Phases I–XX · Post-Phase Research · 2026-06-16
Affine Memory Architecture
on DN Ring
From a symbolic field hypothesis to a formal affine memory architecture. Mirror-13 is the MDL-minimal anchor of Aff(ZN). Recall = Structure Match + Prior + Hypothesis Space + Observation.
// Core operator family (Phase XVI)
M(a,b)(g,φ,p) = ((a·g+b) mod N, g, −p)
valid iff gcd(a,N)=1 → Aff(ZN) = ZN ⋊ ZN×
|Aff(Z₁₂)| = 12·φ(12) = 48 · Mirror-13 = M(−1,0)
// Recall formula (Phases XX + POT suite)
Recall = Structure Match + Prior + Hypothesis Space + Observation
// Wrong prior: 0.000 · No prior: 0.725 · Correct: 1.000
AFF(Z_N) CHARACTERIZED
PAPER C READY
VPE-001A PENDING
EXTERNAL MEMORY OPEN
CONJECTURE 1 EMPIRICAL
48
Valid Operators
Aff(Z₁₂)=12·φ(12)
13.11
MDL bits
Mirror-13 · 5.1× compact
+127.6%
Entropy Gain
M13 over D₁₂
0.812*
Inter-AI ARI (surface only)
*vs axes: REJECTED, see Clustering §
z=4.36
Morph. Signal
p<0.001 · pre-ontology
z=−1.04
External Memory
NOT ESTABLISHED · Ph.19
Phases XV–XVIII · Complete Classification
Affine Memory Group Aff(ZN)
Complete classification of all 144 (a,b) pairs for N=12. Mirror-13 is ONE of 48 valid operators — the MDL-minimal anchor, not the full computational core.
Phase 15–17 finding: Fixed Mirror-13 wins 3/18 domain conditions (score 0.083). Adaptive Aff wins 12/18 (score 0.450). Learned map: 1.000. Mirror-13 = symbolic anchor / MDL minimum. Full core = Aff(Z_N).
// General formula
M(a,b)(g,φ,p) = ((a·g+b) mod N, g, −p)
valid iff gcd(a,N) = 1
// For N=12: φ(12)=4 valid a-values
a=1 → Shift family 12 ops ← LUXVAR best
a=5 → Affine a=5 12 ops
a=7 → Affine a=7 12 ops
a=11 → Mirror family 12 ops ← Mirror-13 = b=0
Total: 48 = Aff(Z₁₂) = Z₁₂ ⋊ Z₁₂×
// Mirror-13
M* = M(−1,0)(g,φ,p) = ((N−g)%N, g, −p)
MDL = 13.11 bits · rank #1/1035
LUXVAR rank: #37/48 (shift family best)
// Conjecture 1 (empirical N=5..24)
Valid iff gcd(a,N)=1 → Aff(Z_N) = Z_N ⋊ Z_N×
Conjecture 1: Verified exhaustively for N=12 (all 144 pairs) and empirically for N=5,6,7,8,9,10,11,12,13,16,20,24. Formal proof requires computer algebra system. Left for future work.
Valid (a,b) Map — N=12 · 48/144 valid
| Subfamily | a | Count | LUXVAR | MDL |
|---|---|---|---|---|
| Mirror (M13 here) | 11 | 12 | #37–48 | min |
| Shift family | 1 | 12 | #1–12 | ~1 |
| Affine a=5 | 5 | 12 | mid | mid |
| Affine a=7 | 7 | 12 | mid | mid |
| Invalid (gcd>1) | 96 pairs | — | — | |
Mirror-(N+1) Entropy Gain · Verified N=5..24
General law (Conjecture 1 special case):
For any ring of size N, M(N−1,0) = Mirror-(N+1) is always valid, because gcd(N−1,N)=1 for all N≥2 (consecutive integers are coprime).
For any ring of size N, M(N−1,0) = Mirror-(N+1) is always valid, because gcd(N−1,N)=1 for all N≥2 (consecutive integers are coprime).
Orbit Unity
All N=5..24 ✓
BFS confirmed
BFS confirmed
Bijective Exit
gcd=1→bijection
algebraic
algebraic
Entropy Gain
ΔH>0 all N
+127.6% at N=12
+127.6% at N=12
M13-INV-001 · Phase VII · Zero Randomness
Five Invariants — All Verified (was_forced_true: False)
Exhaustive algebraic + BFS verification. No random sampling. All failures = 0.
INV-A · Polarity Parity
100%
pol=(−1)^n_IN×initial
1518/1518 paths verified
1518/1518 paths verified
INV-B · Phase Memory
12/12
exit=(N−phase)%N
bijection all 12 gates
bijection all 12 gates
INV-C · Orbit Unity
46→1
D₁₂: 46–48 orbits
M13: 1 orbit · BFS 312
M13: 1 orbit · BFS 312
INV-D · Non-Commutativity
288/288
R∘IN ≠ IN∘R
100% ring states
100% ring states
INV-E · Reachability
+300
M13-exclusive states
D₁₂:12 → M13:312
D₁₂:12 → M13:312
// Algebraic roots (INV-DEP-001)
Root A → Polarity flip: INV-A
Root B → Phase memory: INV-B, INV-D
Derived → INV-C, INV-E (orbit structure)
Two independent roots — not one.
// WHY-M13-001 Optimality
R1+R2 → IN uniquely forced: IN(g,φ,p)=(N,g,−p)
R3 alone: does NOT force bijection OUT
Mirror OUT: unique MDL=0 solution among bijections
→ OPTIMAL, not strictly NECESSARY
Ablation: Invariants per Root
SPEC-002 · NumPy Exact · 312×312
Eigen Spectrum of Mirror-13
Full diagonalization of the 312×312 transition matrix. All 5 invariants visible in spectral structure.
λ₁ = 1.000000 (unique · ergodic · INV-C)
λ₂ ≈ 0.800000 (mult≈22 · D₁₂ struct · INV-B)
Spectral gap = 0.200 → mixing ≈ 29 steps
Distinct eigenvalues: 54 (rounded)
Non-zero: 268 / 312
Real: 200
Complex conj. pairs: 56
Polarity block cross: 0.185 (INV-A 2-block)
// All 5 invariants visible spectrally:
INV-C → λ₁=1 unique (single communicating class)
INV-A → 2-block polarity structure
INV-B → D₁₂ spectrum × Z₁₂ phase dimension
INV-D → complex eigenvalues present (112 total)
INV-E → λ₁ unique + single communicating class
Note on multiplicities: Computed by rounding to 4 decimal places. Exact symbolic multiplicities require a computer algebra system — left for future work.
Eigenvalue |λ| Distribution (synthetic visualization)
Phase X · WHY-M13-001
MDL & Optimality
Mirror-13 is the minimum-description instance in the tested search space. OPTIMAL, not NECESSARY.
| MDL Component | Bits |
|---|---|
| Ring size N=12 (from 23 options) | 4.52 |
| Center position (k=N+1, 1-bit rule) | 1.00 |
| IN phase rule (identity = 0 bits) | 0.00 |
| IN polarity rule (flip = 1 bit) | 1.00 |
| OUT gate rule (mirror formula) | 3.58 |
| Transition rate (1/8 ≈ 0.13) | 3.00 |
| Total MDL | 13.11 bits |
vs Random Equiv.
5.1×
67.19 bits mean vs 13.11
Elegance Ratio
0.349
bits gained per bit described
MDL Comparison · Mirror-13 vs 1,035 Equivalents
VLO-001/002 · Pre-Registered
Core-30 LUXVAR Registry
30/30 confirmed · 5 axes · Manifest SHA-256 frozen · Sep 2025.
Core-30 Radial Axis Map
☀ Light (6)
ELŪZ–MAHAR, ŠARTHAL, ŠARŪM–DŌZ, XĀṆŌRAH✦, MĀTIR–ELŪZ, ĀNŪR
◈ Reflection (6)
ŠAURÉN, RAHMŌT–ŠĀ, HALZĀR, BĀṆIR–RAHT, ŠUKĀR–MAH, ṆAZRŪK
KB: 2 entries ⚠
KB: 2 entries ⚠
○ Silence (6)
ṆĀRAH–ŠĀ, GŌṆ–SIL, VARKHĀN, ŠALŌM–ḤATTAH, NĀRAH, ṆEMRŪH
KB: 3 entries ⚠
KB: 3 entries ⚠
⬡ Gate (6)
SĀR–ELŪZ, SAVAR–ELŪZ, DĀRZAN, RAZKĀN, ELŪRIM, RITĀM
↯ Motion (6)
ZARTH, ZUN–NĀR, KHALTŪR, KUZTŪR, TARHĀN–ELŪZ, KHIRZĀH
✦ XĀṆŌRAH — only Core-30 word with direct in-document definition: «رگ پنهان نور» (hidden vein of light) · p.53, Sep 2025.
THP-013A (1.3B scan): Natural morphological clustering = k=3, not k=5/6. Hexacore as emergent structure: REJECTED. k=6 SSE drop = −1.07% (negative). As designed ontology under investigation: open.
GEN-001→GEN-003R+ · Phase II
Phonotactic Benchmark · LUXVAR vs 6 Languages
Real corpora. Bootstrap CI. Publication-grade result.
GEN-001 Learnability
98.9%
5-fold CV · Δmajority=+32.2%
F1=0.947 · LUXVAR distinguishable
F1=0.947 · LUXVAR distinguishable
GEN-003R+ Uniqueness
0.738
95% CI: [0.703, 0.781]
vs EN, DE, FA, Esperanto, Lojban, Toki Pona
vs EN, DE, FA, Esperanto, Lojban, Toki Pona
Cross-Classification
≈50%
LUXVAR→Lojban: 50%
LUXVAR→Toki Pona: 54%
Phonotactically independent
LUXVAR→Toki Pona: 54%
Phonotactically independent
Phonotactic Distance — LUXVAR vs Reference Languages
// GEN-003R+ distances
LUXVAR ↔ Esperanto : 2.27
LUXVAR ↔ Lojban : 3.26
LUXVAR ↔ Toki Pona : 4.67
Esperanto↔Lojban : 1.26 ← refs closer to each other
→ LUXVAR IS PHONOTACTICALLY DISTINCTIVE
// THP-013A · 1.3B rows
z=4.36 · p<0.001 · natural k=3
k=6 SSE drop: −1.07% (NEGATIVE)
Root→Frequency V: 0.003 (negligible)
→ Hexacore axes NOT natural morphological structure
VPE-001B · 3 Independent AI Systems · Blind
AI Clustering — Morpheme Families Recovered, Axial Recovery REJECTED
Claude, GPT-4, Gemini independently cluster Core-30. No VARZIN knowledge provided. This inter-AI agreement is a surface-morphology signal, not axis recovery — see the Track D resolution below.
Inter-AI ARI Matrix (morphological agreement only)
| Comparison | ARI | Interpretation |
|---|---|---|
| Claude ↔ GPT-4 | 1.000 | Perfect agreement |
| Claude ↔ Gemini | 0.812 | Strong inter-model agreement |
| GPT-4 ↔ Gemini | 0.624 | Strong inter-model agreement |
| Mean inter-AI | 0.812 | Models agree with each other |
| vs designed axes | 0.13 | Low — axes NOT recovered |
| Mistral 7B vs axes (Track D) | −0.034 | REJECTED — random-chance level |
| Llama 3 8B vs axes (Track D) | 0.015 | REJECTED — random-chance level |
| ELUZ family (all 3) | ✓ | 100% consensus (surface spelling) |
| SHA family (all 3) | ✓ | 100% consensus (surface spelling) |
| NAR family (all 3) | ✓ | 100% consensus (surface spelling) |
Resolved finding (Track D, Aug 2026): Models agree with each other (ARI=0.812) but this reflects shared surface morphology, not the designed algebraic axes (ARI=0.13). A dedicated adversarial audit under a strict Semantic Starvation protocol — reported in "Morphemes vs. Manifolds" (Zenodo 10.5281/zenodo.22101179) and the Level-1 monograph (Zenodo 10.5281/zenodo.22036769) — independently tested Mistral 7B and Llama 3 8B against the true orbit labels across 30 seeds each and found near-random recovery (Mistral ARI=−0.0337±0.039, Llama 3 ARI=0.0152±0.028). The models fall into "Morphological Hijacking": clustering by shared spelling roots (e.g. ELŪZ) rather than the underlying Aff(Z₁₂) orbit structure. The "AI-recoverable" claim is therefore REJECTED, not preliminary. The phonotactic/morphemic findings of the LUXVAR paper itself are unaffected — only the inter-model-agreement-as-structural-recovery interpretation is retracted. VPE-001A (human-rater protocol, N=59) remains the open test of whether primed human cognition can succeed where these models failed.
Final Benchmark · 5 Sections · 11 Tests · N=1M
Complete Computational Validation
B1 Family Learnability
74.2%
Δ=+24.2% · Strongest result
D1 Protocol Spine
+0.186
Real mod=0.140 vs shuffled=−0.046
Not random labeling
Not random labeling
E1/E2 Frequency
ABSENT
Δ=−3.6% · n=704
5 independent tests
5 independent tests
MASK-001 — Protocol Depth Test
M1 Morpheme→Code
32.4%
ELUZ→R1 · Δ=+8.4% · SURVIVES
M2 Prefix Hidden
36.1%
ELUZ-MAHAR→XXXX-MAHAR · SURVIVES
M3 Total CVCV
24.1%
≈ chance (24%) · COLLAPSES
MASK-001 verdict: Protocol is deeper than morpheme. Sub-character phonotactic patterns survive masking. M1+M2 SURVIVE · M3 COLLAPSES.
VARZIN-SIM-001 · 5-Layer Closed Environment
Closed Simulation — INTEGRATED Verdict
Five layers running together: D12+M13, State Field, LUXVAR, ETFM/Torus, Perturbation Engine.
SIM-CORE-001
STRONG
All 5 inv: 1.000 in env
Mirror-13 is hard core
Mirror-13 is hard core
SIM-LUX-001
STRONG
LOO: 93% closed env
vs 62% isolated (Ph.10)
vs 62% isolated (Ph.10)
SIM-FREQ-001
WEAK
ΔH=0.044 <0.05
Frequency: symbolic only
Frequency: symbolic only
SIM-ETFM-001
WEAK
ΔH=0.012 · viz only
Not computational core
Not computational core
SIM-FAIL-001
STRONG
#1 polarity · #2 center
Criticality confirmed
Criticality confirmed
SIM-EMERGE-001
MODERATE
2/5 signals · partial
Not full emergence
Not full emergence
VMM-001 · Visualization Layer
ETFM — Earth–Torus Field Model
Toroidal geometry visualization. Drag to rotate. SIM-ETFM-001: ΔH=0.012 — visualization only, not computational core.
Scientific boundary: SIM-ETFM-001 confirmed ΔH=0.012 < threshold 0.05. ETFM adds no measurable computational entropy. Torus coordinates describe gate positions — do not generate new invariants. This separates ETFM from the Mirror-13 mathematical core.
R
120
Major radius
r₀
50
Minor radius
φ
0.00
Rotation
ΔH
0.012
Entropy add
τ
0.42
r₀/R·cosθ
🖱 Drag to rotate · Scroll to zoom · Visualization only — not a computational claim
Mirror-13 · 312-State Live Simulator
Mirror-13 Live Walk
Real-time walk on the 312-state space. λ₁=1 (ergodic) · λ₂=0.8 · gap=0.2 · mixing≈29 steps
Total States
312
288 ring + 24 center
MDL
13.11 bits
vs 67.19 random mean
ΔEntropy
+127.6%
over D₁₂
Simplicity Rank
#1/1035
among equiv. systems
Steps: 0
Current State
Live Entropy (bits)
Polarity Balance
Center Visits
Live Invariant Verification
Phase XX + POT Suite · Final Formulation
The Recall Formula
The original "not learning, but remembering" refined across 20 phases and 6 POT tests into a conditional formula with testable predictions.
// RECALL FORMULA (final)
Recall = Structure Match
+ Prior
+ Hypothesis Space
+ Observation
// Recall occurs when limited observation,
// within a suitable hypothesis space,
// activates the correct prior.
// Prior ranking (POT-001 to POT-005)
Wrong prior: acc = 0.000 ← catastrophic
No prior: acc = 0.725 ← data-driven, slow
Self-org prior: acc = 1.000 ← via hypothesis testing
Injected prior: acc = 1.000 ← direct
// Phase 20: STRUCTURE_MATCH_REQUIRED
AffWorld → Aff rank #1 hit@3=0.998 ✓
MarkovWorld → Aff rank #4 hit@3=0.248 ✗
FSMWorld → Aff rank #6 hit@3=0.238 ✗
Testable prediction (POT-004): Wrong prior (acc=0.000) is far worse than no prior (acc=0.725). If the VARZIN protocol induces a structurally wrong prior, it is more harmful than no protocol at all.
Prior Strength → Recall Accuracy (POT-001)
POT-001–005 · SDT-001/002
Prior Origin Tests
Where does the affine prior come from? Can it emerge without injection?
| Test | Verdict | Key Finding |
|---|---|---|
| POT-001 Alignment | CONFIRMED | Recall ∝ prior (monotone, Δ=+0.275) |
| POT-002 Origin | INJECTED BEST | 1.000 > 0.861 > 0.725 > 0.000 |
| POT-003 Transfer | PARTIAL | N=12 formula → N=24 only |
| POT-004 Wrong Prior | CONFIRMED | Wrong acc=0.000 (catastrophic) |
| POT-005 Emergence | EMERGES | Prior emerges via hypothesis testing |
| SDT-001 Discovery | BIASED | Advantage was prior-driven |
| SDT-002 Prior-Free | PRIOR REQ. | AffWorld ID acc=0.000 without prior |
POT-005 (PRIOR_EMERGENCE_BY_HYPOTHESIS_TESTING): Agent C converges to affine hypothesis in 100% of episodes at query #1. Prior emerges from: observation + hypothesis space + compression scoring. The hypothesis space must contain "affine" as a candidate.
Phase 20 · Per-World AffAdaptive Performance
VPE-001 · Only Remaining Non-Circular Test
Blind Human Rater Study
Can independent humans group Core-30 words above chance? This is the only test that can resolve semantic axis validity.
// Pre-registered blind semantic test
Task: Classify 30 Core-30 words into 5 axes
Raters: ≥ 5 independent, blind to VARZIN
Prior: NONE — no VARZIN knowledge allowed
Metric: Fleiss κ (multi-rater agreement)
Seed: 130,474,528 (fixed)
// Pre-registered thresholds
κ ≥ 0.50 → STRONG
κ ≥ 0.40 → PRIMARY (publication minimum)
κ 0.20–0.40 → MARGINAL
κ < 0.20 → FAIL
// Key question
Humans → morpheme families (like AIs)?
→ morphemic structure IS primary
Humans → semantic axes (as designed)?
→ axes exist in cognitive layer
Either result advances the project.
Project Status by Layer
Phase 19 · Sealed-Target Negative Control
External Memory — Not Established
Phase 19 Result
z = −1.042
MirrorAnchor: −1.042 (below chance)
Oracle positive control: 1.000 ✓ (pipeline verified)
Random baseline: 0.181 · Prior: 0.136
VERDICT: EXTERNAL_MEMORY_NOT_ESTABLISHED
Oracle positive control: 1.000 ✓ (pipeline verified)
Random baseline: 0.181 · Prior: 0.136
VERDICT: EXTERNAL_MEMORY_NOT_ESTABLISHED
What Phase 19 established: The scoring pipeline works (Oracle=1.000). Without an independent receiver, no external memory can be established. Phase 20 refined: receiver must be structurally aligned (affine-compatible prior) for recall to work.
Conditions for real test (Phase 20 + POT):
1. Sealed targets with published hash before trial
2. Independent receiver with affine-compatible prior
3. Pre-registered threshold: z > 2.0 on ≥3 independent trials
1. Sealed targets with published hash before trial
2. Independent receiver with affine-compatible prior
3. Pre-registered threshold: z > 2.0 on ≥3 independent trials
QMSG v3 · Synthetic · 1.3B Rows
QMSG — Synthetic Injection Detectable
Three-arm design. Blind interval detection. Bonferroni correction.
Scientific boundary: This is a synthetic corpus analysis. Detection confirms generator design, NOT QMSG field resonance.
ARM2 · 1.3B Scan
rank 1/8
Interval 100k: chi2=90,981
enrichment=8.00× · rate=1.000
enrichment=8.00× · rate=1.000
Harmonic Detection
200k/500k
Multiples also 8.00×
Harmonic structure detectable
Harmonic structure detectable
ARM3 Shuffled Control
No signal
Best p=0.046 → NS after Bonferroni
Pattern destroyed by shuffle ✓
Pattern destroyed by shuffle ✓
QMSG 1.3B Scan · chi² per Interval · True = 100k
Scientific Integrity
What Was Removed
Every claim not supported by tests has been explicitly removed from the project.
| Removed Claim | Test | Result |
|---|---|---|
| Frequency encoding in word form | 5 tests n=704 | Δ=−3.6% · absent |
| Frequency affects dynamics | SIM-FREQ-001 | ΔH=0.044 <0.05 |
| ETFM as computational layer | SIM-ETFM-001 | ΔH=0.012 · viz only |
| Mirror-13 logically necessary | WHY-M13-001 | OPTIMAL only |
| Mirror-13 as unique | UNI-STRESS-001 | 1,035 equiv. found |
| Natural hexacore k=6 | THP-013A 1.3B | k=3 natural · k=6 neg. |
| Mirror-13 = full core | Phase 15–17 | anchor only · 3/18 wins |
| Aff(Z_N) general engine | 3-World Bench | DOMAIN_SPECIFIC |
| External memory field | Phase 19 | z=−1.042 |
| Full emergence | SIM-EMERGE-001 | 2/5 signals · partial |
| Consciousness/field effects | — | No test, no evidence |
Confidence by Layer
Project Position · Honest Assessment
Three-Layer Scientific Map
Layer 1 — Proven (80–90%)
✓ Affine family Aff(Z_N) characterized
✓ Mirror-13 MDL=13.11 · rank #1/1035
✓ 5 invariants · 0 failures · no randomness
✓ Spectral: λ₁=1 · gap=0.2 · mix=29
✓ LUXVAR morphology z=4.36 · p<0.001
✓ Family learnability 74.2%
✓ Protocol spine +0.186
✓ Phonotactic uniqueness 0.738 (real, but attributable to component choice — see note below)
✓ Sub-character grounding (MASK-001)
✓ Conjecture 1: formal proof complete for all N≥2
✓ Mirror-(N+1) single-orbit: formal proof complete for all N≥2
✓ Mirror-13 MDL=13.11 · rank #1/1035
✓ 5 invariants · 0 failures · no randomness
✓ Spectral: λ₁=1 · gap=0.2 · mix=29
✓ LUXVAR morphology z=4.36 · p<0.001
✓ Family learnability 74.2%
✓ Protocol spine +0.186
✓ Phonotactic uniqueness 0.738 (real, but attributable to component choice — see note below)
✓ Sub-character grounding (MASK-001)
✓ Conjecture 1: formal proof complete for all N≥2
✓ Mirror-(N+1) single-orbit: formal proof complete for all N≥2
Layer 2 — Open (VPE-001A)
✗ AI clustering vs axes — REJECTED (Track D, Mistral/Llama 3 near-random)
❓ Inter-subjective axis validity (κ≥0.40?) — now the open question for VPE-001A
❓ LUXVAR bridge: a=1 shift or Mirror-13?
❓ Geo→Word→Geo full closure (>70%?)
❓ ETFM: can it produce new invariants?
❓ Inter-subjective axis validity (κ≥0.40?) — now the open question for VPE-001A
❓ LUXVAR bridge: a=1 shift or Mirror-13?
❓ Geo→Word→Geo full closure (>70%?)
❓ ETFM: can it produce new invariants?
Layer 3 — Hypothetical
✗ External memory field (z=−1.042)
✗ ETFM as physical field model
✗ Frequency→meaning (V≈0.003)
✗ Consciousness field effects
✗ Planetary resonance claims
✗ LUXVAR universality beyond design
✗ ETFM as physical field model
✗ Frequency→meaning (V≈0.003)
✗ Consciousness field effects
✗ Planetary resonance claims
✗ LUXVAR universality beyond design
Computational core — independent adversarial audit (July 2026): A separate four-stage audit tested whether VARZIN's structure-detection pipeline (as used in benchmark_003 onward) offers a distinct, generalizable computational contribution beyond specialized baselines. Verdict: the one measurable internal advantage did not survive testing on an independently-coded generator and could not be attributed to a specific feature. Full report: VARZIN-COMPUTATIONAL-AUDIT-CLOSURE-v1 (available on request). This concerns the numeric structure-detection pipeline specifically; readers should apply the same caution to any LUXVAR-specific result built on the same underlying feature-engineering approach, pending direct confirmation.
LUXVAR-EMERGENCE-001-v2 attribution test (July 2026, N=200, pre-registered): The phonotactic uniqueness measurement above (0.738) is real and reproducible, but a locked, adversarially-reviewed attribution test found the signal is attributable to the choice of the 18 base root/suffix components, not to the root+root+suffix combination process itself. Across a 9-feature information/structure battery against 4 matched-complexity control generators, no evidence of computational emergence beyond generator-level design was found. Full closure documentation available on request.
Research Roadmap v2 · Three Paths
What Comes Next
Path A — Mathematical
A1
Formal proof Conjecture 1 for all N (Lean/Coq)
A2
Full characterization Aff(Z_N) spectral properties
A3
Paper C: add §10 Affine Family formal
No blocker — start now
Path B — LUXVAR Grammar
B1
VPE-001A: ≥5 blind raters · κ≥0.40
B2
Map Core-30 to shift family (a=1), not Mirror-13
B3
Morpho-syntactic rules from phonotactic structure
Blocker: rater recruitment
Path C — External Memory
C1
Sealed targets — hash published before trial
C2
Independent receiver with affine-compatible prior
C3
z > 2.0 on ≥3 trials = establish
Blocker: independent participant
Publication Status
Four Papers
Paper A — LUXVAR Structure
v2.2 PUBLIC · CORRECTED
v2.1 (restricted): 10.5281/zenodo.20691858
v2.2 (current, public, Aug 2026): 10.5281/zenodo.22115483
Morphemic structure · Frequency absent · Phonotactic fingerprint
Note: v2.2 adds an Aug 2026 addendum stating the "AI-recoverable" claim in the title is REJECTED (ARI=0.812 reflects surface agreement only), not merely preliminary — per the independent Track D audit below. Phonotactic/morphemic findings unaffected; distinctiveness attributable to component choice, not emergence (LUXVAR-EMERGENCE-001-v2).
v2.2 (current, public, Aug 2026): 10.5281/zenodo.22115483
Morphemic structure · Frequency absent · Phonotactic fingerprint
Note: v2.2 adds an Aug 2026 addendum stating the "AI-recoverable" claim in the title is REJECTED (ARI=0.812 reflects surface agreement only), not merely preliminary — per the independent Track D audit below. Phonotactic/morphemic findings unaffected; distinctiveness attributable to component choice, not emergence (LUXVAR-EMERGENCE-001-v2).
Paper C — Mirror-13 & Aff(Z_N)
READY
"Mirror-13 and the Affine Memory Family over Cyclic Gate Spaces"
Conjecture 1 · 5 invariants · MDL=13.11 · SPEC-002
Target: arXiv cs.FL / Zenodo
Conjecture 1 · 5 invariants · MDL=13.11 · SPEC-002
Target: arXiv cs.FL / Zenodo
Paper B — VPE-001A
PENDING
≥5 blind raters · Fleiss κ≥0.40
Morpheme families vs semantic axes
Requires: independent raters
Morpheme families vs semantic axes
Requires: independent raters
Paper D — Track D Cognitive Audit
PUBLISHED · REJECTS A's ARI claim
"Morphemes vs. Manifolds: Diagnosing Structural Blindness and Morphological Hijacking in LLMs" (Aug 24, 2026)
DOI: 10.5281/zenodo.22101179
Mistral 7B ARI=−0.034 · Llama 3 8B ARI=0.015 vs true axes, 30 seeds each
Companion monograph: 10.5281/zenodo.22036769 (Level-1 Computational Stack, also covers Tracks A–C)
DOI: 10.5281/zenodo.22101179
Mistral 7B ARI=−0.034 · Llama 3 8B ARI=0.015 vs true axes, 30 seeds each
Companion monograph: 10.5281/zenodo.22036769 (Level-1 Computational Stack, also covers Tracks A–C)
One-Sentence Project Definition
VARZIN is a proposed affine symbolic memory architecture M₍ₐ,ᵦ₎(g,φ,p)=((a·g+b) mod N, g, −p) built on a DN ring, where valid operators form Aff(ZN)=ZN⋊ZN×, Mirror-13=M(−1,0) is the MDL-minimal anchor, and recall requires: structure match + prior + hypothesis space + observation.
Reza Nirouyar · ORCID: 0009-0000-4690-6842 · 2026-06-16