AI Skill Report Card
Mapping Engine Numogram Correspondence
Mapping Engine-Numogram Correspondence
Quick Start13 / 15
Given two systems — a concrete computational engine (counters, phases, gears) and a symbolic/numerological framework (zones, cycles, currents) — produce a correspondence table where every claimed link is:
- Stated as a precise mathematical object on each side (group, map, invariant)
- Tested (exhaustively if finite, randomly/statistically if continuous)
- Tagged with a verdict and a method tag
# Engine element Numogram element Verdict Tag
1 Node k mod 6 structure Zone 2^(k mod 6) mod 9 EXACT PROVEN CRT: C3×C2 ≅ ℤ/6
9 Stator mod 60 — NO MAP base-60 ≠ base-9 casting out
Never assert a correspondence without stating why — give the group-theoretic or arithmetic reason, not just "both involve the number 6."
Recommendation▾
This is a highly niche/unusual domain (numogram correspondence) — consider whether broader applicability or a more generic 'structural correspondence auditing' framing would increase reuse value.
Workflow14 / 15
Progress:
- [ ] 1. Enumerate the engine's finite/discrete structures (counters, mod-N cycles, group actions)
- [ ] 2. Enumerate the numogram's (or target system's) finite structures (zones, currents, gates, syzygies)
- [ ] 3. For each candidate pair, identify the actual algebraic objects (cyclic groups, CRT factors, quotients)
- [ ] 4. Attempt to construct an explicit map; check it is well-defined and respects structure (homomorphism? bijection? mere containment?)
- [ ] 5. Classify verdict: EXACT PROVEN / PARTIAL PROVEN / PARTIAL COMPUTED / PARTIAL HEURISTIC / NO MAP
- [ ] 6. Write the one-line reason citing the precise obstruction or proof
- [ ] 7. Verify: exhaustive check for finite structures, random/statistical test for continuous or combinatorial claims
- [ ] 8. Flag discrepancies between the live system's actual behavior (code) and any accompanying documentation (papers/specs)
- [ ] 9. Assemble correspondence table + verification log
Verdict taxonomy (use exactly these categories)
- EXACT PROVEN — a true isomorphism or exact identity holds for all elements; proof is algebraic (e.g. CRT, group generator argument) and checked exhaustively.
- PARTIAL PROVEN — an exact algebraic fact exists (e.g. CRT factorization) but it does not describe what the actual system does (wrong index used, different selection rule).
- PARTIAL COMPUTED — numeric/geometric agreement holds under a special condition (e.g. a parameter = 1) but not in general; demonstrate both the matching case and the general mismatch.
- PARTIAL HEURISTIC — some shared numeric pattern (matching digital roots, same period) exists but the underlying mechanisms are different; state what fraction/how many instances agree.
- NO MAP — state the precise obstruction (different group orders, no common homomorphism besides trivial parity map, continuous vs. finite, no shared divisor) — don't just say "doesn't correspond."
Recommendation▾
Add an explicit template for the final output artifact (full correspondence table schema with all columns) rather than just the abbreviated Quick Start example.
Examples17 / 20
Example 1 (EXACT PROVEN): Input: Engine node index k cycles mod 6 via (k mod 3, k mod 2). Numogram Time-Circuit zone = 2^(k mod 6) mod 9. Output:
- Verdict: EXACT PROVEN
- Reason: ℤ/6 ≅ C3 × C2 by CRT. 2^j mod 9 has order exactly 6 (checked: 2^0..2^6 mod 9 = 1,2,4,8,7,5,1). Since 2^j = 4^(2j mod 3)·8^(j mod 2), the C3 part tracks j mod 3 and the C2 part tracks j mod 2, matching the engine's own (k mod 3, k mod 2) decomposition. Because 6 | 30 (node count per tier), the map closes on every lap.
Example 2 (NO MAP): Input: Engine's stator register increments mod 60. Claim: this parallels Numogram's "casting out nines" (digit sum ≡ n mod 9). Output:
- Verdict: NO MAP
- Reason: Casting out nines works only because the base (10) satisfies 10 ≡ 1 (mod 9). The base-60 analogue would cast out 59, not 9. Since 60 ≡ 6 (mod 9) merely lands on a Warp zone by coincidence, no structural property transfers. Verified numerically: base-10 digit sum ≡ n (mod 9) held, base-60 digit sum ≡ n (mod 59) held, but base-60 digit sum mod 9 ≠ n mod 9 in the test case.
Example 3 (PARTIAL COMPUTED): Input: Engine computes an "inferred" 4th point V4 = pNode + qScale·(diagonal vector); Numogram's syzygy closure is x + (9−x) ≡ 0 (always exact). Output:
- Verdict: PARTIAL COMPUTED
- Reason: True parallelogram closure (the numogram's exact analogue) requires qScale = 1. The engine's qScale is a derived scalar (0.65 + 0.45·√A_R/(ΣQ+1)) almost never equal to 1, so V4 ≠ V4* in general. Report both points and their Euclidean distance as evidence (e.g. |V4 − V4*| = 40.99) rather than asserting closure.
Recommendation▾
Include one worked PARTIAL PROVEN and PARTIAL HEURISTIC example in full detail (currently only sketched in the taxonomy) to match the rigor of the other three examples.
Best Practices
- State the group, not just the number. "Both involve 6" is not a proof; "ℤ/6 ≅ C3×C2 via CRT" is.
- Distinguish isomorphism from containment. A subset with matching gcd conditions (e.g. units mod 30 all coprime to 9) is a containment, not a structural equivalence — don't overclaim.
- Always give the exact test performed: exhaustive enumeration for anything finite (state the size, e.g. "30 nodes checked"), random sampling with a fixed seed for continuous/geometric identities (report pass count, e.g. "300/300 random triangles").
- Separate "same mechanism" from "same number by coincidence." Fibonacci's Pisano period 24 and an unrelated lcm(8,3)=24 are PARTIAL HEURISTIC at best — flag "no map between them" explicitly even while noting the coincidence.
- Audit code vs. documentation separately from the math correspondence — note where the live implementation (e.g. stage derived from
pulseCount mod 150) diverges from a spec/paper's stated rule (e.g. edge-triggered counter), even if this is orthogonal to the group theory. - Use BigInt/exact rational arithmetic when verifying modular/integer claims; never rely on floating point for PROVEN-tier verdicts. Reserve floating-point comparisons (with explicit relative-difference tolerances) for COMPUTED-tier live-telemetry checks.
- Report verification results as a pass/fail log, one line per assertion, with the exact numbers used (not just "verified").
Common Pitfalls
- Don't call something an isomorphism when it's only a map on one direction or only injective on a subset (e.g. wheel units mod 30 → zones mod 9 is not a group homomorphism from ℤ/30).
- Don't let digital-root or modular coincidences masquerade as structure — always ask "is there a shared generator/automorphism, or just arithmetic luck?"
- Don't average over cases to call something PROVEN — if it holds in 3 of 5 instances, it's HEURISTIC, not PROVEN, no matter how suggestive.
- Don't conflate continuous engine parameters (frequencies, damping, phase drift) with finite Numogram structure — these default to NO MAP; say so plainly rather than forcing a strained analogy.
- Don't skip stating the obstruction in NO MAP cases — "no correspondence" without the group-theoretic reason (different orders, gcd mismatch, no common quotient beyond trivial) is incomplete.
- Don't fix discrepancies between code and spec silently — flag them as their own category, separate from the mathematical correspondence table.