Li and Boyle (2024) demonstrated that aperiodic tilings — including Penrose,Ammann-Beenker (AB), and Fibonacci geometries — give rise to a remarkable class ofquantum error-correcting codes (QECCs) in which quantum information is encoded throughthe geometric structure of the tiling itself. Their analysis establishes that these codes arerobust against erasure errors, where a finite patch of the tiling is deleted and reconstructedfrom the surrounding geometry. Here we introduce a substrate-level diagnostic for a distinctperturbation class we term silent relational corruption: the scrambling of a graph'sconnectivity topology while preserving local degree statistics. Unlike erasure, silent relationalcorruption produces no detectable gap in the substrate — the perturbation leaves no visibletrace. Although motivated by aperiodic quantum error-correcting codes, the present audit isclassical: it evaluates recoverability of privileged-persistence sites under controlled graphperturbations, not the performance of a quantum code. Using a matched degree-preservingrewiring protocol applied to AB (N=273) and Penrose (N=7,160) substrates generated viacut-and-project methods, we demonstrate that vulnerability to silent relational corruption issubstrate-dependent and predictable from a geometric invariant: the exo/endo persistenceaxis. AB substrates, whose structural logic is stored in hidden perpendicular-spacecoordinates (exo-addressed), retain near-ceiling identity recoverability under full relationalscrambling (AUC = 1.000). Penrose substrates, whose structural logic resides entirely in thelocal relational weave (endo-reconstructive), show substantially reduced identityrecoverability (AUC = 0.800). We further demonstrate that identity persistence and functionalreconstruction are operationally distinct processes — the former is address-led, the latterweave-led — and that hybrid architectures incorporating both channels providecomplementary redundancy under mixed damage. These findings suggest that silent relationalcorruption resistance should be evaluated as a substrate selection criterion for aperiodicerror-correcting architectures. Contact: niedzkatie@gmail.com
Li and Boyle (2024) demonstrated that aperiodic tilings — including Penrose,Ammann-Beenker (AB), and Fibonacci geometries — give rise to a remarkable class ofquantum error-correcting codes (QECCs) in which quantum information is encoded throughthe geometric structure of the tiling itself. Their analysis establishes that these codes arerobust against erasure errors, where a finite patch of the tiling is deleted and reconstructedfrom the surrounding geometry. Here we introduce a substrate-level diagnostic for a distinctperturbation class we term silent relational corruption: the scrambling of a graph'sconnectivity topology while preserving local degree statistics. Unlike erasure, silent relationalcorruption produces no detectable gap in the substrate — the perturbation leaves no visibletrace. Although motivated by aperiodic quantum error-correcting codes, the present audit isclassical: it evaluates recoverability of privileged-persistence sites under controlled graphperturbations, not the performance of a quantum code. Using a matched degree-preservingrewiring protocol applied to AB (N=273) and Penrose (N=7,160) substrates generated viacut-and-project methods, we demonstrate that vulnerability to silent relational corruption issubstrate-dependent and predictable from a geometric invariant: the exo/endo persistenceaxis. AB substrates, whose structural logic is stored in hidden perpendicular-spacecoordinates (exo-addressed), retain near-ceiling identity recoverability under full relationalscrambling (AUC = 1.000). Penrose substrates, whose structural logic resides entirely in thelocal relational weave (endo-reconstructive), show substantially reduced identityrecoverability (AUC = 0.800). We further demonstrate that identity persistence and functionalreconstruction are operationally distinct processes — the former is address-led, the latterweave-led — and that hybrid architectures incorporating both channels providecomplementary redundancy under mixed damage. These findings suggest that silent relationalcorruption resistance should be evaluated as a substrate selection criterion for aperiodicerror-correcting architectures. Contact: niedzkatie@gmail.com
Li and Boyle showed that the Penrose tiling defines a quantum error-correcting code: superpositions of tilings over isometry orbits protect quantum information against erasure of any bounded region. We extend the construction to the aperiodic monotiles discovered by Smith, Myers, Kaplan and Goodman-Strauss. For the Hat, we prove strong local indistinguishability for all Hat tilings, and we prove local recoverability unconditionally for all nonsingular Hat tilings via the torus parametrization of the underlying cut-and-project scheme. The remaining singular case reduces to one sharply posed geometric question -- can a region that is a union of hats be retiled a second way? -- which we verify computationally has no counterexample up to a substantial scale: a certified 2490-tile patch admits precisely one tiling by hats, so all 22490 of its tile-subregions retile uniquely. Unlike the Penrose, Ammann-Beenker and Fibonacci tilings, both monotiles form two local-indistinguishability classes, so their code spaces split into two erasure-correcting sectors carrying a superselected classical label. Whether the label survives depends on which isometries are gauged: the Spectre's classes are exchanged by a 30∘ rotation and merge once all proper isometries are gauged, whereas the Hat's are exchanged only by reflections. Under the physically natural gauge group SE(2), the Hat code therefore stores one robust classical bit -- the handedness of its long-range order, readable in any bounded window with separation Δ=∣K∣5/3 -- alongside its protected quantum sectors. It is the reflexible monotile, not the chiral one, that carries the chirality bit. We give the exact Perron-Frobenius data of both codes, including the per-class reflected-Hat frequencies (3∓5)/6 and the Spectre orientation-class frequencies (5±15)/10.
The Penrose tiling (PT) is an intrinsically non-periodic way of tiling the plane, with many remarkable properties. A quantum error-correcting code (QECC) is a clever way of protecting quantum information from noise, by encoding the information with a sophisticated type of redundancy. Although PTs and QECCs might seem completely unrelated, in this paper we point out that PTs give rise to (or, in a sense, are) a remarkable new type of QECC. In this code, quantum information is encoded through quantum geometry, and any local errors or erasures in any finite region, no matter how large, may be diagnosed and corrected. We also construct variants of this code (based on the Ammann-Beenker and Fibonacci tilings) that can live on finite spatial tori, in discrete spin systems, or in an arbitrary number of spatial dimensions. We discuss connections to quantum computing, condensed matter physics, and quantum gravity.
This paper develops an engineering-facing diagnostic bridge between the recoverability-based theory of structural irreversibility and practical evaluation in quantum error correction and monitored quantum platforms. The central problem addressed in this work is that structural recoverability, threshold-like margins, false-collapse, and Recoverability Index–type diagnostics provide useful theoretical distinctions, but they do not by themselves specify how experimentalists or engineers should organize platform-level data, compare regimes, or report recoverability-relevant behavior. The paper therefore introduces a proxy map from structural quantities to engineering diagnostics. Generalized recoverability is associated with recovery performance, decoder-assisted restoration, logical-state fidelity, and related reconstruction-sensitive quantities. Threshold-like margins are associated with entropy-like, noise-sensitive, or transition-localizing proxies. False-collapse is associated with mismatch windows in which collapse-like or transition-like indicators become strong while recovery-sensitive records still indicate residual recoverability. The paper also proposes a minimal diagnostic architecture for QEC and monitored platforms. This architecture consists of four layers: an appearance-sensitive proxy, a recoverability-sensitive proxy, a threshold-like locator, and a window-based reporting layer. Together, these layers support regime classification, early-warning detection, false-collapse-like mismatch identification, finite-window reporting, and cross-regime comparison. A QEC-oriented realization is presented to show how syndrome histories, logical-failure markers, logical recovery performance, entropy-like proxies, false-collapse window frequencies, and RI-like summaries can be organized into a coherent diagnostic workflow. The goal is not to replace detailed decoder analysis or hardware-specific benchmarking, but to provide a bridge-level language for deciding what should be logged, estimated, compared, and reported when studying the approach to effective irreversibility in error-managed quantum systems. This work should be read as an engineering and experimental companion to the broader recoverability program. Its contribution is to make structural irreversibility operationally legible by mapping theoretical quantities to measurable, computable, and reportable diagnostics in contemporary quantum platforms.
Quantum error correction is priced in combinatorial currency — rate, distance, threshold — while the physical cost of performing correction goes uncounted. Every cycle of active (measurement-based) error correction erases information: syndrome extraction, majority voting, and ancilla re-initialization each reset redundancy, and Landauer's principle prices each erasure at no less than kT·ln2 per bit. This paper derives the resulting thermodynamic floor for correction overhead — (n−k)/k · kT·ln2 per logical qubit per round for an [[n,k]] code — verifies its scaling across code families (linear for repetition, quadratic for surface, constant for constant-rate), and applies the JPCUB joules-per-correct-answer metric to rank correction architectures. A two-level nested (tree-structured) code is compared with flat Hamming codes of equal rate on a clustered NAND-like channel: hierarchical decoding cuts the per-round erasure count by a factor of 1.6 to 3, while the naive construction pays a residual-error penalty — a partial null that leaves the floor claim intact and defines the protocol for an industrial flash-memory test. Biological systems that run robust quantum processes without correction (photosynthetic energy transfer, radical-pair magnetoreception) show the floor is an architecture choice: structure prepays in capital energy what correction pays in operating erasures. Autonomous and dissipative correction is identified as the boundary of the claim. A closing section documents the 2026 quantum-algorithms-and-QEC curriculum snapshot: the energy bill the paper prices is absent from the field's own teaching materials as of 2026-08-26.
Three predictions derived from the Substrate Independent Topology Framework (SITF) concerning topological error protection in quantum computing systems. The predictions are entered into the prior art record before examination of any experimental dataset against which they would be tested. The registration timestamp is the operative prior art date. The three predictions concern: (1) local noise suppression scaling in a superconducting qubit chain engineered to enforce Fibonacci state space structure; (3) the substrate-independent suppression ratio per encoding depth level in Fibonacci-structured quantum systems across hardware platforms; and (4) a hierarchy of topological protection quality indexed to the attractor value of the state space recursion, distinguishing Fibonacci-structured from Ising-structured quantum systems. Each prediction is specific enough to be falsified by existing or readily obtainable experimental data from current quantum hardware platforms. The generative architecture that produces these predictions is not disclosed in this document. This preregistration establishes priority for the phenomenological predictions — what the framework predicts, where, and at what numerical values — without revealing the structural logic that generates them. The two are legally distinct and maintained separately. Note on prediction numbering: This preregistration covers P-QC-01, P-QC-03, and P-QC-04. P QC-02 is held pending completion of the substrate translation derivation required to state it with the operational precision a preregistration demands. P-QC-05 is withheld per standing protocol.
Quantum error correction is priced in combinatorial currency — rate, distance, threshold — while the physical cost of performing correction goes uncounted. Every cycle of active (measurement-based) error correction erases information: syndrome extraction, majority voting, and ancilla re-initialization each reset redundancy, and Landauer's principle prices each erasure at no less than kT·ln2 per bit. This paper derives the resulting thermodynamic floor for correction overhead — (n−k)/k · kT·ln2 per logical qubit per round for an [[n,k]] code — verifies its scaling across code families (linear for repetition, quadratic for surface, constant for constant-rate), and applies the JPCUB joules-per-correct-answer metric to rank correction architectures. A two-level nested (tree-structured) code is compared with flat Hamming codes of equal rate on a clustered NAND-like channel: hierarchical decoding cuts the per-round erasure count by a factor of 1.6 to 3, while the naive construction pays a residual-error penalty — a partial null that leaves the floor claim intact and defines the protocol for an industrial flash-memory test. Biological systems that run robust quantum processes without correction (photosynthetic energy transfer, radical-pair magnetoreception) show the floor is an architecture choice: structure prepays in capital energy what correction pays in operating erasures. Autonomous and dissipative correction is identified as the boundary of the claim.