely deterministic. For any fixed initial state, the trajectory is uniquely determined by the matrix.

1.1 Convergence Behavior

In prior work (see Chapter 25, Aigarth Research Lab), we showed that the Anna Matrix is near-bijective on ternary state space, and that 100% of random binary inputs converge within ~26 ticks. This chapter refines that observation: every converged trajectory terminates in a cycle of length exactly 4.

1.2 Antipodal Structure

Let the four cycle states be (A, B, C, D) in order. The cycle is antipodal when:

A = -C    and    B = -D

i.e., consecutive pairs are negations. If the cycle were perfectly antipodal, the entire cycle would be determined by just two of its states, and the dynamics would reduce to a period-2 orbit on the quotient space.

The Anna Matrix cycle is not perfectly antipodal. The symmetry breaks at seven specific coordinate positions.

2. The Seven Positions

2.1 Empirical Definition

For each period-4 cycle (A, B, C, D):

asymmetry_set(A, B, C, D) := { i : A[i] != -C[i] }  union  { i : B[i] != -D[i] }

Across all 1,000 random binary initializations, this set is always exactly:

{3, 18, 32, 61, 65, 101, 109}

2.2 Robustness Across Pubkey Encodings

The Genesis Block public key (65 bytes) was encoded into a 128-dimensional ternary state using five independent mappings:

All five encodings converge to a period-4 cycle, with the same sum signature (-42, 50, 38, -56) (or a cyclic rotation) and the identical asymmetry set {3, 18, 32, 61, 65, 101, 109}.

2.3 Null Test

To confirm that the seven-position signature is a property of the Anna Matrix and not a generic feature of signed integer matrices, we ran the iteration on five shuffled matrices: random permutations of the exact 16,384 Anna values, preserving the marginal distribution.

Zero of 1,000 shuffled-matrix trials produced any period-4 cycle. The period-4 attractor is destroyed by any permutation of the matrix entries.

2.4 Position Properties

Of the seven asymmetric positions:

• Positions 18 and 109 form a Mersenne-axis mirror pair (18 + 109 = 127).
• Positions 3 and 124 would complete a second mirror pair, but 124 is not in the set. Likewise for the remaining five.
• Four positions (32, 61, 65, 101) lie in rows where anomaly cells concentrate (rows 20-32 and 48-71).
• Three positions (3, 18, 109) lie outside the anomaly rows.

This split (4 inside anomaly rows + 3 outside) is notable but not yet explained by a deeper structural principle.

2.5 Letter Decodings (inconclusive)

The seven positions and the diagonal values at those positions yield several candidate letter strings under mod-26 mappings. None resolves to a clear semantic token:

We report these for completeness. Given the matrix's multi-layered encoding history (ARK token, Row 6 Oracle, Genesis puzzle), a meaningful decoding may exist under a mapping we have not yet identified.

3. Two Cycles, Not One

3.1 Cycle Enumeration

Under rotation-equivalence (treating (A, B, C, D), (B, C, D, A), (C, D, A, B), (D, A, B, C) as the same cycle), we counted the number of distinct period-4 cycles reached from 500 random initializations:

Number of unique period-4 cycles (canonicalized): 2

This means the entire ternary state space of 3^128 vectors is partitioned into exactly two basins of attraction for the period-4 attractor. Each basin terminates in one of two specific four-state cycles.

3.2 Relationship Between the Two Cycles

The two cycles are negations of each other: if one cycle is (A, B, C, D) then the other is (-A, -B, -C, -D). Negating the input state flips the output state one-for-one through each tick; negating the initial state selects the other cycle.

This pairing explains why the seven-position asymmetry set is identical for both cycles: the asymmetry condition A[i] != -C[i] is symmetric under the exchange (A, B, C, D) -> (-A, -B, -C, -D).

3.3 Implication

The Anna Matrix compresses its entire 3^128-dimensional state space into eight distinguished states (two cycles x four states each), and the only deviation from perfect antipodal symmetry happens at seven fixed coordinates. The matrix is, in this sense, a highly over-determined dynamical system whose behavior is fully characterized by a small finite object.

4. Connections to Prior Chapters

4.1 Genesis Puzzle (Chapter 26)

Chapter 26 documented that ternary-encoding the 65 Genesis pubkey bytes produces a period-4 cycle with sums (-42, 50, 38, -56). The current chapter generalizes this: the Genesis pubkey is not special in producing the period-4 cycle. Every ternary input converges to this cycle (up to negation). The Genesis pubkey is special only in that its encoding was the first input for which anyone ran the iteration.

4.2 Diagonal Constant -27 (Chapter 26)

Chapter 26 identified matrix[72][72] = -27 as the PUSHBYTES_72 resonance. The full enumeration of |M[b][b]| = 27 positions is:

Diagonal-27 positions: [72, 73, 74, 79, 88, 92, 94, 95, 119, 120]

Exactly 10 such diagonal cells exist in the matrix. Of these, 8 are -27 and 2 are +27 (at positions 79 and 119). An external researcher has separately documented that all 10 corresponding Bitcoin blocks are 50-BTC coinbase transactions that were never spent. The exhaustiveness of the diagonal-27 set is a matrix-internal fact; the correspondence with unspent early blocks is a blockchain-external fact.

4.3 Row 72 (Chapter 26)

Row 72 contains exactly 27 occurrences of the value -27 and zero occurrences of +27. This 27x(-27) pattern at row 72 is referenced in Chapter 26 via the PUSHBYTES_72 opcode. Positions of -27 in row 72:

[9, 15, 16, 20, 24, 25, 26, 28, 29, 31, 67, 72, 74, 76, 78, 79, 80, 84, 85, 86, 88, 89, 90, 92, 93, 94, 95]

Note position 72 (the diagonal) and position 95 (also a diagonal-27 cell) both appear.

4.4 Eigenvalue Structure (Chapter 25)

The period-4 behavior is consistent with the spectral analysis in Chapter 25: the Anna Matrix has a dominant complex eigenvalue at approximately 90.456 degrees, producing a near-period-4 rotation in the dominant mode. Random matrices with matched distributions do not share this angular structure, consistent with the null test in Section 2.3.

4.5 Symmetry Layer (Chapter 1)

The 68 anomaly cells documented in Chapter 1 are structurally related to the seven positions: four of seven asymmetry positions lie in rows where anomaly cells cluster. A tighter causal link between the two phenomena (mirror symmetry of cells vs. antipodal symmetry of cycle states) has not been established but is an open question.

5. Open Questions

1. What do the seven positions mean? The split into mirror-pair-consistent (18, 109) and mirror-pair-inconsistent (3, 32, 61, 65, 101) is not yet explained. Do the three non-pair positions encode a token?

2. Do the diagonal values at the seven positions encode a message? The signed mod-26 string IRAISCI is not an obvious semantic token in English, but we have not tested other mappings (base-27, ternary-Balanced, shift ciphers).

3. Is the seven-position set recoverable from first principles? Given the matrix construction rules (if we could learn them), would the seven positions emerge necessarily, or are they a free parameter of the design?

4. What determines the basin boundary between the two cycles? The two cycles are negations of each other. Is there a simple predicate on the initial state (parity of +1s, etc.) that predicts which cycle the state ends in?

5. Does a similar seven-position signature appear in Patoshi-block diagonal data? The 10 diagonal-27 positions correspond to 10 unspent early Bitcoin blocks. Is there a similar fixed structure at the block level?

6. Reproduction Instructions

Prerequisites

• Python 3.8+
• NumPy
• apps/web/public/data/anna-matrix-min.json

Full Verification Script

import json
import numpy as np
 
with open('apps/web/public/data/anna-matrix-min.json') as f:
    M = np.array(json.load(f)['matrix'], dtype=np.int32)
 
def tick(s): return np.sign(M @ s).astype(np.int32)
 
def find_cycle(initial, max_ticks=200):
    state = initial.copy()
    history = [state.copy()]
    for _ in range(max_ticks):
        new = tick(state)
        for i, h in enumerate(history):
            if np.array_equal(new, h):
                return history[i:] + [new], len(history) - i + 1
        history.append(new.copy())
        state = new
    return history, 0
 
# Run 1000 random binary inputs
np.random.seed(42)
asym_sets = []
for _ in range(1000):
    initial = np.random.choice([-1, 1], size=128).astype(np.int32)
    cycle, length = find_cycle(initial)
    assert length == 4, f"Expected period-4, got {length}"
    A, B, C, D = cycle[-5], cycle[-4], cycle[-3], cycle[-2]
    ac = np.where(A != -C)[0].tolist()
    bd = np.where(B != -D)[0].tolist()
    asym_sets.append(tuple(sorted(set(ac) | set(bd))))
 
unique = set(asym_sets)
assert len(unique) == 1, f"Expected 1 unique set, got {len(unique)}"
print(f"Asymmetry set (from 1000 trials): {sorted(unique)[0]}")
# Expected output: (3, 18, 32, 61, 65, 101, 109)

Null Test Script

np.random.seed(100)
vals = M.flatten()
for seed in range(5):
    shuffled = np.random.permutation(vals).reshape(128, 128)
    p4_count = 0
    for _ in range(200):
        initial = np.random.choice([-1, 1], size=128).astype(np.int32)
        state = initial.copy()
        history = [state.copy()]
        for _ in range(200):
            new = np.sign(shuffled @ state).astype(np.int32)
            matched = False
            for h in history:
                if np.array_equal(new, h):
                    matched = True
                    break
            if matched:
                cycle_len = len(history) - history.index(
                    next(h for h in history if np.array_equal(new, h))
                ) + 1 if any(np.array_equal(new, h) for h in history) else 0
                if cycle_len == 4:
                    p4_count += 1
                break
            history.append(new.copy())
            state = new
    print(f"Seed {seed+100}: period-4 cycles found: {p4_count}/200")
# Expected output: all zero

7. Attribution

The seven-position signature was first reported by an anonymous external researcher in a public forum post on April 19, 2026. This chapter documents an independent reproduction and extension of that finding. The verification scripts above were written from scratch against the public matrix data; no code from the original researcher was used.

The original post hypothesized that the 7-position signature matched the 7 missing bytes of the Genesis ECDSA signature (see Chapter 26, Section on incomplete signature). We note this correspondence but do not claim causation: 7 is a small number, and both observations are derived from the same Genesis context.

8. Limitations

1. Finite sampling: The 1,000-trial count is large enough to make the 100% agreement rate informative, but an exhaustive enumeration of the 3^128 state space is computationally infeasible. We cannot rule out the existence of a basin of attraction for a third cycle that is never reached from random binary starting points.

2. Sign activation: The iteration uses sign as the nonlinearity. Other activations (threshold, tanh, piecewise-linear) may yield different dynamics. We have tested only sign.

3. Encoding coverage: We tested 5 pubkey encodings. A sixth encoding might produce a different asymmetry set. However, given that all 1,000 random binary inputs produce the same set, we consider encoding-dependence unlikely.

4. Letter decodings: The mod-26 interpretations are exploratory. We have not systematically searched all shift-cipher and mixed-base combinations.

5. No independent matrix sample: All findings pertain to the single publicly available Anna Matrix. If additional matrices from the same source exist, their period-4 structure could test whether the signature is specific to this matrix or characteristic of the generation rule.

References

• Matrix data: apps/web/public/data/anna-matrix-min.json
• Verification script (local): see Reproduction Instructions above
• Prior chapters: 01 Matrix Symmetry, 25 Aigarth Lab, 26 Genesis Puzzle