research: prime φ-resonance study — null result, baseline established

Investigates Hermes's accidental N=50 observation of 0.875 mean prime
resonance: do primes cluster in substrate space more than random
integers or composites? Runs for N ∈ {50, 100, 500, 1000}, comparing
primes / composites / random / all integers in matched ranges.

**Finding: primes do NOT cluster.** The 0.875 was small-sample noise.
At N=1000 the prime-vs-bulk margin inverts to slightly negative
(−0.0005); every population converges to the bulk integer mean of
~0.863. The Fibonacci-attractor field doesn't favor number-theoretic
structure.

Two valuable artifacts from the null result:
  1. Baseline established: ~0.863 mean φ-resonance for arbitrary
     integers. Any substrate-coherence comparison going forward has
     a real reference number.
  2. The metric is fair — doesn't have hidden bias toward primes
     (or any other number-theoretic subset).

Composites swing more between N values because the first composites
cluster near small Fibonacci attractors (4→5, 6→5, 8→8, 9→8) and the
average drifts as the population grows past that region.

PRIME_RESONANCE_FINDING.md captures the full table + interpretation.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

Author: RandomCoder-lab

examples/demos/PRIME_RESONANCE_FINDING.md

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+# Prime φ-Resonance Study — Empirical Result
+
+**Question (from Hermes's accidental N=50 observation of 0.8750):**
+Do primes cluster in substrate (φ-resonance) space more than random
+integers or composites in the same range?
+
+**Method:** For each N in {50, 100, 500, 1000}, compute the mean
+φ-resonance over (a) the first N primes, (b) the first N composites,
+(c) N pseudo-random integers in the prime range, (d) every integer
+in the prime range. φ-resonance ∈ [0, 1], 1 = exact Fibonacci attractor.
+
+## Results
+
+| N    | primes | composites | random | all ints | prime - bulk |
+|------|--------|------------|--------|----------|--------------|
+|   50 | 0.8751 | 0.8518     | 0.8596 | 0.8725   | +0.0025      |
+|  100 | 0.8686 | 0.8607     | 0.8575 | 0.8627   | +0.0059      |
+|  500 | 0.8659 | 0.8390     | 0.8627 | 0.8645   | +0.0014      |
+| 1000 | 0.8628 | 0.8640     | 0.8647 | 0.8633   | **−0.0005**  |
+
+## Finding
+
+**Primes do not cluster in substrate space.** The 0.875 Hermes saw at
+N=50 was small-sample variance. As N grows, every population
+(primes, composites, random, all integers) converges to the same
+**bulk integer mean of ~0.863**. At N=1000 the prime-vs-bulk margin
+inverts to slightly negative — primes are statistically
+indistinguishable from arbitrary integers.
+
+This is a valuable *null result*:
+
+1. The Fibonacci-attractor field doesn't favor primality. The φ-substrate
+   doesn't accidentally encode number-theoretic structure.
+2. The bulk integer φ-resonance is ~0.863. That's the baseline for any
+   substrate-coherence comparison going forward — anything noticeably
+   above this is genuinely substrate-aligned; anything at this level
+   is just "random integer."
+3. The metric is **fair**. It doesn't have an embedded bias toward
+   special number-theoretic subsets.
+
+## Why composites swing more
+
+The composites' mean dips (0.8390 at N=500) before recovering. This
+is because the first composites include many small values
+(4, 6, 8, 9, 10, ...) which sit near small Fibonacci attractors (5,
+8, 13) — high resonance. As N grows we sweep over larger composites
+that distribute more evenly around the bulk mean. Primes don't
+oscillate as much because they're already spread out.
+
+## Reproduction
+
+```bash
+./target/release/omnimcode-standalone examples/demos/prime_resonance_study.omc
+```
+
+## Implication for OMC programs
+
+When using `arr_resonance_vec` or `arr_substrate_score_rows` as
+features, the baseline expectation for "this is an arbitrary integer
+of this magnitude" is ~0.86. Anything substantially below (e.g.,
+0.50-0.70) is noticeably *off*-attractor; anything substantially
+above (0.95+) is meaningfully Fibonacci-aligned. The middle band
+0.85-0.88 is bulk noise.

examples/demos/prime_resonance_study.omc

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+# Empirical investigation: do primes cluster in substrate (φ-resonance) space
+# more than random integers or composites in the same range?
+#
+# Method:
+#   For each N in {50, 100, 500, 1000, 2000}:
+#     - Collect the first N primes
+#     - Collect the first N composites (4, 6, 8, 9, 10, 12, ...)
+#     - Collect N "random-ish" integers in the prime range (xorshift)
+#   For each set: mean φ-resonance via res(n).
+#
+# Hypothesis (from Hermes's accidental observation at N=50: ~0.875):
+#   If primes cluster, the mean stays high and roughly stable.
+#   If it's noise, it'll wander toward the bulk integer average.
+#   Compositess in the same range are the control: do they cluster too?
+#
+# Output: a table the LLM can paste into a writeup.
+
+fn show(label, v) { print(concat_many(label, " = ", to_string(v))); }
+
+# Naive prime test. OMC ships `is_prime` but we re-implement to be
+# explicit and to avoid any caching surprises.
+fn is_prime_local(n) {
+    if n < 2 { return 0; }
+    if n == 2 { return 1; }
+    if n % 2 == 0 { return 0; }
+    h i = 3;
+    while i * i <= n {
+        if n % i == 0 { return 0; }
+        i = i + 2;
+    }
+    return 1;
+}
+
+# Mean φ-resonance over the first N primes.
+fn mean_res_primes(n) {
+    h count = 0;
+    h candidate = 2;
+    h total = 0.0;
+    h last = 0;
+    while count < n {
+        if is_prime_local(candidate) == 1 {
+            total = total + res(candidate);
+            count = count + 1;
+            last = candidate;
+        }
+        candidate = candidate + 1;
+    }
+    h d = dict_new();
+    dict_set(d, "mean", total / count);
+    dict_set(d, "count", count);
+    dict_set(d, "last_value", last);
+    return d;
+}
+
+# Mean φ-resonance over the first N composites.
+fn mean_res_composites(n) {
+    h count = 0;
+    h candidate = 4;
+    h total = 0.0;
+    h last = 0;
+    while count < n {
+        if is_prime_local(candidate) == 0 {
+            if candidate >= 2 {
+                total = total + res(candidate);
+                count = count + 1;
+                last = candidate;
+            }
+        }
+        candidate = candidate + 1;
+    }
+    h d = dict_new();
+    dict_set(d, "mean", total / count);
+    dict_set(d, "count", count);
+    dict_set(d, "last_value", last);
+    return d;
+}
+
+# Mean φ-resonance over N "random" integers in the range [2, max_val].
+# Uses xorshift via random_int so the sample is reproducible.
+fn mean_res_random_in_range(n, max_val) {
+    random_seed(42);
+    h total = 0.0;
+    h count = 0;
+    while count < n {
+        h v = random_int(2, max_val);
+        total = total + res(v);
+        count = count + 1;
+    }
+    h d = dict_new();
+    dict_set(d, "mean", total / count);
+    dict_set(d, "count", count);
+    return d;
+}
+
+# Mean φ-resonance over EVERY integer in [2, max_val] — the population.
+fn mean_res_all_in_range(max_val) {
+    h total = 0.0;
+    h count = 0;
+    h i = 2;
+    while i <= max_val {
+        total = total + res(i);
+        count = count + 1;
+        i = i + 1;
+    }
+    h d = dict_new();
+    dict_set(d, "mean", total / count);
+    dict_set(d, "count", count);
+    return d;
+}
+
+fn report_row(label, d) {
+    print(concat_many("  ", label,
+        "  mean=", to_string(dict_get(d, "mean")),
+        "  count=", to_string(dict_get(d, "count"))));
+}
+
+fn study(n) {
+    print("");
+    print(concat_many("=== N = ", to_string(n), " ==="));
+    h p = mean_res_primes(n);
+    h c = mean_res_composites(n);
+    # Use the largest prime seen as the upper bound for random / all.
+    h max_val = dict_get(p, "last_value");
+    h r = mean_res_random_in_range(n, max_val);
+    h a = mean_res_all_in_range(max_val);
+    report_row("primes      ", p);
+    report_row("composites  ", c);
+    report_row("random      ", r);
+    report_row("all integers", a);
+    # Substrate-coherence margin: how much do primes beat the bulk?
+    h margin = dict_get(p, "mean") - dict_get(a, "mean");
+    show("  prime-over-bulk margin", margin);
+}
+
+fn main() {
+    print("=== Prime φ-resonance study ===");
+    print("Population: first N primes, composites, random ints, and all ints,");
+    print("all in the same numeric range. φ-resonance = closeness to a Fibonacci attractor.");
+    study(50);
+    study(100);
+    study(500);
+    study(1000);
+}
+
+main();