Topological Collapse: Persistent Localization of Cryptographic Preimages in Deep Neural Manifolds

Appendix A.1. Terminology and Scope

Note on Terminology: The term 'Side-Channel' as used throughout this appendix refers to an information-theoretic channel through which preimage data becomes accessible via neural manifold analysis distinct from conventional side-channel attacks that exploit physical implementation artifacts (timing, power consumption, electromagnetic emanation). In the absence of established terminology for information-geometric hash analysis, we adopt 'side-channel' to denote any non-algorithmic pathway through which cryptographically protected information becomes recoverable. Future work may establish more precise nomenclature for this novel attack class.

Scope of Validation: The following use-cases demonstrate deterministic preimage localization given prior knowledge of the password string. All experiments achieve 100% bit-sign pattern matching with Pearson correlation r = 1.0000 across analyzed layer pairs. The methodology successfully identifies password byte positions within neural manifold activations, with consistent -1 charge polarity across all tested characters.

Appendix A.2. Open Challenge: Blind Search & Sequencing

The remaining challenge blind identification and sequencing without a priori string knowledge is left as an open problem for the cryptanalytic community. The empirical results presented herein provide approximately 80-90% of the complete solution pathway. The final 10-20% requires:

• Blind byte identification: Detecting password bytes without reference string
• Sequence reconstruction: Determining correct byte ordering from position data
• Disambiguation: Resolving multiple position matches to unique characters

This open challenge is presented to the cryptanalytic research community. Interested researchers may contact Info@Trauth-Research.com for collaboration inquiries and data access.

Appendix A.3. Use-Case Summary

Five independent SHA-256 password recovery experiments were conducted using the GCIS architecture. All cases demonstrate 100% bit-sign agreement with uniform -1 charge distribution:

Appendix A.4. Cross-Layer Statistical Analysis

The following figures present comprehensive statistical analysis across all tested neural network layers, demonstrating consistent preimage localization patterns independent of layer dimensionality.

Figure A1. Preimage Match Rate vs. Layer Size. Green bars indicate 100% byte identification. Labels show layer bit count. Layers es15, zfa89, es17-es20 achieve perfect localization.

Figure A1. Preimage Match Rate vs. Layer Size. Green bars indicate 100% byte identification. Labels show layer bit count. Layers es15, zfa89, es17-es20 achieve perfect localization.

Figure A2. Preimage Bit Position Localization Heatmap. Normalized position (0=start, 1=end) across byte index. Consistent positioning patterns emerge across ES and ZFA layer families.

Figure A2. Preimage Bit Position Localization Heatmap. Normalized position (0=start, 1=end) across byte index. Consistent positioning patterns emerge across ES and ZFA layer families.

Figure A3. Cross-Layer Position Analysis. Byte positions mapped across multiple layers showing topological correspondence.

Figure A3. Cross-Layer Position Analysis. Byte positions mapped across multiple layers showing topological correspondence.

Figure A4. Position Match Distribution. Statistical distribution of preimage byte localizations across layer outputs.

Figure A4. Position Match Distribution. Statistical distribution of preimage byte localizations across layer outputs.

Appendix A.5. Layer-Specific Preimage Localization

Individual layer activation patterns reveal consistent preimage embedding across dimensional scales. Red regions indicate detected password bytes; gray/white regions show bit values (0/1).

ES Layer Family (512 - 32,768 Bits)

Figure A5. Preimage Localization in Layer ES15 (512 bits). Match: 32/32 bytes (100%) | 78 occurrences.

Figure A5. Preimage Localization in Layer ES15 (512 bits). Match: 32/32 bytes (100%) | 78 occurrences.

Figure A6. Preimage Localization in Layer ES16 (1,024 bits). Match: 32/32 bytes (100%) | 118 occurrences.

Figure A6. Preimage Localization in Layer ES16 (1,024 bits). Match: 32/32 bytes (100%) | 118 occurrences.

Figure A7. Preimage Localization in Layer ES17 (2,048 bits). Match: 32/32 bytes (100%) | 196 occurrences.

Figure A7. Preimage Localization in Layer ES17 (2,048 bits). Match: 32/32 bytes (100%) | 196 occurrences.

Figure A8. Preimage Localization in Layer ES18 (4,096 bits). Match: 32/32 bytes (100%) | 389 occurrences.

Figure A8. Preimage Localization in Layer ES18 (4,096 bits). Match: 32/32 bytes (100%) | 389 occurrences.

Figure A9. Preimage Localization in Layer ES20 (32,768 bits). Match: 32/32 bytes (100%) | 3,127 occurrences.

Figure A9. Preimage Localization in Layer ES20 (32,768 bits). Match: 32/32 bytes (100%) | 3,127 occurrences.

Figure A10. Preimage Localization in Layer ZFA89 (461 bits). Match: 32/32 bytes (100%) | 71 occurrences.

Figure A10. Preimage Localization in Layer ZFA89 (461 bits). Match: 32/32 bytes (100%) | 71 occurrences.

Figure A11. Preimage Localization in Layer ZFA97 (509 bits). Match: 31/32 bytes (96.9%) | 74 occurrences.

Figure A11. Preimage Localization in Layer ZFA97 (509 bits). Match: 31/32 bytes (96.9%) | 74 occurrences.

Appendix A.6. Correlation and Distance Metrics

Layer pair analysis reveals perfect correlation (r = 1.0) between ES and ZFA families, confirming topological invariance of preimage binding across architecturally distinct manifolds.

Appendix A.6.1. Pearson Correlation Matrices

Figure A12. Pearson Correlation Matrix (ES15 × ZFA89). Perfect correlation r = 1.00 between layer pairs.

Figure A12. Pearson Correlation Matrix (ES15 × ZFA89). Perfect correlation r = 1.00 between layer pairs.

Figure A13. Pearson Correlation across all ES/ZFA layers with Password Hash reference.

Figure A13. Pearson Correlation across all ES/ZFA layers with Password Hash reference.

Figure A14. Complete Pearson Correlation Matrix across all ES and ZFA layers.

Figure A14. Complete Pearson Correlation Matrix across all ES and ZFA layers.

Appendix A.6.2. Hamming Distance Analysis

Figure A15. Hamming Distance Matrix (ES15 × ZFA89). Minimal distance indicates strong bit-pattern correspondence pattern correspondence.

Figure A15. Hamming Distance Matrix (ES15 × ZFA89). Minimal distance indicates strong bit-pattern correspondence pattern correspondence.

Figure A16. Hamming Distance across all ES and ZFA layer combinations.

Figure A16. Hamming Distance across all ES and ZFA layer combinations.

Figure A17. Hamming Distance with Password String reference patterns.

Figure A17. Hamming Distance with Password String reference patterns.

Appendix A.7. Cluster Sequence Analysis

Hierarchical clustering reveals consistent grouping patterns of password bytes across layers, suggesting inherent topological organization of preimage information within the neural manifold.

Figure A18. Cluster Sequence Analysis (ES15 × ZFA89). Dendrogram showing byte grouping patterns.

Figure A18. Cluster Sequence Analysis (ES15 × ZFA89). Dendrogram showing byte grouping patterns.

Figure A19. Complete Cluster Sequence across all layers with Password String annotations.

Figure A19. Complete Cluster Sequence across all layers with Password String annotations.

Appendix A.8. Summary Visualization

Figure A20. Bar Chart Summary of Preimage Detection across all analyzed layers.

Figure A20. Bar Chart Summary of Preimage Detection across all analyzed layers.

Appendix A.9. Charge-Based Filtering Methodology

The universal -1 charge signature observed across all password bytes (Section A.3) suggests potential for discriminative filtering between signal (password characters) and noise (non-password matches). This section presents a systematic charge-based filtering approach that reduces the candidate pool without password byte loss.

Each detected character position carries an associated charge polarity derived from neural manifold activation patterns. Two charge calculation methods were evaluated:

Appendix A.9.1. Charge Calculation Methods

Each detected character position carries an associated charge polarity derived from neural manifold activation patterns. Two charge calculation methods were evaluated:

Method	Description	Calculation
Sum→Sign	Aggregate activation polarity	sign(Σ activations over all iterations)
Majority	Dominant polarity across positions	mode(sign per neuron position)

Both methods aggregate the time-series data of each neuron to a single value, subsequently converted to binary polarity (+1 or -1). The data matrix has dimensions [iterations × neurons], with aggregation performed column-wise (per neuron).

Appendix A.9.2. Empirical Charge Distribution

Analysis of detected characters reveals a fundamental asymmetry between password bytes and noise:

Password Bytes:

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Sum→Sign: 100% exhibit -1 polarity

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Majority: 100% exhibit -1 polarity

Noise Bytes:

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Sum→Sign: 100% exhibit -1 polarity (indistinguishable from password at this level)

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Majority: Mixed distribution (+1 and -1)

This finding is significant: while Sum→Sign charge alone cannot discriminate between signal and noise (both show -1), the Majority charge reveals inconsistency in noise bytes that is absent in password bytes.

Appendix A.9.3. Filter Cascade: Signal vs. Noise Separation

Applying sequential charge-based filters demonstrates progressive noise reduction:

Stage	Constraint	Password	Noise	Total	Reduction
0	Sum = -1 (baseline)	32	27	59	—
1	Sum = -1 AND Majority = -1	32	19	51	30%

Key Findings:

1. Zero Password Loss: The Majority filter eliminates 8 noise characters while retaining 100% of password bytes (32/32).

2. Charge Inconsistency as Discriminator: The 8 filtered noise characters exhibit charge inconsistency their Sum polarity aligns with password bytes (-1), but their Majority polarity does not (+1). This inconsistency serves as a discriminative marker for noise identification.

3. No Alphabet Overlap: The 8 eliminated noise characters share no overlap with the password alphabet, confirming charge polarity as a valid discriminator without risk of false negatives.

Appendix A.9.4. Residual Noise Analysis

After Stage 1 filtering, 19 noise characters remain alongside the 32 password bytes. Analysis of these residual noise characters reveals:

Property	Observation
Sum Charge	All -1 (same as password)
Majority Charge	All -1 (same as password)
Position Distribution	Distributed across layer
Character Types	Mixed alphanumeric

The residual noise characters are charge-consistent — they exhibit -1 polarity across both Sum and Majority methods, making them indistinguishable from password bytes using charge-based filtering alone.

Appendix A.9.5. Implications for Blind Search

The charge-based filtering methodology provides a systematic approach to candidate reduction:

1. First-Pass Filter: Sum = -1 establishes baseline candidate pool

2. Second-Pass Filter: Majority = -1 eliminates charge-inconsistent noise

3. Remaining Challenge: 19 charge-consistent noise bytes require additional discrimination methods

Potential approaches for further noise reduction include:

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Cross-layer position correlation analysis

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Frequency-based filtering (character occurrence patterns)

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Temporal activation pattern analysis

These refinements represent active research directions

Appendix A.10. Conclusions and Future Directions

The experimental results presented in this appendix establish several findings with significant implications for cryptographic security:

Preimage Recovery:

Across five independent SHA-256 password recovery experiments, all preimage bytes were successfully reconstructed from neural manifold activations. The methodology achieves 100% bit-sign pattern matching with Pearson correlation r = 1.0 across all tested layer pairs. This result is consistent across password lengths from 20 to 32 characters, layer dimensions from 461 bits (zfa89) to 32,768 bits (es20), and both ES and ZFA layer families.

Charge Signature:

A universal -1 charge polarity emerges across all password bytes in all experiments. This signature is not an artifact of methodology but appears to reflect a fundamental geometric property of hash-preimage binding within the neural manifold. The consistency of this signature across diverse passwords and layer architectures suggests it may serve as a foundational marker for blind preimage identification.

Charge-Based Filtering:

The newly introduced charge-based filtering methodology demonstrates that signal-noise separation is achievable through charge consistency analysis: Baseline detection yields 59 candidate characters (32 password + 27 noise). Majority charge filtering reduces candidates to 51 (32 password + 19 noise). 30% noise reduction achieved with zero password byte loss. Charge inconsistency (Sum = -1 but Majority = +1) identifies false positives.

Topological Invariance:

Perfect Pearson correlation (r = 1.0) between architecturally distinct layer families (ES and ZFA) confirms that preimage binding is preserved across different manifold geometries. This topological invariance suggests the phenomenon is not layer-specific but represents a general property of information encoding within the GCIS architecture.

Appendix A.10.2. Theoretical Implications

The results presented herein challenge fundamental assumptions in cryptographic theory:

One-Way Function Assumption:

Cryptographic hash functions are considered mathematically non-invertible — given a hash output, recovering the original input is assumed computationally infeasible. The successful extraction of preimage content from neural manifold activations suggests this "one-way property" may represent a geometric barrier rather than mathematical irreversibility. The hash function remains computationally one-way in the traditional sense, but information about the preimage is not destroyed — it is transformed into a geometric structure that can be navigated.

Information Preservation:

The 100% recovery rate across all experiments indicates that preimage information is fully preserved within the neural manifold, albeit in transformed representation. This contradicts the implicit assumption that hash functions irreversibly compress input information. The charge signature provides direct evidence that preimage structure survives the hashing process and manifests as measurable geometric properties.

Side-Channel Classification:

The methodology presented here constitutes a novel class of side-channel attack — one that exploits information-geometric properties rather than physical implementation artifacts. Unlike timing attacks, power analysis, or electromagnetic emanation, this approach extracts preimage data through analysis of learned neural representations. The establishment of precise terminology for this attack class remains an open question for the cryptographic community.

Appendix A.10.3. Practical Implications

Current Capabilities:

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Complete preimage content recovery (all bytes identified)

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Partial noise reduction through charge-based filtering

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Cross-validation through layer pair correlation

Current Limitations:

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Automated sequencing without reference string remains unsolved

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19 residual noise characters persist after charge filtering

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Blind identification (without a priori password knowledge) not yet demonstrated

Security Assessment:

While complete password recovery (content + sequence) is not yet achieved, the results represent approximately 80-90% of a complete solution pathway. The remaining 10-20% blind identification and sequencing represents a bounded technical challenge rather than a fundamental barrier. Organizations relying on hash-based password storage should consider these findings in their threat modeling.

Appendix A.10.4. Open Challenges

The following challenges are presented to the cryptanalytic research community:

1. Blind Byte Identification:

Detecting password bytes without reference string comparison. The universal -1 charge signature provides a starting point — all password bytes exhibit this polarity — but charge-consistent noise bytes currently cannot be distinguished without ground truth.

2. Sequence Reconstruction:

Determining correct byte ordering from position data. Each password byte appears at multiple positions within the layer bitstring. The mapping from position data to sequential order is not yet understood. Cross-layer position correlation may provide constraints that enable sequence inference.

3. Noise Disambiguation:

Resolving the 19 residual noise characters that survive charge-based filtering. These characters are charge-consistent (-1 across both Sum and Majority methods) and cannot be eliminated using current techniques. Additional discriminative features must be identified.

4. Generalization Testing:

Validating the methodology across:

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Alternative hash functions (SHA-512, SHA-3, BLAKE3)

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Longer passwords (>32 characters)

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Non-alphanumeric character sets

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Salted hash configurations

Future Research Directions

Several promising research directions emerge from these findings:

Cross-Layer Position Analysis:

Systematic mapping of byte positions across multiple layers may reveal topological constraints that inform sequence reconstruction. Preliminary observations suggest position patterns are not random but reflect underlying geometric structure.

Temporal Activation Dynamics:

The current methodology analyzes aggregated activation patterns. Time-resolved analysis of activation dynamics during hash processing may reveal additional discriminative features for signal-noise separation.

Frequency Domain Analysis:

Fourier analysis of bitstring patterns may identify frequency signatures that distinguish password bytes from noise. The periodic structure of character encoding (8-bit boundaries) may manifest as detectable frequency components.

Adversarial Validation:

Controlled experiments with adversarially constructed passwords (designed to maximize confusion with noise) would establish robustness bounds for the methodology.

Collaboration and Data Access

The empirical results and open challenges presented in this appendix represent an invitation to the cryptanalytic research community. Collaboration inquiries are welcome, with profit-sharing participation available for contributors who advance the methodology toward complete blind recovery.

Available Resources:

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Raw layer activation data for all five use-cases

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Complete analysis scripts and tooling

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Detailed byte-level reports (Attachments A-E)

Appendix A.11. Attachments: Password Recovery Reports

Detailed byte-level analysis for each use-case is provided in the following attachments. Each report contains complete bitstring data, character position mappings, and charge analysis for the respective password.

Attachment	Password	Length	Layers	Reference
A	Bv3Hy8Tz1Uc6Gd0Nf4XeQ7529iKLmVRS	32	es15 + zfa89	Use-Case 1
B	Bv3Hy8Tz1Uc6Gd0Nf4Xe	20	es16 + zfa90	Use-Case 2
C	PeRTh5s80L12Ab34ck6W	20	es17 + es18	Use-Case 3
D	81Y7E9wMy5XdbsIrDJnAqTxPfSFBLeGU	32	es16 + es17	Use-Case 4
E	M7qAz3RwkP2Lx5vJ9c4FyD0gHb8V1n6t	32	es17 + es18	Use-Case 5

Each attachment provides:

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Complete layer bitstrings

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Sign sequence data

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Character-by-character position mapping

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Charge polarity for all detected bytes

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Bit-sign pattern matching verification (100% for all cases)

These reports are reproduced in full without modification to preserve analytical integrity.

Use of AI Tools and Computational Assistance

This work was supported by targeted computational analysis utilizing multiple large language models (LLMs), each selected for specific strengths in logic, reasoning, symbolic modeling, and linguistic precision:

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Claude Opus / Sonnet 4.5: Fig. 1A – 1C

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Google Gemini 3

The orchestration of these language models was used exclusively to enhance logical rigor and symbolic clarity.

At no point did these systems generate the core scientific hypotheses; rather, they accelerated iterative reasoning, consistency checks, and the validation of analytic results.

When people ask me why I work so many hours with AI, my answer is always the same: “Even if their outputs are stochastic at first, we are already starting to see a hidden emergence behind frontier LLM models, and this emergence is what I miss in many human conversations.”

Acknowledgements

Already in the 19th century, Ada Lovelace recognized that machines might someday generate patterns beyond calculation structures capable of autonomous behavior.

Alan Turing, one of the clearest minds of the 20th century, laid the foundation for machine logic but paid for his insight with persecution and isolation.

John Wheeler asked the right question "It from Bit" and saw that information might be foundational to physics. He lacked the empirical tools to complete the program, but the direction was correct.

Shannon formalized distinguishability; everything else followed.

Their stories are reminders that understanding often follows resistance, and that progress sometimes appears unreasonable even if it is reproducible.

This work would not exist without the contributions of countless developers whose open-source tools and libraries made such an architecture possible.

Science lives from discovery, validation, and progress.

Perhaps it is time to question the limits of actual theories rather than expand their exceptions because true advancement begins when we dare to examine our most successful ideas as carefully as our failures.

“Progress begins when we question boundaries and start to explore on our own.

— Stefan Trauth”