The Time-Symmetric Parity Constraint
A testable approach to cosmology based on a single geometric constraint.
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About this website
Most visitors will find it helpful to begin with the Pocket Guide, which provides a short, non-technical introduction to the ideas behind TSPC and explains how the various papers relate to one another. Those wishing to examine the mathematics or the observational evidence in greater depth can then proceed directly to the research papers.
This website has been organised to make the development of the Time-Symmetric Parity Constraint as accessible as possible. The principal research papers are accompanied by explanatory notes, supporting material and references, allowing readers to explore the theory at whatever level they choose.
Pocket Guide to the Time-Symmetric Parity Constraint (PDF)
The navigation panels remain the same throughout the website, making it easy to move between the main papers, supporting notes and related material without losing your place. Each page also contains links to associated papers and downloadable PDF versions where available.
It’s all Greek to me!
The papers supporting the proposition described on this website are for the most part technical, and contain a substantial amount of mathematics. For those who have an interest in the Universe but are not partial to algebraic descriptions, here is an introduction to TSPC with some historical antecedents but without the maths.
About the author
Ian Salisbury is an architect and an independent consultant specialising in construction and professional services. His practice includes architectural consultancy, expert witness appointments, arbitration, adjudication and mediation.
Copyright
Unless otherwise stated, the content of this website, including all research papers, text, illustrations, diagrams and other original material, is © Ian Salisbury 2025, 2026. All rights reserved.
The author generally asserts his moral rights to be identified as the creator of these works in accordance with Part I, Chapter IV of the Copyright, Designs and Patents Act 1988.
The papers may be downloaded, stored and cited for private study, research and scholarly reference, provided that they are reproduced accurately, remain unaltered and are accompanied by appropriate acknowledgement of authorship and source.
No part of this website or its contents may be reproduced, republished or distributed for commercial purposes, or incorporated into derivative works, without the prior written permission of the copyright holder, except where permitted by law.
Website designed and developed by Good Consideration LLP, 2026
Preface
The Preface offers a brief orientation to the TSPC Library. It explains how the collection is organised, why the papers are largely self-contained, and the ideas that connect them into a single programme of investigation.
Pocket Guide
The Pocket Guide provides a concise introduction to the Time-Symmetric Parity Constraint (TSPC) for readers who want an overview before exploring the technical papers. It outlines the central idea, explains how a single geometric constraint offers an alternative perspective on inflation, dark matter and dark energy, and summarises the principal observational tests.
The guide is intended for scientists, students and interested non-specialists. No specialist mathematical background is assumed.
Library
The Library contains the collection of papers describing the Time-Symmetric Parity Constraint. Beginning with the foundational formalism, they develop the theory, examine its cosmological implications, test its predictions against observation, and conclude with broader methodological and philosophical reflections. Although presented as a coherent programme, each paper may be read independently.
1. Introduction
This introductory paper argues that many of the outstanding problems in modern cosmology may arise not from missing matter, dark energy or inflation, but from an unquestioned assumption: the adoption of the Friedmann-Lemaître-Robertson-Walker (FLRW) geometry as the foundation of the standard cosmological model. It introduces the Time-Symmetric Parity Constraint (TSPC), which replaces this assumption with a single algebraic constraint on spacetime, without modifying general relativity or introducing new fields or particles. The paper establishes the conceptual framework for the programme that follows and explains how its predictions are subjected to empirical test.
2. Reassessing late-time cosmic acceleration in light of supernova progenitor age bias
This paper reopens the observational case for late-time cosmic acceleration by examining supernova progenitor-age bias as a physical consequence of stellar evolution rather than merely a nuisance parameter. The resulting uncertainty does not establish an alternative cosmology, but it removes the need to treat persistent acceleration as an unquestionable starting premise. That opens a different line of enquiry: whether phenomena conventionally attributed to additional cosmological components might instead arise from the admissible structure of spacetime itself. It is from that question that the TSPC investigation begins.
Reassessing late-time cosmic acceleration in light of supernova progenitor age bias
3. The Time-Symmetric Parity Constraint
This is the core theoretical statement of TSPC. A single global symmetry condition is imposed on spacetime, restricting the class of admissible cosmological solutions. The paper shows how this constraint reshapes expansion, structure, and temporal behaviour without introducing dark sectors or modifying gravity.
The Time-Symmetric Parity Constraint
4. Late-Time Acceleration without Dark Energy
Here it is shown that acceleration-like behaviour can arise geometrically over a limited epoch. No dark energy field or cosmological constant is required. What appears as acceleration is reinterpreted as a consequence of constrained spacetime evolution rather than an ongoing dynamical driver.
Late-Time Acceleration without Dark Energy
5. The Cosmological Constant Problem Reconsidered
This paper reframes the cosmological constant problem – a mismatch between the theoretical and observed values of vacuum energy of some 120 orders of magnitude – as a geometric misinterpretation rather than a tuning failure. By removing the need for a fundamental vacuum energy, the problem dissolves rather than being solved parametrically. The emphasis shifts from cancellation to exclusion.
The Cosmological Constant Problem Reconsidered
6. Dark Matter without Dark Matter
Galactic and lensing anomalies are addressed without invoking unseen particles. The paper shows how a slightly stiffened geometry reproduces the same phenomenology attributed to dark matter. The result challenges the assumption that missing mass is the only available explanation.
Dark Matter without Dark Matter
7. Inflation without Inflation
Inflation was introduced to explain why the Universe appears so flat, so uniform, and so finely ordered despite the apparent limitations of conventional Big Bang cosmology. This paper argues that these long-standing puzzles arise only because spacetime has been treated as unconstrained. Under the Time-Symmetric Parity Constraint, global temporal symmetry fixes the large-scale geometry from the outset, naturally accounting for flatness, horizon uniformity, and the low-entropy origin without invoking an inflation field or a brief epoch of exponential expansion. Inflation therefore becomes an unnecessary hypothesis rather than an established physical requirement. The paper concludes by identifying the observational tests that distinguish this structural explanation from conventional inflationary cosmology.
8. The Origin of Time’s Arrow
The Origin of Time’s Arrow addresses one of cosmology’s oldest unresolved questions: why time appears to flow irreversibly from past to future. Rather than treating the thermodynamic arrow as a finely tuned initial condition or relying on cyclic entropy resets, it argues that temporal asymmetry emerges naturally from the Time-Symmetric Parity Constraint. A single algebraic restriction on spacetime geometry produces a universe that is globally time-symmetric while giving rise to the familiar local arrow of time. The paper sets out the theoretical framework, contrasts it with conventional approaches, and identifies clear observational tests by which the proposal may be confirmed or falsified.
9. Linked Tensions in H0 and S8
The persistent discrepancies in the Hubble constant (H₀) and the structure-growth parameter (S₈) are widely regarded as two of the strongest challenges to the standard cosmological model. This paper argues that they are not independent anomalies but linked consequences of a single geometric constraint. Within the Time-Symmetric Parity Constraint, a small algebraic modification of gravitational coupling simultaneously raises the expansion rate and enhances structure growth, without introducing new fields or particles. The framework makes a clear, falsifiable prediction: future surveys should reveal a correlated pattern in H(z) and structure growth, or rule the hypothesis out.
10. The Metastability of Entropic Zero
Why did the universe begin in such an extraordinarily low-entropy state? Conventional cosmology generally accepts this as an unexplained initial condition. This paper argues instead that the entropy minimum is a structural consequence of the Time-Symmetric Parity Constraint. The states T = 0⁻ and T = 0⁺ emerge as metastable geometric extrema from which entropy necessarily rises towards a maximum at T tip before falling symmetrically. In doing so, the paper ventures close to the traditional boundary between physics and metaphysics, but seeks to replace a unique cosmological miracle with a simple, constrained and ultimately testable geometric framework.
The Metastability of Entropic Zero
11. Introductory Note: Empirical Tests of Global Geometric Constraint
This note introduces the programme of empirical tests that follows. Each paper examines a distinct observational consequence of the Time-Symmetric Parity Constraint using existing astronomical data. Together they illustrate the framework’s commitment to falsifiability by identifying clear opportunities for observational confirmation or, equally importantly, refutation.
12. Void Geometry and and Alcock–Paczyński Distortion
Can the large-scale geometry of the Universe be tested directly? This paper argues that cosmic voids provide one of the cleanest opportunities to do so. Because stacked voids behave as near-geometric objects, the Alcock–Paczyński relation becomes a direct probe of the mapping between redshift and physical distance. Within the Time-Symmetric Parity Constraint, this mapping is not freely adjustable. Persistent, redshift-dependent distortions therefore represent resistance to the framework itself rather than invitations to further parameter fitting. Void geometry thus becomes an important empirical test of whether the proposed global spatial constraint can survive confrontation with observation.
Void Geometry and Alcock–Paczyński Distortion
13. HerS-3 and the Geometry of Lensing: a Baryonic Alternative to ΛCDM Lensing
HerS-3 has been presented as one of the clearest examples of a gravitational lens requiring an unseen dark matter halo. This paper argues that the evidence is less decisive than commonly claimed. Within the Time-Symmetric Parity Constraint, the anomalous lensing can instead be explained by ordinary baryonic matter that has escaped detection because of cosmological dimming, dust, or diffuse intra-group light. Comparable lens systems demonstrate that complex image geometries need not imply exotic matter. The paper proposes specific JWST and ALMA observations capable of distinguishing between a genuinely light-free halo and an overlooked baryonic component, making HerS-3 a direct empirical test rather than a confirmation of ΛCDM.
HerS-3 and the Geometry of Lensing: a Baryonic Alternative to ΛCDM Lensing
14. Gravitational Waves and the Geometry of Dark Matter: a geometric alternative to dark-matter microphysics
This paper explores an unexpected use of gravitational-wave astronomy: not as a probe of hypothetical dark-matter particles, but as a test of spacetime geometry itself. It argues that the observed evolution of black-hole merger rates can be explained by the combined effects of TSPC’s parity-induced geometric weighting and well-established baryonic processes, without invoking exotic dark-matter microphysics. The paper identifies clear observational signatures that distinguish the two interpretations and shows how future observatories, including the Einstein Telescope, Cosmic Explorer and LISA, could decisively test them. Gravitational waves thus become an independent empirical probe of the geometric foundations of cosmology rather than the properties of an unseen dark sector.
15. The CMB and Radio-Source Dipoles as Projections of Global Symmetry: a geometric alternative to cosmological bulk flow
This paper asks whether one of cosmology’s most persistent anomalies, the mismatch between the Cosmic Microwave Background dipole and the much stronger radio-source dipole, points to previously unrecognised large-scale motion or to something more fundamental. Within the TSPC framework, both signals are interpreted as complementary projections of a single global parity structure embedded in spacetime, rather than evidence of motion through an isotropic universe. The paper derives testable predictions for forthcoming radio surveys and shows how this anomaly could become a decisive empirical test of the theory’s geometric foundations.
16. Cosmic Chronometers and Differential Ageing
This paper asks whether cosmic chronometers can do more than refine estimates of the Universe’s expansion history. Within TSPC they are treated as differential clocks, testing whether the observed relationship between redshift and elapsed proper time remains compatible with a globally constrained temporal geometry. Rather than being absorbed into increasingly flexible cosmological fits, persistent discrepancies become direct tests of the framework itself. Together with the companion papers on void geometry and gravitational waves, cosmic chronometers form part of a deliberately selective programme of empirical falsification, aimed at exposing structural weakness rather than improving parameter estimates.
Cosmic Chronometers and Differential Ageing.
17. Conclusion: Recognising the Blind Spot at the heart of Cosmology
The concluding essay reflects on what follows if the preceding analyses withstand continued observational scrutiny. Rather than beginning with a preferred cosmological model and seeking evidence to support it, the Time-Symmetric Parity Constraint has followed the opposite course: proposing a single geometric constraint and then repeatedly testing it against independent observations and then repeatedly exposing it to independent observational tests designed to reveal where it fails. So far, no decisive inconsistency has emerged. This resilience does not establish TSPC as correct, but it does make it increasingly difficult to dismiss as speculative. If that position is maintained, dark energy, dark matter, inflation and time’s arrow may come to be understood not as separate physical ingredients, but as different manifestations of one underlying geometric constraint, with profound implications for ΛCDM and the future direction of cosmology.
Conclusion: Recognising the Blind Spot at the heart of Cosmology
18. The Galactic-Centre Gamma Excess Reconsidered
Rather than proposing another dark-sector explanation, this supplement asks whether the reported Galactic Centre gamma-ray excess can be understood as a local consequence of the Time-Symmetric Parity Constraint. It argues that, if TSPC is a genuine geometric constraint on spacetime, its effects should not be confined to cosmological scales but should also appear within galaxies wherever curvature and baryonic density are greatest. The paper develops this reinterpretation, identifies observational tests capable of distinguishing it from dark-matter annihilation or pulsar models, and extends TSPC from a cosmological framework into the realm of galactic astrophysics without introducing new particles or additional free parameters.
The Galactic-Centre Gamma Excess Reconsidered
19. The Radial Acceleration Relation at the Edge off Galaxy Formation: a geometric reading
This paper examines new observations showing that the well-known relationship between visible matter and gravitational acceleration begins to break down in the faintest dwarf galaxies. Rather than interpreting these departures as evidence for increasing dark-matter dominance, the paper argues that they may instead signal the onset of curvature effects predicted by the Time-Symmetric Parity Constraint. It proposes that, at extremely low baryonic densities, geometry rather than matter becomes the principal determinant of gravitational behaviour. The paper concludes by identifying a programme of observations capable of distinguishing between conventional dark-matter explanations and the geometric interpretation advanced by TSPC.
The Radial Acceleration Relations at the Edge of Galaxy Formation
20. DESI DR2: a geometric reassessment
DESI DR2 has provided the most precise map of cosmic geometry yet obtained, prompting renewed claims for evolving dark energy, interacting dark sectors, and other extensions to ΛCDM. This paper reviews those interpretations against the observations themselves, arguing that the data measure geometry rather than ontology. It compares the principal DESI-based analyses with the predictions of the Time-Symmetric Parity Constraint, showing that the observations remain compatible with a single geometric constraint without introducing new particles or fields. The paper also considers recent supernova and CMB-temperature results, identifies clear observational tests capable of falsifying TSPC, and argues that DESI has strengthened the case for treating cosmic geometry, rather than unseen substances, as the primary object of explanation.
21. Undressing the Emperor
Undressing the Emperor steps back from individual cosmological puzzles to examine a deeper question: why has the FLRW cosmological model itself remained largely beyond critical scrutiny for a century? Tracing its historical development from mathematical convenience to accepted orthodoxy, this essay argues that modern cosmology routinely modifies matter content, dark sectors and free parameters while leaving its underlying geometry effectively immune from falsification. Against this, TSPC treats geometry as a testable scientific hypothesis rather than a protected assumption. The paper is both a historical critique and a methodological manifesto, arguing that genuine scientific progress depends upon exposing our deepest assumptions to empirical risk.
22. The Extragalactic Dipole. Clock-skew, luminosity-distance anisotropy and volume growth
The first Observational Consequences paper examines one of cosmology’s more intriguing anomalies: the unexpectedly large dipole seen in radio and quasar catalogues, which exceeds that predicted from the Solar System’s motion alone. Rather than proposing a specific cosmological model, the paper derives a general first-order framework showing that any such excess must arise through just three observable channels: clock-skew, anisotropic luminosity distance, or anisotropic volume growth. This model-independent kernel provides a rigorous diagnostic for testing geometric cosmologies, including TSPC, and marks an important transition from theoretical formulation to concrete observational prediction by identifying precisely how small distortions of the past light cone would appear in large extragalactic surveys.
23. An Audit of TSPC: 3 observational tests
This paper steps back from theory-building to ask a more demanding question: how have TSPC’s published predictions fared against subsequent observations? It reviews three key tests, transient cosmic acceleration, differential ageing inferred from cosmic chronometers, and large-scale anisotropy, in the light of recent data from DESI, Pantheon+, chronometer surveys, and dipole studies. The audit finds no decisive confirmation, but neither does it identify any clear falsification. Instead, several developments have shifted in directions anticipated by TSPC, while remaining compatible with conventional cosmology. The result is a measured assessment of the theory’s current evidential standing: the observational landscape has moved slightly towards TSPC, but the decisive tests still lie ahead.
24. TSPC II: Why the Standard Model cannot explain the Universe
This paper extends the structural critique developed in the earlier TSPC essays into the increasingly influential field of quantum cosmology. It argues that many concepts now invoked to explain the Universe, including inflationary potentials, vacuum states, axion fields, moduli and string landscapes, originated as operational constructs within quantum field theory before acquiring an ontological role in cosmology. The paper asks whether this represents a continuation of the same pattern seen in the standard cosmological model: the progressive extension of theoretical machinery to preserve an assumed framework. It concludes that lasting progress is more likely to come from identifying a genuine geometric constraint on spacetime than from multiplying explanatory ingredients.
25. TSPC III: Dangerous Observations in Cosmology
This paper examines a growing class of observations that challenge not the numerical parameters of the standard cosmological model, but its underlying assumptions. Focusing on the cosmic dipole anomaly, rest-frame non-convergence and related indications of large-scale anisotropy, it argues that these are “dangerous observations” because they test the internal consistency of the standard inference chain itself. The paper distinguishes genuine foundational tests from ordinary parameter tensions and considers them in the light of the Time-Symmetric Parity Constraint. Rather than claiming these observations confirm TSPC, it asks whether they are more naturally accommodated by a framework based on structural constraint than one dependent upon explanatory flexibility.
26. TSPC IV: Discriminator Observables in cosmology
What observations could genuinely falsify a theory of spacetime, rather than merely refine its parameters? This paper argues that many celebrated cosmological “tests” function primarily as consistency checks within an assumed framework, leaving foundational geometric assumptions largely unchallenged. It introduces a hierarchy of “discriminator observables”: measurements capable of distinguishing between competing spacetime structures instead of simply adjusting model parameters. Using TSPC as an illustrative example, it identifies six such tests and emphasises that a viable theory must expose itself to decisive empirical failure. The paper is therefore methodological rather than model-specific, proposing a clearer standard for what genuine falsification in cosmology should look like.
27. When Flexibility Replaces Explanation: forecasting cosmology in the era of generational surveys
Forecasts are indispensable to modern cosmology, but they also shape how future observations will be interpreted. This essay argues that, as models have expanded from ΛCDM to increasingly flexible frameworks such as wCDM and w₀wₐCDM, explanatory emphasis has shifted from structural necessity towards parametric accommodation. Generational surveys are increasingly judged by how well they constrain these enlarged parameter spaces rather than by whether additional freedom is required at all. The paper contrasts this trend with geometry-first approaches that impose hard constraints on admissible spacetimes, asking whether cosmology risks allowing methodological flexibility to replace genuine explanation.
When Flexibility Replaces Explanation
28. Reduction, Proliferation or Constraint? On competing foundational commitments in cosmology
What, in the end, is the Golden Fleece that cosmology seeks? This paper argues that it is not a final theory, but a forgotten criterion for judging competing explanations. By comparing three distinct strategies, proliferation, reduction, and constraint, it explores the different prices each pays for cosmic order, whether in ontology, interpretation, or structural restriction. The paper concludes that progress depends not only on fitting observations, but on recovering explanatory discipline: a willingness to expose foundational commitments to decisive empirical test.
Reduction, Proliferation or constraint
29. The Persistent Return of Torsion
Why does torsion keep returning whenever physicists attempt to derive symmetry breaking from geometry alone? This paper argues that the answer lies not in torsion’s physical necessity but in the structural requirements of spontaneous symmetry breaking itself. Across a wide range of unification programmes, torsion repeatedly assumes the role of a surrogate scalar, only for the same difficulties to reappear in new mathematical language. The analysis is deliberately structural rather than technical, exposing a recurring pattern that reaches well beyond any individual model. For readers interested in the conceptual foundations of unification, it offers a challenging perspective on why some of the field’s most sophisticated geometric programmes continue to arrive at remarkably similar destinations.
31. Highly Redshifted Neutral Hydrogen
A proposed lunar observatory, CosmoCube, may soon make much better measurements of the highly redshifted 21-cm signal from neutral hydrogen. Such observations are routinely described as probing the cosmic “Dark Ages”, but that description already assumes a cosmological interpretation. This paper separates what would actually be observed from what ΛCDM infers from it, and asks whether the unusually precise spectral predictions of the standard model could distinguish particulate dark matter from the geometrical alternative proposed by TSPC. The answer is revealing: CosmoCube could provide a stringent test of a ΛCDM prediction, but TSPC cannot yet make the corresponding high-redshift prediction needed for a genuine comparison. The paper identifies precisely what further development would be required.
32. TSPC Under Audit
TSPC was deliberately constructed as a tightly constrained framework, intended to fail rather than acquire additional freedom when observation proves incompatible with it. This paper puts that principle into practice. Earlier claims that TSPC could explain flat galactic rotation curves are re-examined and rejected: a scale-independent strengthening of gravity changes its magnitude but cannot produce the required radial behaviour. Galactic rotation is therefore removed from the phenomena claimed to be explained by TSPC, without modifying the underlying constraint. The possibility that the missing gravity instead represents genuinely missing mass is then reconsidered, with particular attention to primordial and quiescent black holes and the strong constraints imposed by microlensing. Finally, the wider TSPC programme is audited by distinguishing what is predicted, consistent, untested, unsupported or falsified. The result is a narrower framework: its proposed explanation of apparent late-time cosmic acceleration remains unconfirmed, but no presently identified observation is found to falsify it.