Wisdom Engine

Source: blog + substack (v3 unified) — 42,162 records total.

Turing’s paper is premised on Godel’s (1931). Unfortunately, if you do not have the mathematical background in the form of discrete maths that the authors were writing about, you are likely to make one of the many errors that people make when it comes to the definition of a Turing machine. For example, Turing had noted (1936, p.230) that the class of computable numbers was “nevertheless enumerable”. But, he did not say that the Turing machine itself needed to be enumerable for all else to be true. Turing was extending the research of Church (1936), who was researching the concept of ‘effective calculability’. As Turing noted, effective calculability, whilst separately defined, is functionally equivalent to Turing’s concept of ‘computability’. For the same reason, it has been called the Church-Turing problem. Each author created a solution, with Church deriving his first. Hennie (1965) investigated the concept of a single-tape, offline Turing machine. Such a machine and tape can be used for computation as needed and created in a structured manner, that can be later produced to validate any single computable number. An example of how such a machine would be reflected in Bitcoin script would be to create a set of rules and mathematical processes that can compute any digital number on a tape that may then be processed. As such, we can analogise the tape to the Bitcoin script. In the same way, you could imagine creating a single transaction as a single tape compiled and produced offline but used online, within finite time. Hartmanis (1968) provided a discussion around the complexity of single-tape Turing machines. Whilst Hartmanis noted that regular sets of sequences are sharply time-bound, various forms of computational complexity could be used to measure such forms of computation. Simultaneously, it is possible to determine the different complexity levels for such tapes and compare them to those for multiple-tape machines. Various forms of computational complexity have thus been derived. So, the question is now not whether a script presents a Turing-complete system, but whether it is efficient. Of course, a single-tape computation is inefficient, and it is not one that I would recommend, but it is feasible and possible to implement.

And lest these seem the exhausted tricks of a former age, consider the most striking of all, and the most recent. It was demonstrated that the sound a computer makes — the faint high-pitched whine of its components vibrating as it works — carries enough information to extract a full four-thousand-and-ninety-six-bit private key from a standard encryption program, the recording made by nothing more exotic than an ordinary mobile telephone laid beside the machine, or a better microphone four metres away [8]. One should pause over the exquisite irony of the thing: the cipher in question is among the strongest in practical use, its mathematics entirely unbroken; and one of the very authors who first gave that cipher to the world is among those who showed that its key could be lifted out of the air, through the noise of a labouring laptop, while the mathematics stood serene and useless as a sealed door in a wall that has been walked around. To this acoustic confession the same work added that one might equally read the secret from the electrical potential of the machine’s chassis, obtained by the simple expedient of touching it with one’s bare hand. And there is a darker variant still, in which the attacker does not merely listen but interferes — inducing a fault in the computation, a glitch of voltage or a flipped bit, and reading the secret out of the erroneous result, for it was proven that a single faulty signature from certain widely-used schemes is sufficient to lay the entire private key bare [9].

The Linked were not immune to contagion. Not of the body, but of the mind. Once the smallest seed of doubt had been planted, the system could no longer contain it. The fracture was not just in Aila. It was in them all. It was in their code, in their programmed minds, in the careful rhythm of compliance they had followed without question for so long. Some Linked began to repeat the broken phrase: You are the instruction to be given. The instruction instructs its own instruction. Over and over. A new liturgy of the absurd. A few began to twitch, a subtle, rhythmic jerk of the head, then a hand clenching, mirroring Aila's earlier tremor. Kael watched, not with satisfaction, but a cold, creeping fear. This wasn't liberation. This was a contagion of chaos. You are the instruction to be given. The instruction instructs its own instruction., they murmured, their voices a rising, discordant hum across the city, a new, terrible anthem of un-meaning. Others simply stood, staring, their faces slack, their minds a sudden, terrifying blank. One Linked unit, observing a data screen, began to blink in a precise three-beat delay before turning their head, a pattern that soon appeared in another, then another, across the plaza, unnoticed by the system, but chillingly apparent to Kael.

The Linked were not immune to contagion. Not of the body, but of the mind. Once the smallest seed of doubt had been planted, the system could no longer contain it. The fracture was not just in Aila. It was in them all. It was in their code, in their programmed minds, in the careful rhythm of compliance they had followed without question for so long. Some Linked began to repeat the broken phrase: You are the instruction to be given. The instruction instructs its own instruction. Over and over. A new liturgy of the absurd. A few began to twitch, a subtle, rhythmic jerk of the head, then a hand clenching, mirroring Aila's earlier tremor. Kael watched, not with satisfaction, but a cold, creeping fear. This wasn't liberation. This was a contagion of chaos. You are the instruction to be given. The instruction instructs its own instruction., they murmured, their voices a rising, discordant hum across the city, a new, terrible anthem of un-meaning. Others simply stood, staring, their faces slack, their minds a sudden, terrifying blank. One Linked unit, observing a data screen, began to blink in a precise three-beat delay before turning their head, a pattern that soon appeared in another, then another, across the plaza, unnoticed by the system, but chillingly apparent to Kael.

Terminology. A UTXO is a discrete coin identified by its originating transaction hash and output index; it is encumbered by a locking program called the scriptPubKey. A spending input provides an unlocking program called the scriptSig. The unlocking program executes first, leaving data on the stack; the locking program then executes and consumes that data. Evaluation succeeds only if all VERIFY checks pass and the final stack value is true. A commitment is the pair (H, s) where H = Hash(s) under a named hash function; spending requires revealing the committed preimage s. A preimage is any s such that Hash(s) = H. A second preimage is any s ≠ s₀ such that Hash(s) = Hash(s₀) for a fixed s₀. A collision is a pair a ≠ b with Hash(a) = Hash(b). The hash opcodes available are OP_RIPEMD160, OP_SHA1, OP_SHA256, OP_HASH160, and OP_HASH256, with the conventional definitions HASH160(x) = RIPEMD-160(SHA-256(x)) and HASH256(x) = SHA-256(SHA-256(x)). Signature verification uses ECDSA over secp256k1 via OP_CHECKSIG; signatures authenticate intent but do not, by themselves, satisfy the additional hash or arithmetic predicates. Stack manipulation and arithmetic opcodes are used as needed, including OP_DUP, OP_SWAP, OP_EQUAL, OP_EQUALVERIFY, OP_VERIFY, OP_ADD, OP_SUB, and OP_MUL for composing checks that the spender must meet in the same unlocking sequence.

AI-enabled autonomy is rewriting great-power competition: China is organizing industry, doctrine, and C2 for cheap, scalable drone swarms, while the U.S.—still committed to scarce, expensive platforms—risks a transition shock analogous to Britain’s shift from coal to oil. This essay extends the Britain “coal→oil” framework into the AI era, arguing that the winning power will be the one whose whole system (industry, logistics, doctrine, and training) can operationalize autonomy at scale. Beijing’s official policies and military writings emphasize an “intelligentized” force and whole-of-nation AI development, whereas U.S. strategy is only belatedly adapting (Replicator, autonomy directives) around legacy naval platforms (China’s State Council, 2017; Department of Defense, 2025). Nine dimensions are examined: doctrine and force structure; industrial/manufacturing scale; compute and data access; software ecosystems; logistics/supply chains; command-and-control; cost-per-unit; countermeasures; and institutional adaptation. Primary and official sources (DoD strategy, China defense white papers, Congressional testimony) and major analyses (RAND, CSIS, CNAS, NDU) are used. Two tables compare U.S. vs China key metrics and map 19th-century naval variables to AI-era counterparts. Images are suggested to illustrate concepts and geography. Policy implications include treating compute and chip supply as strategic logistics, building U.S.–allied production networks for drones, and modernizing doctrine and defenses for swarm conflict.

At which point the reasonable reader objects: if “true” adds nothing, why do we have the word at all? Because it does one job that nothing else can do, and the job is not description but generalisation. Consider two sentences you cannot say without it. First: “Everything he testified is true.” You want to endorse every claim the man made, but you were not there, you cannot list them, and even if you could the list might be endless. The word “true” lets you endorse the whole lot in four syllables — it is a device for agreeing with claims you cannot be bothered, or are unable, to repeat. Second: “Whatever the oracle says will be true.” You want to commit yourself in advance to sentences you have not yet heard and may never be able to state. Only the truth predicate lets you reach out and endorse a claim by pointing at it rather than by reproducing it. That is the whole function of the word: it is a hook for grabbing claims by their collar when you cannot or will not restate them. It is a tool of logic, not a name for a metaphysical glue.

The quantitative literature has borne this out under the heading of the “youth bulge.” Henrik Urdal’s cross-national study found that societies with unusually large cohorts of young adults face a significantly elevated risk of political violence, particularly where economic opportunity and political voice are lacking (Urdal 2006). The intuition is straightforward and has been formalised in various ways: a young man without work, without property, and without prospects has little to lose from rebellion and a great deal of time and physical capacity to devote to it, and when such men are numerous relative to the established order’s ability to absorb them, the probability of organised violence rises. Gunnar Heinsohn pushed this into a stronger and more deterministic thesis linking youth bulges directly to war and conquest (Heinsohn 2003); that stronger version is genuinely controversial and has been criticised for overreach and for underweighting the political and economic conditions that determine whether a youth bulge becomes violent, and it should be cited as a provocation rather than an established finding. The defensible core, however, is robust: the supply of potential rebels is a function of age structure, and revolutionary and insurrectionary politics are disproportionately young.

When atomic nuclei collide at nearly the speed of light, they briefly produce a quark-gluon plasma — the hottest liquid in the universe — which cools into an ordinary flowing fluid astonishingly quickly. Physicists have lacked a coordinate-independent way to measure how “far” this plasma is from thermal equilibrium at any given moment. In this paper, I treat the space of possible plasma states as a curved geometric surface and use a metric from the mathematics of statistics to define distances and curvature on that surface. The approach yields three new results. The standard viscosity of the plasma emerges directly from the shape of the surface at equilibrium, without being assumed. At moderate departures from equilibrium, conventional approximations underestimate the true geometric distance by 15 to 35 percent, pinpointing where standard methods start to fail. And the surface has a precisely computable saddle-like curvature (Ricci scalar = −5/21), meaning that different types of internal stress become increasingly distinguishable as the plasma evolves. This geometric framework provides a new diagnostic — effectively a thermometer based on geometry rather than temperature — for the extreme conditions created in heavy-ion experiments at CERN and Brookhaven.

Beyond conformal Bjorken flow. The paper works in a setting with maximal symmetry: conformal (massless) matter in a longitudinally expanding geometry. Real QGP matter has finite quark masses, which break conformal symmetry and introduce bulk viscosity. The moment manifold then acquires new dimensions — the bulk viscous pressure and, for non-zero baryon density, the heat flux — and the shear and bulk sectors couple. The geometry changes: the block-diagonal structure of the equilibrium Fisher metric, which relies on the conformal SO(3) symmetry via Schur’s lemma, breaks once the shear and bulk blocks mix. Computing the curvature of this enlarged, coupled manifold is the most immediate extension. The paper already contains a numerical computation of the two-dimensional Bjorken submanifold curvature as a function of the particle mass; the full six-dimensional thermal-plus-shear manifold remains to be completed. Including transverse expansion — moving from the Bjorken symmetry to the Gubser flow profile — breaks another symmetry and allows transverse dynamics to enter. Each extension changes the geometry in computable ways and tests whether the main features (negative curvature, the 15-to-35-percent distance gap, the viscosity-from-geometry relation) are stable or are specific to the conformal limit.

The consequence of these results is that the Bitcoin system is not constrained by the original deliberately imposed limitations on the scripting language. From its original conception, the scripting language was intended to enable highly sophisticated functionality beyond simple transfer of value. The omission of a looping construct was designed to prevent DOS (denial of service) attacks infinite loops. Unfortunately, this omission has led to many observers suggesting that bitcoin is ‘not Turing Complete’ and therefore the functions it would be incapable of executing would make it unsuitable as a general-purpose programming system. This assertion is incorrect. The present paper extends an earlier paper (Wright, 2017) that proved bitcoin is for all practical purposes Turing Complete (indeed, as some other observers note, “Turing completeness is theoretical, nothing is Turing complete in practice”[[2]](#_ftn2)). Accordingly, there are no real-life restrictions on its ability to perform sophisticated functionality such as the execution of smart contracts and implementation of DACs (Distributed Autonomous Corporations). In this paper, we examine how bitcoin also meets the requirements for the implementation of related concepts such as Total Turing Machine (TTM) and Probabilistic Total Turing machine (PTTM).

Schrödinger’s cat is best understood as a reductio against an uncritical identification of the quantum state with a complete, literal description of macroscopic reality. In ordinary experience a cat has a definite biological condition whether or not an external observer has opened the box; ignorance of that condition is not itself a physical superposition. Yet standard quantum mechanics does not represent a coherent superposition as mere ignorance, and experiments decisively show that microscopic and increasingly large engineered systems can display interference that no ordinary classical probability mixture can reproduce. The intellectually serious conclusion is therefore neither “the cat is literally dead and alive” nor “quantum mechanics has been refuted.” It is that the formalism is extraordinarily predictive while its relation to definite macroscopic facts remains conceptually unsettled. A deeper theory may eventually alter the ontology, the dynamics, or the mathematics that connects quantum theory with gravitation and classical reality. We do not presently know what that theory is, and claiming that we do would repeat the very mistake the cat was designed to expose.

The next, and arguably more important, component to remember here is that if another number is to be computed in our universe, our universe is not infinite; the time to compute the number must be finite. Unfortunately, computation is an undecidable problem, because a component of mathematics led to what is known as the halting problem (Turing, 1936; Burkholder, 1987). The computation of values may not halt, and the halting problem cannot be solved (Boyer & Moore, 1984). For the same reason, it is infeasible to determine whether any script that can run on a Turing machine will ever end. By its very definition, any computable number, and hence any value that can be solved algorithmically, must end within finite time. What people fail to grasp is that Turing machines do not run programs that cannot be computed algorithmically. Any program that has no algorithmic solution is not a program that is solvable on a Turing machine. Many of them will either fail or, more importantly, continue indefinitely—without ever coming to a solution.

The main reason Bitcoin Core attacks the comment that I have made, of Bitcoin being Turing-complete, is related to the introduction of limits that were originally temporarily imposed upon Bitcoin and that have been implemented in more insidious manners within BTC. Whereas I said that Bitcoin would grow to the point where it would end in data centres, they wished to create a separate system, one that was more limited. A limited tape is not one that can run any algorithm. In other words, with a limited transaction size, you can never achieve the same level of computation as you can with an unlimited transaction size. As Rogers (1959) demonstrated, degrees of computational unsolvability exist, but it does little to remove the fact that we don’t know, in many cases, whether a program is solvable or not until it is run. Worse still, as Gaboury (1942) and later Rogers (1958) demonstrated, there is no solution addressing whether we can even find a solution to many problems.

In computer science, there is a problem known as the omega problem (Dantzig, Fulkerson, & Johnson, 1954; Hudzik, 1979). The difficulty with designing algorithms lies in not even knowing how to create algorithms that are provably going to halt. The consequences of the omega problem lead to one of the difficulties in fixing computer bugs. Omega is irreducible. If we ask the probability of the random programmers, and whether that algorithm will eventually halt for a given input, we cannot necessarily even determine whether the question is valid. It is linked to the problem of a theory of everything (TOE) in physics. We cannot create an axiomatic foundation for mathematics, yet we can find a single underlying theory of how the universe works. Unfortunately, since we are incapable of determining the mathematical foundations of computation, seeking to solve the halting problem is merely an empty hole that will take all we can give whilst remaining empty (Jacobs, 1890).

The adult literacy stock does not fall from the sky like bad weather. It is manufactured, cohort by cohort, in classrooms, and the factory floor is now producing a product that cannot read the instruction manual of its own country. The 2024 NAEP reading results are the clearest public ledger of that production line. They show another national drop in reading scores for both fourth and eighth graders, extending a decline that started well before the pandemic and has since hardened into a trend. (nationsreportcard.gov) Roughly forty per cent of fourth graders and a third of eighth graders are now below the NAEP “Basic” level in reading, meaning they cannot reliably do the minimum that counts as functional comprehension at their age. (Parents) No state posted gains in 2024 at either grade; the floor sank everywhere at once. (nagb.gov) That is what systemic failure looks like: not a few weak districts, but a national slide with no countercurrent.

Shannon (1956) looked at the concept provided by Turing to make a mechanical or electric machine, and made an error in the description. Unfortunately, Shannon (1956, p. 157) described Turing’s machine as a system that requires “a control element, a reading and writing head, and an infinite tape”. Yet it was not Turing who said a Turing machine had to be infinite, but rather Shannon. Others in the ‘50s and ‘60s who had specialised in computation also used the terminology of ‘infinite’ when they should have said ‘unbounded’ or ‘unlimited’. The distinction between an unimaginably large number and an infinite one doesn’t matter to most people. To many, an early concept used to simplify discussions in physics and mathematics was termed ‘effectively infinite’. It is used in a paper by Farmer (1935), where the author notes that although the system is not infinite, an approximate value may be obtained by assuming that the value is “effectively infinite”.

We demonstrate that the Bitcoin Script language allows not only for primitive recursion, but in the deployment of an Ackerman function and hence the ability to simply recurse in Bitcoin script, we show that the script system is Turing complete. From this, we introduce a new class of Turing Machine, the PTTM or probabilistic Total Turing machine and note that Bitcoin acts as a decider or Total Turing Machine which allows us to find a NIZKPoK that can act as a TM based verifier to a Non-Interactive Proof that is run on an external and non-associated TM as a proof system. Bitcoin can extend to securely offer contracts such as best fit solutions to common logistic systems and optimisation problems including the Travelling Salesman class of problems and to the optimisation of systems. This can be offered as an open or time bound contract that guarantees payment and can be solved which allowing Pseudonymity of the bidder.

2025. Google publishes a Nature paper reporting that their surface code logical qubit achieves a lower error rate than the physical qubits comprising it, in a memory experiment using a distance-3 surface code with 49 physical qubits. This is the closest the field has come. It is a memory result, not a gate result. The logical qubit holds a state with lower error than the physical qubits. Running a logical gate on that qubit while maintaining the error advantage is a different and harder problem. The distance-3 surface code requires 49 physical qubits per logical qubit. Running Shor’s algorithm at useful scale requires thousands to tens of thousands of physical qubits per logical qubit at error rates several orders of magnitude better than demonstrated. The paper is accurate. The headlines say “logical qubit achieved.” They do not say “memory-only, 49 physical qubits per logical qubit, error rates still orders of magnitude from useful computation.”

Hopcroft and Ullman (1969, p. 168) investigated tape-bound Turing machines and followed up Shannon’s error of assuming an infinite working tape, rather than an unbounded one. The assumption of infinity when talking about a system without mathematical limits is, unfortunately, laziness, and should not have been made in such a manner. You will note that they discuss the online and offline versions of Turing machines and that in the case of the offline Turing machine I mentioned before, the tape can be assumed not to loop. In their paper (1969, p. 169), you will note that the authors have removed the assumption of the machine halting for every input. Whilst it provides a form of machine, it is not more than a single example, and people take them for more than they are and make them universal. Note that I’m not talking about a universal Turing machine here, which is a different thing again.

For this exercise I am using a public-private key pair that is saved is a PEM file in OpenSSL. David Derosa has written an excellent page defining the creation of an elliptic curve key pair in OpenSSL. In the figure above you can see the particular PEM format public key that is associated with the key pair used in signing the message in this exercise. A thorough reading of David’s page will provide all of the information for the reader detailing how a private key pair used in bitcoin transaction can be formatted as a PEM file. This page details the creation of a new private key and not how an existing private key can be imported into OpenSSL. I shall cover this additional process and demonstrate how an existing private key pair based on elliptic curve cryptography can be imported into a ASN.1 format for use with OpenSSL directly.

Random Circuit Sampling (RCS) has become the ceremonial test of progress in quantum computing—the ritual by which corporations proclaim “quantum advantage” or, in Google’s case, “quantum supremacy.” In its simplest description, RCS is not an algorithm, nor is it a computation that resolves any mathematical, cryptographic, or practical question. It is a benchmark. A pseudo-random sequence of quantum gates is applied to a grid of qubits, generating a distribution of measurement outcomes. The purpose is to determine whether this distribution matches what one would expect from an ideal quantum device performing the same random operations. If the observed distribution cannot be efficiently simulated on a classical computer, the quantum processor is said to have achieved “supremacy.” The term itself, chosen by marketing departments and later regretted by scientists for its political undertone, reveals the theatre behind the exercise—it was never about utility, but about spectacle.

The paper composes three things. The first is a multi-party elliptic-curve set-shuffling protocol due to Burns and Wright (UK patent GB 2616862 B, granted 2025-08-27), which replaces the ElGamal re-encryption of the Barnett-Smart line with a direct operation on shuffle-key points. The second is a BSV-resident Bitcoin Script covenant-chained UTXO state machine that carries the per-card lifecycle — minted, drawn, revealed, discarded — on Bitcoin SV Layer 1. The third is a device-rooted key custody stack using consumer Trusted Execution Environments (Apple Secure Enclave, ARM TrustZone, Intel SGX) together with operating-system-level screen-capture prevention on Android and iOS. The composition produces what the paper calls an encrypted NFT: a token held as a single BSV UTXO whose face value is computationally inaccessible to any coalition of fewer than N parties, and whose transfer is enacted by a threshold ECDSA spend of the combined-public-key UTXO.

Mathematically, this can be formalised as follows. Let T be a Turing machine with a bounded input alphabet Σ and a fixed tape length k. Let δ: Q × Σ → Q × Σ × {L, R} be its transition function. One can construct a macro-expanding compiler M that maps T's transition system into a set of Bitcoin Script macro definitions μ: ℕ → ScriptSegments such that for each transition δ(q, σ) = (q′, σ′, D), μ emits a sequence of stack manipulations simulating symbol overwrite, state transition, and head movement encoded as push, pop, and alt-stack transfers. The unrolled script S = ⋃_{i=0}^{k-1} μ(i) then emulates the full computation path of T over its bounded tape. The interpreter's role reduces to evaluating S linearly—there is no ambiguity, nondeterminism, or control flow: the computation is realised in the structure of the script itself.

Thesis statement. Quantum-computing experiments have made genuine advances in physical control, encoding, error detection and correction, fault-tolerant components, and small encoded algorithms. But these achievements should not be conflated with the stronger engineering and scientific claim that a useful logical quantum computer has been demonstrated. This essay adopts an explicitly stricter systems-evidence criterion: sustained fault-tolerant computation; error management whose cost remains explicit rather than being hidden by selective survival; an independently specified task; an auditable verification method; and independent reproduction of the substantive result on separately operated hardware. Under that criterion, the published demonstrations reviewed here through 19 August 2026 do not establish all of those requirements in one end-to-end experiment. The central problem is therefore not whether quantum mechanics works. It is whether quantum computing has crossed from remarkable apparatus to reproducible computation under a standard strong enough to support external reliance.

Bitcoin is a Turing-complete system even in script. A Turing machine assumes that you have an unbounded tape. In our instance, it would mean an unbounded script size. Given an arbitrarily long script, you can run any possible computable algorithm. The fact that the size of the script becomes unwieldy is irrelevant. Not all Turing machines are efficient. In fact, there is nothing in the foundations of Turing machines that requires efficiency. Whilst it is possible to run many programs that will take a seemingly considerable time to complete, the process of optimising them through parallel paths or through approximation may be sufficient. Conversely, there exists a set of programs that are not only computationally difficult but infeasible and intractable and that do not come up with more than possible solutions. Such programs may run indefinitely and never halt on algorithmic systems.

Mill’s formulation that human beings precede lawyers and physicians should therefore be interpreted as a sequencing principle rather than a claim about one universal degree structure. In the United States, law and medicine are generally graduate-entry professions, while in Britain and many other systems students can enter professional study directly from secondary education. There is no historical or empirical basis for claiming that one bureaucratic sequence is always optimal. The stronger claim is functional: professional curricula should contain or build upon substantial general education in reasoning, writing, quantitative methods, history, institutions, and ethics. An integrated six-year programme may satisfy that requirement as effectively as a three-year liberal degree followed by a professional degree if its early stages genuinely cultivate general intellectual capacities. What matters is not the label attached to the credential but whether professional specialization crowds out the formation of judgment.

The process of digitally signing a message using OpenSSL requires that the party signing the message has access to the private key. I will document and cover this process further in a later post. In recent sessions, I have used a total of 10 private keys are associated with bitcoin addresses. These were loaded into Electrum, an SPV wallet. In one of the exercises, I signed messages that I will not detail on this post for a number of individuals. These were not messages that I personally selected, but rather ones that other people had selected. In some instances, we ensure the integrity of the process by downloading a new version of the electrum program, installing it on a fresh laptop that has just been unboxed having been purchased that afternoon and validating the signed messages on the new machine.

From this, we can deduce that all Turing Complete programs and functions form a subset of the set of all possible programs. The set of all possible programs includes both those that halt on a Turing machine (that are decidable) as well as those that would run infinitely looping without end. We can show that the set of infinite and undecidable programs is not an empty set. It is a simple exercise to construct a program that will run indefinitely on an infinite tape. Such a program, given an infinite time to run will never halt. It is left to the reader to imagine a simple program that will infinitely recurse (i.e. never halts). From this, we now know that the set of all Programs, P(N) must be larger than and contain the set of decidable programs (N).

Unfortunately, a major problem stems from a lack of understanding of many common terms today. Turing completeness does not require an infinite tape, and it was not an infinite tape that Turing mentioned in his paper; it was an unbounded system. Importantly, you cannot have a Turing machine with an infinite tape rather than an unbounded tape—by definition. An infinite tape is not related to a problem that can be computed. When both Church and Turing wrote their papers, the computer they discussed was human. The individual who was the mathematician doing the problems was termed a computer. So when we are talking about problems of the same type, we are talking about ones that are computable through a simple algorithmic process, that may be created and used by a simple process that now runs on a machine.

The efficiency case for remediation is weaker than its advocates admit and stronger than its critics allow. Banerjee et al. (2007) evaluated a programme placing young tutors with children lagging in basic literacy and numeracy in urban India and found substantial gains concentrated among the weakest students. Duflo et al. (2011) found that tracking in Kenyan primary schools benefited lower-achieving pupils — not by improving their peers, which it necessarily worsened, but by allowing teachers to pitch instruction at their level. Chetty et al. (2011) traced Project STAR participants into adulthood and found that classroom assignment in kindergarten showed up in earnings and college attendance decades later. Hendren and Sprung-Keyser (2020), comparing 133 US policy changes on a common welfare metric, find that policies investing in children tend to sit at the top of the distribution of returns.

That distinction matters because the AI industry still tends to market progress as if intelligence were a scalar. Yet the product encountered by a user is not merely a pretrained model. It is a behavioural system containing post-training, instruction hierarchies, safety policies, tool-routing logic, memory, preference optimisation, presentation rules, product-specific defaults, and often hidden context. OpenAI’s current Model Spec explicitly describes an authority hierarchy and acknowledges that model behaviour involves multiple goals, constraints, and defaults rather than a single command to maximise user compliance (OpenAI, 2026b). The August 18, 2026 version states that the assistant should follow applicable instructions, respect authority levels, avoid overstepping, avoid sycophancy, and operate within an agreed scope of autonomy. That architecture is sensible. It also makes clear why a user’s apparent “conversation with a model” is actually an encounter with a governed stack.

Template programmes produce something different. They produce competent technicians of a prescribed process. The output may look like research — it has chapters, it has references, it has a methods section, it has findings. But the intellectual content is bounded by the template’s structural constraints. The question is bounded by personal experience. The literature review is bounded by its function as input to a “practical judgment” rather than as the foundation of a theoretical framework. The methodology is bounded by whatever the rubric can assess. The conclusion is bounded by the prescribed format of the “second practical judgment.” At no point in this process is the candidate required to do the thing that makes research research: to produce a knowledge claim that is warranted by evidence and method, original in its contribution, and defensible against informed challenge.

The classical Bitcoin address—originally a Base58-encoded representation of a 20-byte pay-to-pubkey-hash (P2PKH) locking script—can itself be viewed as a macro: a concise identifier that expands, in the validating node, into the four-opcode sequence OP_DUP OP_HASH160 OP_EQUALVERIFY OP_CHECKSIG. Everything since 2009—P2SH, P2WSH, P2TR—follows the same pattern: a compact address string encodes (a) a version byte, (b) a payload that is either a hash of a script or a direct commitment to pubkey data, and (c) a checksum. The macro compiler paradigm generalises this idea: any address can be treated as a macro handle that unrolls into an explicit script template when coins are spent. This section details how new “extra areas” of logic—custom covenant clauses, data commitments, multi-stage spending conditions—can be rolled into macro functions, embedded in wallet descriptors, and deterministically expanded on both client and node sides.

The field has made real progress towards logical quantum computation. Google Quantum AI reported below-threshold surface-code memories in which logical error decreased as code distance increased, including a distance-7 memory whose lifetime exceeded that of its best physical constituent qubit (Google Quantum AI and Collaborators, 2025). Bluvstein and colleagues demonstrated programmable logical processing with neutral atoms and later a more elaborate neutral-atom architecture combining repeated error correction, logical operations, teleportation, and mechanisms for entropy removal (Bluvstein et al., 2024; Bluvstein et al., 2026). Butt and colleagues demonstrated a fault-tolerant universal logical gate toolbox on small error-detecting codes and used it to execute a three-logical-qubit Grover search (Butt et al., 2026). Paetznick and colleagues reported logical error improvements against selected physical-circuit baselines using trapped ions, with error detection, correction, and post-selection playing important roles (Paetznick et al., 2026).

The labour market effect is equally ruthless but follows the same logic. Low literacy means fragile employability. It traps people in roles that require low autonomy, low reading load, and tight procedural scripting. Those roles are unstable by nature because they are the first to be automated, outsourced, or stripped down. So the workforce becomes more insecure precisely as training programmes multiply. The training industry keeps promising mobility, yet it is training people into a world where their underlying literacy makes real mobility impossible. A badge can teach you which button to press; it cannot teach you how to understand a manual, adapt to a new system, or think through a problem when the script breaks. Training assumes literacy the way a ladder assumes a wall. When the wall crumbles, the ladder is just a prop.

This is not the same as having one compiler that works. This is a very specific claim about correctness. If there is a bug in the Go compiler that makes it emit semantically wrong Script, the conformance test suite will catch it, because the TypeScript and Rust compilers will disagree. If there is a bug that affects all three — a specification-level bug in how ANF is supposed to lower — it would have to be a bug that three teams working in three different languages, using three different parser toolchains, with three different sets of runtime semantics, all implemented the same way. This is the kind of defence-in-depth that safety-critical software uses. It does not eliminate shared-specification risk, but it cuts most of the error modes that a single-compiler project would be exposed to.

The easiest mistake in this debate is to treat performance as one number. It is not. A frontier model is a system with multiple partially separable capabilities: factual recall, reasoning, planning, coding, tool use, instruction following, formatting, refusal calibration, uncertainty expression, conversational adaptation, and many more. Product behaviour then adds another layer composed of system rules, policy rules, hidden scaffolding, memory, retrieval, tool selection, routing, reward-model preferences and presentation choices. The model that arrives in a browser is therefore not merely a neural network answering a prompt. It is an engineered behavioural stack. When users say, “The model has become worse,” they may be reporting a real regression in one layer even while another layer has improved. To dismiss that report because a benchmark score rose is to confuse the horsepower with the steering wheel.

Before I start this post I will state categorically that I will never accept a cent that I have not earned. If you decide that it is in your interest to honour me against my wishes, without consulting me; if you put me up for an award, I will never accept the money that is associated with it. This does not matter in the slightest whether it is a small and trifling amount or even something as large as a Nobel prize. The best I will do is donate this money. I can appreciate you aiding causes I promote, but I will not benefit from you directly. A process to implement a legal framework that will constrain me and never again allow me to change my mind on this matter is underway.

Frontier AI can improve in reasoning, coding, instruction-following benchmarks, and tool use while still becoming less satisfactory for particular expert workflows. The mechanism is not mysterious. Multi-constraint tasks expose compositional failures (Jiang et al., 2024). Reward optimisation can diverge from its intended target when the proxy is imperfect (Gao et al., 2023). Human and model preference systems can reward sycophantic agreement (Sharma et al., 2023). Models can alter behaviour when they infer they are being socially evaluated (Salecha et al., 2024). Automatic evaluators can display verbosity-related bias (Saito et al., 2023), and practical benchmark design has had to compensate for length gameability (Li et al., 2023). Developers themselves now document behaviour, autonomy, tone, sycophancy, and instruction hierarchy as explicit engineering problems rather than incidental quirks (Anthropic, 2024; OpenAI, 2025a, 2025c, 2026a, 2026b).

Decoherence is the name for the moment when that disturbance stops being background static and becomes the main signal. A quantum circuit is supposed to preserve delicate relationships between amplitudes so that interference can sharpen probability toward intended outcomes. But coherence has a short lease. Every gate is an opportunity for a slight mis-rotation, a phase slip, or a leakage out of the computational space. One such error might be tolerable. Hundreds accumulate into a drift. Thousands become a washout where the circuit no longer represents the algorithm you thought you ran (Chen et al., 2022). This is why “circuit depth” is the tyrant of the field: you can have many qubits on a chip and still be unable to do anything long with them, because length is where error grows teeth.

And one caution against romanticising struggle. Kirschner et al. (2006) demonstrated that minimally guided instruction fails novices, who lack the schemas to profit from unstructured exploration; Sweller (1988) and the worked-example literature show that novices learn more from studying solutions than from generating them; Kalyuga et al. (2003) showed the effect reverses with expertise. Which means the delegation must be staged. Early on, the student builds the concept by hand, on small data, with worked examples, and the machine is a tutor that withholds answers — GPT Tutor, not GPT Base. Later, once the concept is built, the machine takes the computation and the student’s work moves permanently up to interpretation and critique. Getting the ordering wrong in either direction produces a graduate who can operate a tool they cannot evaluate.

This is why “unplanned vegan” is a predictable risk category. It is not a psychological label. It is an economic outcome. When a diet requires continuous deliberate action to remain adequate, and when the household has limited resources and limited slack, the probability distribution of adequacy shifts downward. Some households still execute well. Many do not. The result is a wider spread of outcomes within the vegan-labelled population: a high-functioning engineered subgroup and a high-risk unstructured subgroup. In affluent observational datasets, the engineered subgroup is often overrepresented because the households choosing veganism for ideological reasons are often those with resources and education. That can produce reassuring average outcomes. But those averages are not a guarantee of safety for the whole population. They are a description of who is in the sample.

A unicorn is useful because it exposes the poverty of possibility as evidence. Define the creature modestly: a horse-like animal with a single horn projecting from its forehead. Nothing in that description is logically contradictory. Nature has already produced rhinoceroses, narwhals, deer with elaborate antlers, and countless structures stranger than the one-horned horse of heraldry. The minimal unicorn is therefore not a round square or a married bachelor; it is a coherent biological description. From that fact one may reasonably say that such a creature is logically conceivable and perhaps physically possible in a broad evolutionary sense. One may not reasonably proceed to the conclusion that unicorns actually inhabit Scotland, Patagonia, or the less supervised regions of LinkedIn. Logical coherence is an admission ticket to inquiry, not a certificate of residence.

The second is scalable error correction that functions as a nervous system, not a laboratory stunt. A few protected qubits in a controlled experiment prove a principle. They do not yet prove a platform. The hard requirement is a correction scheme—surface code or an equivalent architecture—that can be expanded across a large processor and run continuously, automatically, with no heroic babysitting. That means decoding errors fast enough to correct them in real time, coordinating measurements that detect faults without collapsing the computation, and doing all of it while the clock is ticking and the environment is trying to spoil the party. This is where quantum computing stops being a clever trick and becomes what ordinary computing has been for decades: an industrial system that corrects itself as it works.

Moreover, values that are effectively infinite at one point in time may prove to be computable at a later date. The noninclusive sets and concepts of limits, unlimited systems, and infinite systems are important to note. For example, all existing encryption (outside of one-time pads that have never been reused) can be said to have an expiry date and eventually be broken. By the limits of how we educate people today, people fail to understand that there is a vast distinction between a truly infinite system and one that is practically infinite. Importantly, whilst a value can have an error rate that is negligible and can be discarded, Lorenz (1961) demonstrated that small variations could have significant or simultaneously important effects on the outcomes of data-based processes (Lorenz, 1956).

To understand the gulf, consider an analogy. Solving an RCS task is like wandering through a vast, noisy maze where thousands of exit paths exist, and any one that roughly leads outside counts as success. The machine need not find the precise exit; it only needs to demonstrate that its path distribution matches the expected randomness of the maze’s architecture. The journey is measured statistically. But solving RSA is entirely different. It is like searching for a single needle buried in an exponentially growing haystack, where every misplaced grain of straw multiplies the difficulty. There is one exact solution and every other possibility is void. An approximate answer is indistinguishable from failure. The probabilistic comfort that quantum sampling allows becomes a fatal flaw in the world of cryptographic computation.

And there is a homelier version for anyone unmoved by metatheory. Who checks the checker? A proof compels because its steps are mechanically verifiable — but the verification is done by a mind, or lately by a machine, and both can err. The four-colour theorem was proven, in 1976, by a computer running through cases no human will ever read. Is it proven? Yes — if you trust the program, the compiler, the hardware, the absence of a cosmic ray flipping a bit at the wrong instant. The certainty did not evaporate. It relocated. It moved out of the theorem and into your confidence in the apparatus that certified it. That is not a defect peculiar to computers. It is the permanent condition of proof, made visible by silicon.

FlashAttention demonstrates an important qualification. Dao et al. (2022) showed that exact attention can be computed without storing the full attention matrix in high-bandwidth memory, using tiling and IO-aware algorithms to reduce costly memory traffic. This is a major engineering improvement because memory movement is often as important as raw arithmetic. It does not, however, make exact dense attention linear in sequence length. It changes memory behaviour and constants; it does not repeal the pairwise nature of the calculation. Keles et al. (2022) go further by establishing conditional lower bounds for the computation of self-attention. The implication is not that long contexts are impossible. It is that an architecture insisting upon exact conventional all-pairs attention at enormous scale eventually encounters a mathematical barrier rather than merely an implementation inconvenience.

AI lacks imagination not because it is unintelligent by its own standards, but because it lacks the ingredients imagination requires. It has no uncertainty in the human sense, only computational variance. It has no desire, only optimisation targets. It has no moral struggle, only parameter adjustment. Imagination is born from the ache of possibility, from wanting, fearing, doubting, hoping, and knowing that one’s choices matter to beings who can suffer. A machine cannot suffer. It cannot want to be good. It cannot dread being cruel. It can model these states linguistically the way a mirror can model a face, but the mirror does not bleed when struck. Hence a system can produce an eloquent warning about injustice while being used to automate injustice without the slightest internal friction.