Exploring quantum information, finite registers and their possible physical interpretation.
The papers below examine finite registers, quantum information, matter and gauge structure, lattice dynamics and cosmology. Each paper distinguishes exact finite results from the assumptions needed for a physical interpretation. Abstract pages provide publication details, free PDFs and archive links.
Looking for our applied AI work? Papers on AIModern physics is fascinating, sometimes extremely weird, and often hidden behind deeply opaque mathematics. It attracts a great deal of popular interest on social media and in popular-science books and articles, but these are usually too superficial to give much real understanding and are often riddled with conceptual errors. The serious student will find a good selection of textbooks that give a rigorous academic presentation of the subject, but these are generally written for the professional physicist and can leave a reader bewildered unless they have studied mathematics to postgraduate level.
At Neuro-Symbolic we needed material to bring able graduates up to speed on the subject, and we found very little in the gap between popular and professional science. These four books are our attempt to bridge that gap. They may also be of interest to others, perhaps those studying at undergraduate level or starting a new project in the field. We have done our best to check that the material is correct, and we have said where a topic is still under debate and not yet accepted physics. Any mistakes are our own, and we very much welcome corrections from other researchers.
The books use a question-and-answer format. We found this a good way of presenting the material in workshops, and it seemed to help participants engage with it.
States, measurement and matter
A question-led introduction to quantum physics · 27 questions · 65 pages
One two-route experiment, taken one question at a time, leads to amplitudes, states, measurement and the no-cloning rule, and then to particles, spin, entanglement and the probability rule. No calculus: arithmetic and a right-angled triangle are enough.
Mass, charge and the rules of change
A question-led introduction to particle physics · 31 questions · 65 pages
Mass, generations, colour and charge; what the weak interaction changes; why neutrinos are so hard to catch; what makes a particle stable; and the open questions behind the Standard Model's familiar answers. Uses the ideas of Volume 1 with the arithmetic kept visible.
Clocks, causal paths and black holes
A question-led introduction to relativity and gravity · 16 questions · 37 pages
Why observers agree on the speed of light, why gravity bends light and why it is so weak; photon time, gravitons and the limits of a classical source; black holes as causal structures, evaporation, and whether acceleration can make empty space feel warm. No tensor calculus.
Evidence, inflation and unanswered questions
A question-led introduction to cosmology · 13 questions · 35 pages
Why the universe is expanding and what the hot Big Bang actually explains; what inflation would explain and what could convince us it happened; what the dark components mean; and why the history has a direction. Observation, inference and proposal are kept apart.
Each book has its own page with its question list and two sample answers; the series page at /books/ gathers all four. The four are also combined into a single Kindle book on Amazon for 99p, for reading in an e-reader or on a phone.
How far can stable records take an information-first account of physics? This essay separates established results within quantum mechanics, operational reconstruction principles and the additional assumptions of a register-and-ledger model. It explains the scope of Born-rule and thermodynamic arguments and distinguishes syndrome readout from protected quantum information. An optional construction uses bond-centred bipyramids. Records alone do not select quantum theory, the eight-bit cell, three spatial dimensions or an interaction strength; the value of the framework is in making those choices and the remaining physical questions explicit.
Read the technical paper (PDF) View on Zenodo Abstract & citationSeparate studies of lattice actions, algebraic dictionaries and a conditional counting coefficient. None supplies a joint matter–gauge–geometry action or derives universal gravitational response.
David Elliman, Neuro-Symbolic Ltd · October 2026 · Revision 6
Two separate actions are studied on a bond-centred lattice and an assumed four-dimensional extension. For the linearised gravity action PL-1N, an exact certificate establishes leading constraint closure, while finite spectral sampling finds two massless transverse-traceless modes and no sampled physical instability. Source normalisations are imposed, and the finite-spacing time-dependent response differs from general relativity. Separately, the Wilson-type PP+ fermion action has one free light Dirac field, time-cell reflection-positivity evidence and conditional gauge-coupled positivity results. Its leading free anisotropy coefficient is +1/24. Interpreting that coefficient as an electron dispersion constraint requires a physical particle identification and interacting matching assumptions. These results do not establish a positive one-step transfer in general, a chiral Standard Model, or a joint matter–gauge–geometry action with universal gravitational response.
David Elliman, Neuro-Symbolic Ltd · September 2026 · Version 2.0
A checkpoint in the programme's quantum-gravity problem that now turns on a negative result. The native discarded-component matrix has rational rank 670, giving original-active dimension 694 alongside the certified rank-24 projection; but an explicit nonzero original-active vector is also a defining composition defect, so it vanishes in the correction quotient and complete original-active survival is ruled out. The restricted AB/AH splittings stand, and for a separately specified contact-field model over a three-variable Laurent ring an exact null column and a contact-only equivariant left inverse are verified, producing local dictionaries for every target in the ring. The strain, curvature, tensor-pole, scalar-pole, lensing and binary-radiation calculations are retained as conditional benchmarks. The paper states its boundary plainly: no physical dictionary is selected, no temporal dynamics established, and no Einstein limit, Newton constant, universal matter coupling or full gravitational gauge system is derived.
David Elliman, Neuro-Symbolic Ltd · September 2026
A finite event census gives a conditional coefficient under a specified monitoring-and-erasure identification and a supplied map to far-field gravity. The reported leading and next-order values, 205.554 and 206.554, are compared with 206.49 assembled from a QCD anchor and cosmological inputs. The input uncertainties exceed the small numerical residual; the comparison does not select the next-order term or determine Newton’s constant. Counting one service slot does not establish physical erasure, heat, reset or invocation rate. The surviving result is a finite counting calculation within that service model, not a derived gravitational action or absolute physical scale.
Studies of matter registers and conditional cosmology. Their abstracts state the assumptions and scope of each model.
David Elliman, Neuro-Symbolic Ltd · October 2026
A finite matter register supports a useful label dictionary and exact charge and anomaly identities once its gauge representations are assigned. This paper separates those results from the missing dynamical ingredients: a chiral gauge theory, a mass-generating action, CP selection, universal propagation and spectroscopy. Graph calculations retain their stated graph scope, and charged intermediate channels require explicit charge carriers. Spinor and fermionic constructions provide declared matter premises. The dictionary organises possible matter content, but it does not by itself determine the interactions or measured particle parameters.
David Elliman, Neuro-Symbolic Ltd · October 2026
Finite service-clock and covariance calculations suggest conditional formulas for the scalar tilt and amplitude. A supplied pressureless reservoir can improve a CMB forward calculation, but register counts do not establish its stress tensor, transport or gravitational source. The late dark-energy model fails its registered joint test, while the cosmological-constant and Hubble-scale identifications require additional, unselected assumptions. The paper distinguishes these finite results and model comparisons from a completed dark-sector cosmology, and makes no primordial-tensor null prediction.
Methods for auditing claims and measurements: a truth-maintenance system that relocates trust from the AI generator to an evidence-bound verifier, and a measurement discipline requiring an explicit model of what an apparatus reads. These methodology papers are AI research in their own right, so they also appear on the Papers on AI page.
David Elliman, Neuro-Symbolic Ltd · June 2026
PTMS is a truth-maintenance tool for tracking claims, dependencies, evidence and retractions in an AI-assisted research corpus. Its checks can expose inconsistent references, unsupported status changes and claims relying on retired premises. The reported corpus sizes and check counts describe the paper’s historical snapshot. Passing these checks establishes neither the truth of a physical interpretation nor experimental support for the programme; scientific validation remains a separate task.
Download PDF Abstract & citationDavid Elliman, Neuro-Symbolic Ltd · July 2026
This paper separates a count in a finite model from an observable measured by a calibrated apparatus. Specified instrument models illustrate how a record count can remain invariant while a response changes. A physical application must supply the carrier, interaction, response map and calibration. The fine-structure and gravity examples do not establish those maps: the former numerical agreement depends on a geometry no longer identified with the native substrate, and the gravity identification remains conditional. The contribution is the measurement discipline and its finite examples.
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A protocol for recording acceptance criteria, evidence dependencies and failed predictions before interpretation can shift. The paper documents dated examples from the programme. Its procedures make changes and failures inspectable; they cannot certify honesty, correctness or a successful physical theory. The registrations remain historical evidence, while the research branches are frozen.
An account of the distinction between a model’s record ledger and empirical access through calibrated response channels. It argues for explicit measurement maps, calibration and tests capable of failing. The fine-structure numerical example belongs to a retired physical identification and does not validate the principle. The contribution is a methodological proposal about assessing finite information models, not a derivation of physical observables or a replacement for quantum theory.
Companion studies grouped by topic: finite quantum instruments, assigned matter registers and grammars, conditional cosmology, lensing analysis methods and standalone mathematical results.
Finite quantum-instrument results and their limits. Codes, channels and probability assumptions are stated rather than inferred from records alone.
A computer-science introduction to a finite quantum-record model: quantum states, typed operations, measurement instruments and readable records. The worked examples separate internal labels from states and from the records produced by a supplied apparatus. Compact descriptions apply to the selected structured state families, not to arbitrary quantum states. The model supplies a Hilbert space, subsystem structure and dynamics; the representation does not derive those ingredients, explain Bell correlations by a classical local mechanism, or establish a physical origin for spacetime.
A finite audit asks whether a measurement ledger and a proposed gravitational billing ledger describe the same process. The specified histories admit an exact jump–tape–commit correspondence, but that agreement does not select a billing law, an absolute unit or a gravitational action. Two supplied pre-commit completions agree on the committed records and differ in entanglement. This is a counterexample to inferring the underlying dynamics from that record agreement, not a demonstrated unification of measurement and gravity.
A finite transition table is not yet a measurement record. This report separates six objects often collapsed in discrete models — transition support, effects, instruments, environmental export, invocation cadence and autonomous dissipative dynamics — and gives exact finite witnesses for the failed implications between them: calibration fixes the visible effects but not the instrument, environment Gram matrices realise the same dynamics with no, partial or projective export, endpoint maps do not fix cadence, and a lawful Lindblad family still has a free rate. The worked example is the programme's eight-register cell, exhaustively checked over 256 states and 2,048 events, with programme-level interpretations kept as non-claims.
An exact finite register study separates available transitions, exported environment records and the resulting preserved coherences. Channels sharing the same exit edges can preserve different coherences, so transition support alone does not select the monitored channel. The stated polarity-resolved instrument distinguishes more labels than coarser alternatives. The result concerns these specified instruments on the owned register; a physical system–environment interaction selecting one of them remains an additional input.
The sequel to Described Twice? and Pointer States Are Not Enough: once an exact ledger of reads, commits and service events exists, does it fix what happens next? In a finite model with 48 preparations and a ten-outcome apparatus, no, three times over. Two reversible rules reproduce the same first-use specification and disagree on reuse, so the first-use map does not select the law. With one law fixed, two histories with identical counts assign different probabilities to the next outcome. And at eight preparations the two laws agree on every single-read distribution and differ only in the correlations between reads. Exact, verifier-backed counterexamples, plus a table of the inputs still owed. Published here in full as searchable HTML.
Specified monitored quantum instruments distinguish latched record observables from response observables. Under the paper’s stated absorbing-pointer and dressing assumptions, the finite models establish invariance properties for record statistics while responses can change. Pole, contact-term and accumulation statements retain the hypotheses of their individual models. The calculations do not imply that every physical observable has this form, or derive a physical coupling from a record count.
Finite instrument examples show why an integer ledger and a calibrated response are different objects. The response depends on the declared dynamics and measurement channel, even when a record count is unchanged. The definitions of SI units do not imply that physical dimensionless constants are rational counts. In particular, the paper’s fine-structure example is not a successful native-lattice prediction. The useful result is the separation of bookkeeping, response and calibration.
For the specified monitored recovery instruments, a register-access theorem limits which records can be generated by the registers that an instrument reads. Static validity predicates do not by themselves determine a recovery channel, an environment record or a rate. The generation example supplies a finite covariance and contact inventory; it does not derive lepton pole masses, a CP phase or a baryogenesis source. In particular, the exact fixed two-constant mass-spectrum claim is excluded under its stated no-theory-error assumptions (G97), while fitted-phase identities are a separate question. The instrument and its physical selection remain inputs.
Within a supplied complex Hilbert-space and quantum-instrument setting, repeatable syndrome readout produces orthogonal record sectors. With the required dimension and additive, non-contextual probability assumptions, Gleason-type reasoning constrains the probability measure. An isometric record map separately displays the quadratic amplitude form, but this is not an independent uniqueness derivation of probabilities. The eight-bit minimum is conditional on the one-code read/write premise; identifying a uniform service weight with the physical fine-structure constant is a further assumption. Neither quantum theory nor the physical interaction strength follows from classical records alone.
Specified stabilizer instruments give finite examples of syndrome export and redundant records. These calculations illustrate how a chosen environment interaction can distribute information about selected observables. They assume the quantum algebra, code, channel and interpretation of the records; stable records alone do not select a unique quantum code or derive objective spacetime. The paper’s former truncated-cubic geometry does not describe the programme’s bond-centred substrate. Its surviving contribution is the scoped quantum-information calculation.
Finite matter dictionaries and classification results, followed by work on record grammars and their physical interpretation. The representation assignments and dynamical assumptions are stated separately.
A finite audit classifies Boolean matter-register rules within a specified eight-bit architecture. With generation and chirality conventions, charge readouts, representation assignments and a sterile-label exclusion family fixed, the recorded scan isolates a Standard-Model-like rule under its declared tests. The classification is restricted to that family. Eight primitive roles are architectural assumptions, a seven-bit compressed encoding is possible, and the bit-pairing filter is distinct from a test of physical vectorlike gauge content. The result constrains a supplied model rather than deriving the Standard Model from records alone.
A finite encoding reproduces charge and anomaly-cancellation identities for assigned Standard Model representations. The exact checks test consistency of the supplied particle dictionary and can expose changes that spoil it. They do not derive the gauge group, representations or chirality from the code, nor construct a nonperturbative chiral gauge measure. The contribution is a compact, auditable encoding of these identities.
A finite executable grammar encodes selected stabilizer records, Wilson holonomies, non-stabilizer resources and detector-readout examples. Its checks concern the specified objects and compression scheme. They do not show that arbitrary quantum states admit such a compact description, or that the grammar selects nature’s Hilbert space, gauge theory, interaction coefficients or spacetime. Physical interpretations require a supplied action and measurement map.
Finite triality constraints and leading strong-coupling Wilson calculations are studied on a specified cell complex. Vanishing total triality is a necessary charge-selection condition; it is not by itself an SU(3) singlet condition or a proof of physical QCD confinement. The earlier glued complex predates the disjoint register-ownership rule, and the tested graph lengths do not distinguish the proposed Y and Δ interpretations. The finite algebraic and graph results therefore remain conditional examples, without a derived proton or physical string tension.
An attribute grammar organises supplied particle types and interaction rules, with type checks for allowed vertices and charge identities. It formalises a chosen particle model; it does not derive the Standard Model gauge group, generations, statistics, couplings or masses from a code. Restrictions on a fourth generation or sterile readout are properties of the declared grammar. The contribution is a way to represent and check those rules, with the code and physical assignments explicit.
A computational account of how typed grammars, packed forests and dynamic programming can organise perturbative amplitude calculations for supplied field-theory rules. The examples distinguish diagram syntax from the algebra used to evaluate it. Their scope does not select a physical Lagrangian, determine interaction coefficients or certify every loop calculation. The contribution is the parsing and evaluation method within its stated examples.
The qgrammar software implements typed particle records, vertex rules, anomaly checks, a small parser, repair candidates and export scaffolds for a supplied matter model. Its certificates check the encoded identities and typing rules; they do not certify every requirement of a consistent interacting chiral quantum field theory. Particle content, allowed productions and numerical couplings remain inputs. The tool is a front-end consistency aid, not a derivation of the Standard Model or a replacement for full amplitude and event-generator software.
Completed pipeline and selection studies. Their data checks do not validate the framework’s proposed gravitational coupling.
A galaxy–galaxy lensing analysis reports pipeline checks, coordinate-frame diagnostics and instrument validation for a specified data selection. These methodological results remain distinct from the physical model that originally motivated the campaign. The programme’s tick-based gravitational derivation failed its registered equivalence-principle test, so the pipeline does not validate that derivation or establish a unique alternative to dark matter. This is a record of the stated analysis, not an announcement of a continuing observational campaign.
A follow-up lensing analysis studies isolation selection and reports a deep-tail slope for its stated catalogue cuts. The registered primary precision threshold was missed: seven points qualified and the eighth had a 0.105-dex uncertainty against a 0.10-dex limit. Supplementary slope comparisons and full-profile fits favour different aspects of the tested models. They do not constitute a general exclusion of dark-matter or modified-gravity theories, and the framework’s original gravitational coupling remains unestablished. The data work and frozen decision protocol are historical results; no further campaign is commissioned here.
Standalone technical results that stand on their own mathematics rather than on the substrate canon — exact averages, closed forms, and reproducible numerical benchmarks.
A pure-mathematics companion: does the Gauss law generate every differential equation satisfied by the gauge invariants, or are there hidden constraints beyond it? For a finite connected oriented graph with the vertex gauge group acting on the affine SL3 link carrier, the question is whether the annihilator of the invariant ring equals the left ideal generated by the infinitesimal Gauss operators. Equality is not formal; it fails in elementary scaling actions. Schwarz's theorem settles the general case; the work here verifies its largeness hypotheses for graph gauge actions, shows the moment components form a regular sequence, and proves a functoriality theorem: enlarging the chart contracts both ideals exactly onto their small-chart counterparts.
A self-contained mathematical result with no substrate content: a closed form for the mean power sums of normalised random weights, and the lattice geometry those sums encode. Take independent Beta(a, a) weights, normalise them to sum to one, and ask for E[R_n], with R_2 the inverse participation ratio. A Mellin representation removes the random denominator while keeping its fluctuations, reducing the expectation to one integral valid for every a > 0, dimension d ≥ 1 and integer n ≥ 2. At a = 1/2 the arcsine law is the push-forward of a uniform lattice momentum under k → sin²(k/2), so E[R_n] becomes a Brillouin-zone average of a hypercubic invariant; in two dimensions the n = 2 and n = 3 averages are exactly 1 − 1/π and 1 − 3/(2π).