The Block Entropic Information Pressure Engine

A Geometric Ontology Unifying Quantum Behaviour, Gravity, and Information

Preface

This work belongs to a class of theoretical efforts that has become increasingly rare in modern physics: the construction of a replacement ontology. By this is meant not a reinterpretation of existing equations, nor the extension of a formalism into a new regime, but the explicit proposal of new primitives from which established theories are recovered as descriptive limits. Such efforts arise only when conceptual tensions persist despite continued formal success.

The last widely accepted ontological replacement in physics was general relativity, in which force was subsumed into geometry and spacetime itself became the fundamental structure. Since then, progress has been dominated by increasingly powerful calculational frameworks whose empirical success has outpaced ontological clarity. Quantum mechanics, general relativity, and information theory now coexist as highly effective but conceptually incompatible descriptions, each resting on distinct primitives and explanatory assumptions.

Several credible attempts at ontological replacement have been made in the intervening century. Bohmian mechanics offered a deterministic substrate beneath quantum statistics but retained configuration-space wavefunctions and left relativistic unification unresolved. Everett’s formulation removed collapse while declining to specify the ontology of branching, displacing rather than resolving the measurement problem. Causal set theory replaced spacetime with order but struggled to recover quantum matter without grafting external structure. Loop quantum gravity quantised geometry while preserving the conceptual role of spacetime and time itself. Information-theoretic approaches identified information as fundamental yet rarely completed the transition from metaphor to explicit geometry.

What unites these approaches is not failure in the pejorative sense, but incompleteness. Each halted at a boundary it did not cross: retaining time as fundamental, preserving probability as primitive, introducing mediating fields without ontological status, or declining to specify how geometry itself arises. In many cases this restraint was deliberate. Ontological commitment is risky; formal productivity is safer.

As a result, contemporary theoretical physics is rich in calculational techniques but poor in fully articulated pictures of what the world is. Replacement ontologies are rare not because they are unnecessary, but because they are difficult to complete and easy to dismiss.

The present work proceeds in the opposite direction. It begins by committing explicitly to an ontology and accepting the consequences of that commitment. Information is taken as the sole primitive; its admissible expressions are constrained by a small set of universal biases. Geometry is not postulated independently but arises as the minimal structure capable of realising those constraints. Time, probability, force mediation, and even the distinction between matter and antimatter are treated as emergent descriptions rather than foundational inputs.

Whether this ontology is correct is an empirical question. The purpose of this work is not to claim inevitability, but to restore ontological seriousness to a domain where it has gradually been replaced by instrumental success. The rarity of such efforts is not an argument in their favour, but it is a necessary precondition for conceptual progress. Physics advances not only by refining what works, but by occasionally asking what what works means.

This paper should therefore be read neither as a challenge to the empirical success of existing theories nor as an attempt to subsume them by fiat. It is an explicit proposal for a different starting point—one that aims to be internally coherent, geometrically concrete, and empirically falsifiable—and to be judged on those grounds alone.

The traditional route of arXiv publication, peer review, and journal submission is not open to a writer who is neither a professional physicist nor to a paper that is heterodoxical. Perhaps these factors help explain the ossification of the science.

Note on structure.
In the print manuscript, this work includes a formal table of contents. In the web version, section navigation is provided by page scrolling, Ghost’s built-in heading anchors, and external linking.

1. Introduction

Contemporary physics rests on three powerful but ontologically incompatible descriptive frameworks. Quantum mechanics encodes physical behaviour in probabilistic amplitudes defined on abstract Hilbert spaces. General relativity models gravitation as curvature of a four-dimensional spacetime manifold governed by dynamical field equations. Information theory, when invoked at all, is typically treated as an external bookkeeping tool rather than as a physical primitive. There is no single object in the standard formalism that simultaneously carries information, exhibits quantum behaviour, and participates in relativistic structure.

This paper proposes an alternative starting point. The Block Entropic Information Pressure Engine (BEIPE) is a replacement ontology in which information is taken as the sole physical primitive. Its admissible expressions are constrained by a small set of universal biases, and the cumulative structure of those constraints is encoded geometrically. Geometry does not act dynamically; it records constraint. Time, probability, force mediation, and even the distinction between matter and antimatter are treated as emergent descriptions rather than foundational inputs.

The Universe in BEIPE is described by a fixed, four-dimensional, non-orientable manifold (the Mobifold) populated by continuous informational structures. The fundamental physical object is a Worldline+: a Lorentzian worldline-plus consisting of a persistent worldline together with transverse geometric modulation and a long-lived imprint on an ordering field. Worldline+s persist and reconfigure under universal informational biases and are ordered by an entropic scalar field and its associated gradient. Together, these encode admissible continuation without introducing a fundamental temporal parameter. The experienced arrow of time is an emergent description of ordered re-expression along Worldline+s, not evidence of underlying temporal evolution.

Admissible continuation in BEIPE is governed by alignment with the local gradient of the entropic ordering field. The fundamental relation is that the geometric rate of re-expression of a Worldline+ is everywhere aligned with the direction of decreasing entropic admissibility.

$$ \dot{\gamma}^\mu(A) = -\alpha(A)\,\nabla^\mu S $$

In this expression, the left-hand side denotes the admissible geometric continuation of a Worldline+ at a given entropic age. The gradient term represents the local direction of entropic ordering, while the scalar prefactor encodes bias-constrained unpacking. No temporal parameter is introduced; ordering arises entirely from the geometry of the entropic field itself.

Observable separation along a Worldline+ is obtained by integrating admissible continuation against the inverse magnitude of the entropic gradient. This yields an effective distance measure without invoking metric expansion, proper time, or dynamical spacetime evolution.

$$ A(r) \propto \int_0^r \frac{dr'}{|\nabla S(r')|} $$

This relation makes explicit that large apparent separations arise from prolonged traversal of shallow entropic gradients rather than from accelerated expansion of space.

Non-orientability arises as the global geometric encoding of an unavoidable structural fact: when admissible continuation is exhausted at entropic stillness, persistent informational structures cannot terminate, reverse, or persist unchanged. They must instead undergo orientation-inverted re-expression, a process termed Translation. Matter and antimatter appear as global orientation phases of the same informational structures. Spin-one-half behaviour follows from rotation in the two-dimensional normal plane of a Worldline+. Orientation inversion is enforced wherever entropic ordering collapses. These features arise as consequences of continuity-preserving re-expression in a block universe, rather than from imposed symmetries or dynamical postulates.

At observable scales, BEIPE reproduces familiar physical phenomena through geometric projection rather than force mediation. Quantum interference, entanglement, and correlation bounds arise from projection of higher-dimensional rotation into three-space. Gravitation emerges from persistent deformation of the entropic ordering rather than from spacetime curvature sourced by stress–energy. Electromagnetic and nuclear interactions arise from compatibility, instability, and routing constraints among Worldline+ geometries within a shared substrate. No mediating particles, gauge fields, or probabilistic axioms are required at the ontological level.

Cosmology in BEIPE is governed by the large-scale structure of the entropic ordering field. Redshift is interpreted as entropic separation rather than metric expansion. Apparent anisotropies arise from projection of the entropic gradient into an observer’s tangent three-space. Global dipole structure reflects orientation-reversing holonomy rather than peculiar motion. Under mild assumptions on bias-constrained unpacking, the magnitude of the entropic gradient cannot steepen indefinitely. Instead, it flattens at large entropic age, yielding late-time cosmological behaviour without invoking dark energy or accelerated expansion. These asymptotic inequalities lead to concrete, falsifiable predictions distinguishing BEIPE from standard cosmological models.

The purpose of this paper is therefore fourfold: to formalise the BEIPE ontology and its axioms; to derive its geometric structures and physical consequences; to analyse its asymptotic behaviour; and to identify empirical signatures by which it may be tested. The framework is deterministic and non-probabilistic at the ontological level, and explicitly falsifiable.

2. Axioms

The BEIPE framework rests on a small set of ontological axioms. These do not emerge from the formalism; they define the structure within which the formalism operates.

Axiom 1 — Information and Universal Biases

All physical entities are represented as geometric Worldline-plus structures. Each is characterised by a unique cross-sectional informational profile that encodes all physical properties, including mass, spin, energy, and topological features. At the geometric level, a Worldline-plus is structured as a persistent curve in the Mobifold with a one- or two-dimensional cross-section specified by its informational profile.

The admissible expressions, continuations, and reconfigurations of informational profiles are governed by four universal informational biases:

• Continuity: Information resists termination, conserving its profile.
• Directional: Information aligns with the skein’s left-handed chirality.
• Spatial: Information favours separation unless compatible.
• Expressional: Information seeks minimal complexity.

These biases determine all else.

Axiom 2 — Block Universe

The Universe is a static, four-dimensional, non-orientable manifold with a radial, closed-loop, Möbius-like topology, comprising Obverse and Reverse phases connected cyclically. All events—past, present, and future—coexist simultaneously within this fixed geometric structure.

The shape of the manifold, termed the Mobifold, derives from the universal informational biases. Time is not a fundamental coordinate but an emergent perception arising from ordered informational descent along the entropic ordering structure.

Axiom 3 — Entropic Gradient

The entropic ordering field and its associated gradient are geometric quantities defined on the Mobifold that encode the ordering of admissible Worldline-plus expressions under the universal informational biases. They do not represent thermodynamic entropy or dynamical evolution, but instead provide a purely structural ordering of informational continuation.

The gradient arises as the geometric encoding of bias-constrained information. Neutrinos dominate this large-scale ordering and may therefore be regarded as the effective substrate of the gradient.

Between the two global orientation phases of the Mobifold, the gradient carries opposite sign: positive in the Obverse phase and negative in the Reverse phase. This sign structure fixes the orientation of admissible descent without introducing a fundamental time parameter.

All quantities are treated purely geometrically. No thermodynamic units are assigned to the entropic ordering field or its gradient.

3. Core Definitions

Before developing the geometric framework in detail, it is useful to fix a set of core definitions for the main objects and quantities that recur throughout the paper.

Definition 3.1 — Worldline+

A Worldline+ is a Lorentzian worldline-plus: a geometric entity realised as a continuous, differentiable curve defined on the Mobifold and ordered by entropic age. In addition to its persistent curve-like structure, a Worldline+ possesses a two-dimensional cross-section characterised by a width, a real depth, and a geometric information profile.

The Worldline+ modulates across the four dimensions of the manifold. Its information profile encodes all physical properties usually attributed to a particle, including mass, spin, energy, and internal structure.

$$ \gamma : A \rightarrow \mathcal{M} $$ $$ I = \{W,\; y_{\mathrm{real}},\; \sigma,\; \ldots\} $$

Here the information profile consists schematically of a transverse width, a real depth, and topological features such as chirality or braiding. The precise contents of the profile vary by Worldline+ type but are conserved under admissible re-expression.

Definition 3.2 — Entropic Age

Entropic age is an ordering parameter defined along a Worldline+ with respect to the entropic ordering structure of the Mobifold. It measures the depth of progression along admissible geometric ordering rather than duration or elapsed time.

Entropic age is defined only up to monotone reparameterisation and serves solely to order admissible continuation within the block universe.

$$ A(r) \propto \int_{0}^{r} \frac{dr'}{|\nabla S(r')|} $$

This expression provides a schematic representation of entropic ordering depth along the effective radial direction of the Mobifold. It encodes ordering structure rather than physical time. Durations arise only when specific Worldline-based processes are used as clocks by observers, yielding observer-dependent measures.

Definition 3.3 — Neutrino Skein

The neutrino skein is a geometric mesh formed by neutrino Worldline+s. It provides the large-scale structural substrate from which the entropic ordering gradient emerges.

Each skein element possesses a minimal information profile characterised by an extremely small transverse width, zero real depth, and a fixed chirality.

$$ I = \{W \sim 10^{-18}\,\mathrm{m},\; y_{\mathrm{real}} = 0,\; \sigma\} $$

Neutrino Worldline+s dominate the global ordering structure because of their ubiquity and minimal informational complexity. In BEIPE, they function not as force mediators but as the geometric substrate of ordering itself.

Definition 3.4 — Translation

Translation is the compulsory orientation-inverted re-expression of a Worldline+ when admissible continuation within its current orientation is exhausted. It occurs at entropic stillness, where continuation aligned with the prevailing ordering ceases to be defined.

Because Worldline+s are persistent informational structures rather than objects in motion, they cannot reverse direction, stall, or terminate. Continuity bias therefore enforces re-expression into the opposite global orientation phase of the Mobifold. Translation is topological rather than dynamical and preserves the informational profile of the Worldline+.

$$ \nabla S \rightarrow 0 $$

This condition marks the collapse of admissible ordering and triggers enforced orientation inversion rather than motion through spacetime.

Definition 3.5 — Reverse Phase

The Reverse phase is the region of the Mobifold characterised by decreasing entropic ordering along the effective radial direction and by the presence of antimatter Worldline+s. The Obverse phase corresponds to the opposite ordering orientation and is associated with matter Worldline+s.

$$ \nabla S < 0 $$ $$ \nabla S > 0 $$

These two global phases are not separate universes but orientation-distinct regions of a single non-orientable manifold.

Definition 3.6 — White Hole

A white hole is a geometric transition point in the Reverse phase where Worldline+s emerge from a region of entropic stillness while conserving their informational profile.

In the BEIPE framework, white holes are the Reverse-phase counterparts of black-hole curvature wells in the Obverse phase and play a role in global information conservation rather than mass ejection.

4. Ontology of Worldline+s and the Mobifold

This section introduces the fundamental ontological elements of BEIPE: the Mobifold, the skein, the entropic ordering field, and the objects termed Worldline+s. These definitions supply the structural background for all subsequent results.

4.1 Why Global Orientation Fails

In BEIPE, global orientation fails as a direct consequence of continuity-preserving re-expression at entropic stillness. When admissible continuation collapses and Translation is enforced, orientation inversion becomes unavoidable along admissible paths. A space in which such inversions occur cannot support a globally consistent orientation.

Non-orientability is therefore not an independent assumption or modelling choice. It is the geometric encoding of a structural fact about admissible continuation in a universe composed of persistent informational Worldline+s.

Continuity, Stillness, and Forced Re-Expression

Worldline-plus structures are not objects in motion but continuous informational structures embedded in a block universe. As such, they cannot reverse direction, pause, or terminate as primitive operations. Admissible continuation is defined entirely with respect to the ordering imposed by the entropic ordering structure.

At entropic stillness, the ordering that defines admissible continuation collapses. For a persistent Worldline-plus, continuation within the original orientation ceases to be defined. Continuity bias forbids termination, and reversal is not meaningful in the absence of motion. The only admissible continuation is therefore orientation-inverted re-expression, termed Translation.

Because neutrino Worldline-plus structures are long-lived and ubiquitous, entropic stillness is encountered somewhere along admissible continuation—locally in curvature wells and globally in the asymptotic limit of maximal entropic separation. Translation is therefore unavoidable within the ontology.

$$ |\nabla S| \rightarrow 0 $$

This condition marks entropic stillness: the point at which admissible ordering collapses and orientation-preserving continuation is no longer defined.

Failure of Global Orientation

A space in which admissible continuation necessarily includes orientation inversion cannot admit a globally consistent orientation. Any attempt to transport an orientation frame along admissible paths that pass through Translation regions returns an inverted frame. Orientation-reversing transport is not exceptional but intrinsic.

This failure is not a geometric assumption or modelling choice. It is a structural consequence of forced Translation arising from continuity-preserving re-expression.

Residual Description

Once forced Translation is acknowledged, the appropriate geometric description is a four-dimensional, non-orientable manifold in which orientation inversion is realised directly by global transport. No additional labels, fields, or dynamical mechanisms are required.

Non-orientability does not enable Translation; it records it.

Core Insight

Translation is intrinsic to informational persistence at entropic stillness. Because Translation is unavoidable and orientation-reversing, global orientation cannot be preserved.

Non-orientability is therefore not a modelling preference but the geometric signature of continuity-preserving re-expression in a block universe.

The Mobifold is the name given to the global structure that encodes this fact.

4.2 The Mobifold

The Universe in BEIPE is modelled as a four-dimensional non-orientable manifold, termed the Mobifold. The Mobifold possesses a single global orientation-reversing cycle analogous to the half-twist of a Möbius strip, but embedded in four dimensions. Traversal of this cycle reverses the orientation of transported normal vectors.

Definition — Mobifold

A Mobifold is a smooth, connected, four-dimensional manifold equipped with a metric and a non-trivial global orientation structure. Parallel transport around a fundamental cycle of the manifold induces orientation reversal of tangent frames.

$$ w_1(\mathcal{M}) \neq 0 $$

This non-orientability is not an auxiliary assumption. It is the global geometric encoding of unavoidable Translation under continuity-preserving re-expression. Phase inversion, chirality, and the matter–antimatter distinction follow from this encoding rather than motivating it.

4.3 The Skein and Geometric Entropy

The Mobifold is formed by the neutrino skein together with the universal informational biases of its constituent elements. Together, these provide both the embedding environment for Worldline-plus structures and the geometric tension that defines the entropic ordering of the manifold.

Manifold as behavioural geometry

Ontologically, BEIPE treats the behaviour of information as primary, not information as an abstract substance. The Mobifold is the geometric structure required to consistently realise four fundamental informational behaviours: persistence, descent, branching and recombination, and inversion.

Non-orientability, a two-dimensional normal bundle, and a global entropic ordering structure are not imposed assumptions but geometric consequences of these behaviours.

The neutrino skein provides the dominant continuous physical instantiation of this geometry. In this work, analysis is carried out entirely at the geometric level of the Mobifold and its skein, because it is at this level that quantum behaviour, interactions, and cosmology become calculable.

The entropic ordering gradient appears repeatedly throughout BEIPE. It determines the stability of Worldline-plus structures, the apparent flow of time, cosmological redshift, and the conditions under which Translation occurs.

4.4 Worldline-Plus Structures

Worldline-plus structures are the fundamental physical entities of BEIPE. A Worldline-plus is not a point particle but a one-dimensional informational thread with a two-dimensional geometric cross-section. Such structures are continuous, cannot terminate, and must descend the entropic ordering to remain stable.

Worldline-plus revisited

A Worldline-plus is a continuous, differentiable curve embedded in the Mobifold and ordered by entropic age, with a two-dimensional cross-section determined by its informational profile.

Each Worldline-plus possesses two essential geometric structures:

• Informational structure: a one-dimensional encoded content capable of expressing two-dimensional geometric patterns.
• Normal-plane rotation: because a curve embedded in a four-dimensional manifold has a two-dimensional normal space, a Worldline-plus possesses a rotational degree of freedom whose projection into three-space appears as oscillatory modulation.

4.5 Matter, Antimatter, and Orientation

Non-orientability of the Mobifold implies the existence of two global orientation phases, termed Obverse and Reverse. A Worldline-plus in the Obverse phase is perceived as matter. The same structure, after traversing the Mobifold’s orientation-reversing cycle, appears as antimatter.

Orientation phases

Transport of a Worldline-plus around a fundamental cycle of the Mobifold reverses the orientation of its normal bundle. A Worldline-plus in the reversed orientation is observed as antimatter. This follows directly from the Mobifold’s topological class and is independent of the internal composition of the structure.

4.6 The Ontological Picture

The ontology of BEIPE may be summarised as follows:

1. The Universe admits a non-orientable four-manifold description termed the Mobifold.
2. The neutrino skein provides geometric tension and defines the entropic ordering structure.
3. Physical objects are Worldline-plus structures: one-dimensional informational threads embedded in the Mobifold.
4. Worldline-plus structures must descend the entropic ordering to remain stable.
5. Rotation in a Worldline-plus’s two-dimensional normal plane projects into three-space as transverse modulation.
6. Orientation inversion across the Mobifold yields antimatter.

These components define the background against which the dynamics of Worldline-plus structures, Translation, and physical interactions are derived in the following sections.

5. Information Biases and Their Geometric Consequences

So far the discussion has been framed in terms of the Mobifold, the skein, the geometric entropy field, and the one-dimensional objects termed Worldline-plus structures. Ontologically, however, these are not independent ingredients. They are all geometric representations of a single primitive: information.

Information is the substrate from which the Mobifold, the skein, the entropic ordering field, and Worldline-plus structures are composed when coarse-grained into geometry. Its role in the framework is defined entirely by a small set of universal biases.

5.1 Continuity Bias and Persistence

Information exhibits a bias toward continuity: geometric representations of information resist termination. As a consequence, information cannot appear as isolated points or fragments within the Mobifold. It must be realised as continuous paths.

This continuity bias gives rise to the persistence of Worldline-plus structures. A Worldline-plus is the geometric expression of continuous information: a one-dimensional curve embedded in the Mobifold, carrying a two-dimensional cross-section in its normal plane. Its uninterrupted extent reflects the impossibility of information beginning or ending abruptly within the skein. This does not rule out reconfiguration, catastrophic or otherwise.

In this sense, what is conventionally called a “particle” in BEIPE is not a fundamental object but a particular persistent configuration of information: a stable cross-sectional geometry carried along a continuous Worldline-plus.

5.2 Directional Bias and Chirality

Directional bias in BEIPE is set by neutrino chirality. Neutrino Worldline-plus structures are intrinsically chiral, and because the neutrino skein is pervasive, its handedness defines the background orientation against which other Worldline-plus geometries are realised.

Where a Worldline-plus possesses no intrinsic chirality of its own, coupling to the neutrino skein induces one. In this way, chirality is propagated through the skein: neutrinos do not merely exhibit handedness, they impose it. Worldline-plus structures with inherent chirality retain it; those without acquire the neutrino bias.

5.3 Spatial Bias and Reconfiguration

Information is subject to a spatial bias: Worldline-plus geometries favour spatial separation unless compatibility permits closer association. This bias governs how information is distributed across the skein and constrains which geometric configurations are stable.

The consequence of spatial bias is reconfiguration. Compatible Worldline-plus structures may branch, merge, or braid into configurations of lower geometric tension, while incompatible structures remain separated. Branching corresponds to a single informational source being expressed along multiple compatible paths. Recombination occurs when compatible Worldline-plus structures merge into a shared geometry. In all cases, information is conserved; only its spatial expression is altered.

Spatial bias also governs high-density packing. As the entropic ordering field approaches its maximal value, separation becomes geometrically constrained and compatible Worldline-plus structures are forced into dense, vent-like configurations. In this regime, packing dominates over isolation and reconfiguration pathways multiply. Orientation reversal is therefore more readily accessed near maximal entropic ordering, not due to a new mechanism, but because spatial bias suppresses separation and lowers the geometric cost of inversion.

$$ S \rightarrow S_{\max} $$

In the limiting case, reconfiguration is catastrophic. When matter-aligned and antimatter-aligned Worldline-plus structures are forced into geometric coincidence, their opposing chiralities are incompatible and no stable joint configuration exists. The structures collapse and their informational content is redistributed into other permitted configurations. Matter–antimatter annihilation is such a catastrophic reconfiguration.

5.4 Expressional Bias and Entropy

Information is subject to an expressional bias toward minimal scale. Among all geometrically admissible representations, information is stably expressed only in configurations of minimal structural complexity.

The consequence of this bias is an ordering of admissible configurations. Highly structured, tightly constrained expressions are unstable unless externally enforced; simpler expressions persist. This ordering defines the direction of geometric entropy. Entropy does not describe disorder or probability, but the monotonic progression of information toward minimal expressible scale.

When combined with spatial bias, expressional bias produces unpacking. As geometric constraints relax, information reconfigures into progressively simpler spatial expressions. Unpacking therefore does not terminate at a finite separation or configuration: it proceeds until the minimal expressible state is reached.

$$ S_{\max} $$

This limiting value defines maximal entropic ordering. Expressional bias alone fixes the arrow of entropy; no additional principle is required.

5.5 Summary

In BEIPE, information is realised geometrically through a small number of universal biases. Continuity bias ensures the persistence of Worldline-plus structures. Directional bias, carried by the neutrino skein, fixes chirality and its propagation. Spatial bias governs separation, packing, and reconfiguration of Worldline-plus geometries. Expressional bias orders admissible configurations by minimal scale and fixes the direction of geometric entropy.

The consequences described in this section follow directly from these biases when information is realised as Worldline-plus structures in a non-orientable manifold. The remainder of the paper develops the entropic, dynamical, and physical implications of this structure.

6. Geometric Entropy and the Entropic Gradient

In BEIPE, geometric entropy provides the ordering structure that governs admissible informational expression. Although termed entropy for continuity with existing physical language, it is not a thermodynamic quantity. Instead, it encodes the ordering of admissible geometric configurations induced by the information biases, in particular the expressional bias toward minimal scale acting through spatial reconfiguration.

The entropic gradient provides the geometric representation of this ordering and specifies the direction along which stable Worldline-plus progression is realised. This section introduces the formal definitions of geometric entropy and its gradient, together with the structural consequences that follow from treating them as geometric rather than statistical objects.

6.1 Geometric Entropy

Definition — Geometric Entropy

Geometric entropy is a scalar field defined on the Mobifold that provides a geometric ordering of admissible configurations of the neutrino skein. Regions of higher geometric constraint or curvature correspond to lower entropy, while smoother, less constrained regions correspond to higher entropy.

$$ S : \mathcal{M} \rightarrow \mathbb{R} $$

In this work, geometric entropy is treated as a dimensionless or curvature-normalised quantity. It is not assigned thermodynamic units. Conventional thermodynamic entropy is regarded as an emergent, coarse-grained quantity that may be related to geometric entropy in suitable limits, but is not identical to it.

Geometric entropy does not encode disorder, probability, or statistics. It encodes expressional ordering: configurations of higher structural complexity are geometrically constrained, while configurations of lower complexity are more broadly admissible. In this sense, geometric entropy functions as a height coordinate on the landscape defined by the skein.

6.2 The Entropic Gradient

Definition — Entropic Gradient

The entropic gradient is a vector field on the Mobifold derived from the geometric entropy field. It provides the local direction of admissible ordering and specifies the orientation along which stable Worldline-plus progression must align.

$$ \nabla_{\mu} S = g_{\mu\nu}\,\partial^{\nu} S $$

Because geometric entropy is treated as a purely geometric scalar, the entropic gradient has the dimensions of an inverse length. It is not a force and does not correspond to motion through space. Instead, it encodes the expressional ordering induced by the information biases.

The direction of decreasing geometric entropy defines the admissible direction of stable Worldline-plus continuation.

Worldline-plus descent condition

A Worldline-plus structure must align its admissible continuation with the geometric ordering specified by the entropic gradient. This condition expresses stability rather than dynamics and applies everywhere except during Translation events.

$$ \dot{\gamma}^{\mu}(A) = -\alpha(A)\,\nabla^{\mu} S $$

Here the scalar factor encodes bias-constrained unpacking. No temporal parameter is introduced; ordering arises entirely from geometry. This relation expresses admissibility, not evolution.

Observable separation along a Worldline-plus may be represented schematically as an accumulation of ordering depth rather than duration.

$$ A(r) \propto \int_{0}^{r} \frac{dr'}{|\nabla S(r')|} $$

This expression encodes ordering depth along the effective entropic direction of the Mobifold. The parameter used to label progression serves solely to order admissible continuation within the block universe and is not identified with physical time or clock readings.

6.3 Curvature and Geometric Entropy

Although BEIPE does not assume Einstein’s field equations, curvature plays an intrinsic role in shaping admissible configurations of the skein and therefore in the local structure of geometric entropy.

Variations in geometric entropy are modulated by curvature and by skein tension. These inputs shape the expressional landscape without introducing dynamical field equations or thermodynamic assumptions.

$$ \Delta S = f\!\bigl(R L^{2},\,\tau\bigr) $$

Here curvature, characteristic geometric length scale, and skein tension appear only in dimensionless combinations. This ensures that geometric entropy remains a pure ordering quantity.

The explicit functional form is developed later in connection with curvature eigenmodes and hadronic spectra. The essential point is that curvature modulates admissible expressional structure without acting as a dynamical driver.

6.4 Entropic Stillness and Translation

A defining feature of BEIPE is that Translation—the orientation-inverted re-expression relating matter and antimatter—is enforced when admissible ordering collapses.

When entropic ordering vanishes locally, continuation aligned with the existing orientation ceases to be defined. Because Worldline-plus structures are persistent informational entities rather than objects in motion, termination and reversal are forbidden. The only admissible continuation is therefore orientation inversion.

Definition — Entropic Stillness

A region of the Mobifold is said to be in entropic stillness if the entropic gradient vanishes throughout that region.

$$ \nabla S = 0 $$

In regions of entropic stillness, admissible progression aligned with the entropic ordering is undefined. Translation is therefore compulsory.

More generally, Translation is enforced whenever the entropic gradient becomes sufficiently small that accumulated normal-plane rotation can no longer be carried forward as admissible continuation. Orientation inversion is not an additional mechanism; it is the only remaining continuity-preserving re-expression.

Curvature wells, large-scale cosmological voids, and the asymptotic flattening of entropic ordering at late cosmic stages are natural realisations of entropic stillness.

6.5 Summary of Entropic Structure

The central consequences of this section may be summarised as follows:

1. Geometric entropy is a scalar field representing the ordering of admissible configurations of the neutrino skein, shaped locally by curvature and skein tension.
2. The entropic gradient provides the geometric representation of this ordering and fixes the direction of admissible Worldline-plus progression.
3. Alignment with the entropic ordering is required for stability; exhaustion of admissible continuation enforces re-expression.
4. Regions of entropic stillness correspond to collapse of admissible ordering and compulsory Translation.
5. Curvature and skein tension shape geometric entropy without introducing dynamical evolution or fundamental time.

These elements formalise the entropic structure implied by the information biases and prepare the ground for the treatment of Worldline-plus dynamics in the following section.

7. Worldline+ Dynamics: Progression, Modulation, and Stability

A Worldline-plus is a one-dimensional informational object embedded in the four-dimensional Mobifold. Its observable behaviour is the continuous re-expression of its informational profile under the universal informational biases.

Two geometric structures are used to describe this re-expression:

1. the ordering of admissible continuations encoded by the entropic ordering structure, and
2. the rotation of the Worldline-plus’s two-dimensional normal plane.

These structures do not act as dynamical agents. They provide geometric representations of how admissible Worldline-plus expressions are organised. When projected into three-space, constrained re-expression appears as oscillatory behaviour commonly associated with quantum phenomena. This section formalises these effects in geometric rather than probabilistic terms.

7.1 Progression and Stability

A Worldline-plus is said to progress when its tangent realises a locally admissible continuation of its informational profile. The ordering of admissible continuations is represented geometrically by the entropic ordering structure.

Formally, a Worldline-plus exhibits progression when its admissible continuation aligns with this ordering.

$$ \dot{\gamma}^{\mu}(A) = -\alpha(A)\,\nabla^{\mu} S $$

Here the scalar factor encodes the degree to which the realised continuation lies within the locally ordered admissible class.

Stability requires uninterrupted satisfaction of continuity bias. A Worldline-plus remains stable only while its ordering parameter increases monotonically.

$$ \frac{dA}{d\tau} > 0 $$

If no admissible continuation exists within the current ordering class—equivalently, if the admissibility factor tends toward zero—then continuity bias cannot be satisfied by further progression in the current orientation class. In this case, the Worldline-plus must undergo re-expression.

Time as descent.
Subjectively, this regime corresponds to the experience of temporal passage. The felt flow of time reflects the ongoing satisfaction of continuity bias through successive admissible re-expressions of Worldline-plus structures. Physical clocks are Worldline-plus-based processes whose tick rate reflects local patterns of admissible continuation rather than an external temporal parameter.

7.2 Normal-Plane Rotation and Modulation

Because a Worldline-plus is a curve embedded in a four-dimensional manifold, the space normal to its tangent is two-dimensional. The informational profile admits rotational structure within this normal plane.

A convenient schematic representation of this structure is given below.

$$ \Phi(A) = \alpha_{1}\cos(\omega_{1} A)\, N_{1} + \alpha_{2}\cos(\omega_{2} A + \phi)\, N_{2} $$

Here the normal-plane basis vectors form an orthonormal frame, and the rotation parameters characterise the informational profile.

An observer does not perceive the full four-dimensional rotation. What is observed is the projection of this normal-plane structure into the observer’s three-space.

$$ \phi_{\mathrm{obs}}(A) = P_{3}\!\left( \Phi(A) \right) $$

The observed modulation is not a wave propagating through spacetime. It is the shadow cast by higher-dimensional rotation under admissible continuation.

7.3 Informational Bounds and Oscillation Geometry

Although a Worldline-plus is one-dimensional, its informational profile admits a two-dimensional cross-section in the normal plane. In the simplest case, this cross-section forms a square informational envelope.

The maximum possible displacement of the projected modulation is therefore bounded geometrically.

$$ |\phi_{\mathrm{obs}}(A)| \leq \frac{W}{\sqrt{2}} $$

This bound is purely geometric and strictly non-statistical. It arises from the finite extent of admissible informational expression and is independent of probabilistic interpretation.

In later sections, this bound becomes the geometric origin of observed limits on quantum correlations.

7.4 Holonomy and Translation

When admissible continuation within an orientation class is exhausted, continuity bias requires re-expression of the same informational content. In a non-orientable setting, such re-expression necessarily lies in the opposite orientation class. This transition is termed Translation.

Geometrically, accumulated holonomy of the normal frame records that admissible continuation has closed upon itself. Holonomy does not cause Translation; it encodes the fact that continuity-preserving re-expression within the original orientation class is no longer available.

Translation is therefore an informational necessity whose geometric signature is orientation inversion between the Obverse and Reverse phases.

7.5 Summary of Worldline+ Dynamics

The behaviour described in this section may be summarised as follows:

• Worldline-plus progression corresponds to satisfaction of continuity bias through admissible re-expression, ordered geometrically.
• Rotational structure within the normal plane projects into three-space as modulating or wave-like behaviour.
• The amplitude of this projected behaviour is bounded by the geometry of the Worldline-plus informational cross-section.
• When admissible continuation within an orientation class is exhausted, continuity bias selects Translation as the only remaining re-expression.
• Translation is an informational re-expression whose geometric signature is orientation inversion between global orientation phases.

8. Translation and Orientation Inversion

Translation is the compulsory re-expression of a Worldline-plus structure when admissible continuation within its current orientation ceases to be defined. Because Worldline-plus structures are persistent informational entities rather than objects in motion, they cannot reverse direction, fall back, or terminate.

When entropic ordering collapses, continuity bias therefore admits only a single continuation: orientation inversion into the opposite global phase.

8.1 Orientation Phases

The admissible expressions of a Worldline-plus separate into two global orientation classes. These correspond to matter and antimatter as observed in three-dimensional projections.

Definition — Obverse and Reverse Phases

A Worldline-plus is said to be in the Obverse phase if its transported normal frame agrees with a fixed local orientation. It is said to be in the Reverse phase if its transported normal frame is inverted under admissible continuation.

Matter corresponds to the Obverse phase. Antimatter corresponds to the Reverse phase.

Phase invariance under local projection

Local physical laws derived from informational geometry are identical in the Obverse and Reverse phases. The distinction between phases is global rather than local. In three-dimensional projections, the two phases are locally indistinguishable, differing only in how admissible continuation realises global orientation structure.

8.2 Translation as Orientation Re-Expression

Definition — Translation

A Translation of a Worldline-plus is the compulsory orientation-inverted re-expression between the Obverse and Reverse phases, enforced when continuity bias forbids termination and admissible continuation within the current orientation is exhausted.

$$ \gamma : \text{Obverse} \longleftrightarrow \text{Reverse} $$

Holonomy and exhausted progression

As admissible progression along geometric entropy ordering slows, rotation in the normal plane continues to accumulate. When admissible continuation within the current orientation is no longer available, continuity bias enforces re-expression through orientation inversion.

Orientation-reversing holonomy does not cause Translation. It records that admissible continuation has closed upon itself in a manner incompatible with preservation of global orientation.

Translation does not involve motion through spacetime, nor the annihilation of mass. It is a change in how the same informational content is expressed.

8.3 Entropic Stillness and Translation

Translation occurs at entropic stillness, defined as the regime in which admissible ordering collapses.

Definition — Translation Region

A region of the Mobifold is said to be a Translation region if the magnitude of the entropic ordering gradient falls below a small threshold set by the effective separation, or tension, of the neutrino background.

$$ |\nabla S| < \varepsilon $$

Translation condition

In a Translation region, admissible continuation within the current orientation becomes unavailable. Since a Worldline-plus cannot terminate or reverse, Translation is compulsory for any sufficiently minimal structure whose continuation has reached an expressional dead end.

Regions of exact entropic stillness enforce Translation most generally. Typical Translation regions include curvature wells, such as black-hole interiors, and the asymptotic regime of maximal entropic separation previously described as cosmic heat death.

8.4 Information Preservation

Information continuity

Translation preserves informational content. Matter and antimatter are therefore orientation-inverted expressions of the same information.

$$ I_{\text{Obverse}} = I_{\text{Reverse}} $$

Where annihilation is observed, it corresponds to catastrophic spatial reconfiguration rather than destruction of informational content.

8.5 Global Consequences of Translation

Translation is not exceptional. Because neutrino Worldline-plus structures are long-lived and ubiquitous, entropic stillness is encountered somewhere along admissible continuation, both locally through curvature wells and globally in the asymptotic limit.

The resulting ubiquity of orientation inversion means that no globally consistent orientation can be assigned across the full manifold. The geometric encoding of this fact is non-orientability.

8.6 Summary of Orientation Dynamics

The central consequences of this section may be summarised as follows:

1. Worldline-plus expression admits two global orientation phases: Obverse, associated with matter, and Reverse, associated with antimatter.
2. Worldline-plus structures are not objects in motion and cannot reverse, fall back, or terminate.
3. When admissible ordering collapses at entropic stillness, continuity bias enforces Translation.
4. Translation preserves informational content; matter and antimatter are orientation-inverted expressions of the same information.
5. The ubiquity of Translation implies the absence of global orientation, encoded geometrically as non-orientability.

These results complete the intrinsic account of orientation inversion required for the interaction and quantum-geometry sections that follow.

9. Inter-Worldline+ Geometry and the Four Forces

In BEIPE, the four fundamental interactions arise neither from mediating particles nor from independent dynamical fields. They arise from how informational Worldline-plus structures constrain one another’s admissible continuation routes within a shared substrate, and from how those constraints are geometrically encoded in the Mobifold.

Worldline-plus interactions are therefore not forces in the dynamical sense. They are expressions of how:

• informational geometries fit together or conflict,
• admissible continuation routes interfere, align, or exclude one another,
• ordering constraints encoded by the entropic structure are locally reshaped, and
• stability or instability under normal-plane rotation is resolved.

The four interactions correspond to four qualitatively distinct regimes of inter-Worldline-plus geometry, treated in turn below.

9.1 General Framework

Whenever Worldline-plus structures approach, their informational geometries define sets of admissible continuation routes through the skein. These routes may:

• interlock compatibly,
• compete for the same geometric degrees of freedom,
• destabilise under accumulated curvature, or
• re-route one another through substrate-level constraint coupling.

Compatibility leads to braided stability, corresponding to the Strong interaction.
Instability leads to discrete reconfiguration, corresponding to the Weak interaction.
Long-range coupling of continuation routes gives rise to electromagnetic behaviour.
Persistent reshaping of entropic ordering produces gravitation.

9.2 The Strong Interaction: Compatibility and Braided Stability

Definition — Compatibility

Two Worldline-plus structures are said to be compatible if their informational geometries admit a continuous joint embedding whose combined configuration reduces overall entropic tension.

$$ S(\gamma_{1} \cup \gamma_{2}) < S(\gamma_{1}) + S(\gamma_{2}) $$

Braid formation

Compatible Worldline-plus structures may form a stable braided configuration whose combined curvature lies at a local minimum of the entropic landscape. Within such a braid, informational continuity is preserved while entropic tension is collectively reduced.

Strong interaction as braided stability

The Strong interaction is the stabilising geometry of compatible braids. Confinement follows from continuity bias: a braid cannot be separated into isolated components without violating Worldline-plus contiguity. Observed resonance spectra correspond to curvature eigenmodes of the entropic ordering on braided configurations, rather than to bound states mediated by exchange particles.

9.3 The Weak Interaction: Instability and Reconfiguration

Definition — Instability

A Worldline-plus configuration is said to be unstable if accumulated curvature increases while admissible progression diminishes along entropic ordering.

$$ \frac{dK(\gamma)}{dA} > 0 \qquad\text{and}\qquad \frac{d\alpha(A)}{dA} < 0 $$

Reconfiguration pathways

Unstable configurations admit discrete reconfiguration pathways selected by continuity bias. Each pathway rearranges informational geometry into a set of more stable Worldline-plus structures together with minimal-information carriers that preserve informational continuity.

Weak interaction as geometric reconfiguration

Weak processes arise from bias-selected geometric reconfiguration of unstable Worldline-plus structures. Decay products correspond to stable informational geometries. Emitted neutrinos correspond to minimal-information carriers that relieve excess curvature while maintaining continuity. Parity violation reflects the intrinsic chirality of admissible continuation in a non-orientable manifold.

9.4 Electromagnetism: Route Coupling in the Skein

Definition — Route Alignment

Each Worldline-plus structure induces a constraint imprint on the skein by virtue of its admissible continuation routes. Electromagnetic behaviour arises from the geometric alignment or misalignment of these route imprints.

$$ \mathcal{A} = g^{\mu\nu}\, R_{\mu\alpha}(\gamma_{1})\, R_{\nu}^{\ \alpha}(\gamma_{2}) $$

Attraction and repulsion

When route alignment reduces constraint cost, joint continuation is favoured and attraction results. When route alignment increases constraint cost, separation is favoured and repulsion results. The sign and magnitude of the alignment parameter therefore characterise electromagnetic behaviour.

Electromagnetism as pathway interaction

Electromagnetism arises from coupling between admissible continuation routes through the shared skein. A Worldline-plus modifies the local constraint geometry of the substrate; other Worldline-plus structures register this modification as re-routing of their available paths. Long-range behaviour reflects extension of such constraint imprints across the skein rather than exchange of mediators. Apparent inverse-square laws arise from geometric dilution of route compatibility across the Mobifold.

9.5 Gravitation: Grooving of the Entropic Ordering

Definition — Groove

A groove is a persistent modification of the local entropic ordering induced by a Worldline-plus structure.

$$ \partial_{\mu} S = \partial_{\mu} S_{0} + \delta_{\mu}(\gamma) $$

Admissible descent

Nearby Worldline-plus structures realise admissible continuation paths ordered by the entropic structure. This relation is not dynamical: the entropic gradient does not act on Worldline-plus structures but records the ordering of permissible descent selected by informational bias.

Gravitation as ordering deformation

Gravitation is the large-scale consequence of persistent grooves in entropic ordering induced by stable, long-lived Worldline-plus structures. Orbital dynamics and lensing arise from geometric reshaping of admissible descent paths, not from force mediation or spacetime curvature. In the weak-field regime, observed gravitational behaviour is treated as empirical calibration of the scalar ordering field.

9.6 Summary of Interaction Ontology

The four interactions arise as follows:

1. The Strong interaction is braided stability arising from compatibility of informational geometries.
2. The Weak interaction is bias-selected geometric reconfiguration of unstable Worldline-plus structures.
3. Electromagnetism is long-range coupling of admissible continuation routes through the shared skein.
4. Gravitation is persistent deformation of entropic ordering that structures admissible descent.

All four interactions arise from informational biases acting on Worldline-plus geometry, with the Mobifold encoding their geometric expression. No mediators, gauge fields, or independent forces are required at the ontological level.

10. Quantum Behaviour From Informational Geometry

Quantum phenomena arise in BEIPE not from probabilistic postulates, wavefunctions, or operator algebra, but from the informational and geometric behaviour of Worldline-plus structures embedded in the Mobifold. Apparent quantum features—superposition, interference, entanglement, spin, and the Tsirelson bound—arise from three ingredients:

1. rotation in the normal bundle of Worldline-plus structures,
2. projection into an observer’s three-space, and
3. geometric constraints on the transverse informational profile.

10.1 Modulation as Projection of a Four-Dimensional Rotation

A Worldline-plus is a one-dimensional structure embedded in a four-dimensional non-orientable manifold and therefore possesses a two-dimensional normal plane. Rotation within this plane is a geometric property of the structure itself.

When projected into an observer’s three-space, this higher-dimensional rotation appears as oscillatory behaviour, here termed modulation. Quantum “wave” behaviour is therefore not fundamental, but the observable shadow of higher-dimensional geometry.

Projected rotation

The observable modulation corresponds to projection of normal-plane rotation into three-space. This replaces the wavefunction with a concrete geometric object.

$$ \phi_{\mathrm{obs}}(A) = P_{3}\!\left( \Phi(A) \right) $$

Oscillatory behaviour is therefore a projected geometric effect, not a complex amplitude defined on an abstract Hilbert space.

10.2 Informational Geometry and Bounds on Oscillation

A Worldline-plus is a one-dimensional carrier of information whose informational profile is realised transversely as a geometric cross-section in the normal plane. This cross-section encodes shape, topology, and internal geometric structure, not merely area.

Some Worldline-plus structures admit only topological or one-dimensional transverse structure. Others admit extended two-dimensional informational envelopes.

Definition — Informational Envelope

The informational envelope is the geometric cross-section defined by the informational profile of a Worldline-plus. It encodes the full transverse structure of the object, including shape and topology. For minimal informational complexity, the envelope may be represented as a square of fixed width.

Diagonal amplitude bound

Where a two-dimensional informational envelope is admitted, the observable modulation is geometrically bounded.

$$ |\phi_{\mathrm{obs}}(A)| \leq \frac{W}{\sqrt{2}} $$

This bound is purely geometric and strictly non-statistical. It expresses the fact that the largest possible projection of a square envelope is its diagonal. No probabilistic interpretation is required.

Quantum correlation bound

When two correlated Worldline-plus segments are compared at different projection orientations, the resulting correlations take a cosine form characteristic of quantum systems.

$$ C(\theta) = \tfrac{1}{2}\cos(2\theta) $$

The maximum magnitude of the correlation is fixed by the diagonal envelope bound and reaches the value associated with Tsirelson’s limit. This reproduces the quantum correlation bound without invoking Hilbert space, nonlocality, or stochastic axioms.

Worldline-plus structures with purely one-dimensional or topological transverse geometry do not realise this bound directly, but may participate in composite systems whose joint geometry does.

10.3 Interference as Projection Geometry

Interference patterns arise from projection of a single Worldline-plus’s higher-dimensional rotation through multiple observational pathways.

Interference principle

Observed intensity reflects squared projected amplitude, not superposition of propagating waves.

$$ I(\theta) \propto |\phi_{\mathrm{obs}}(\theta)|^{2} $$

Interference therefore reflects differing projection angles of the same higher-dimensional rotation, rather than interaction of multiple wavefronts in space.

10.4 Spin and the Double-Cover Structure

Spin-one-half behaviour emerges naturally from rotation in a non-orientable normal plane.

Spin as rotation in the Mobifold

Because normal-plane rotation is defined in a non-orientable two-plane, a Worldline-plus returns to its original orientation only after a full double rotation. The Mobifold therefore admits the double-cover structure characteristic of spin-one-half systems.

Spin is not an intrinsic angular momentum appended to a point particle, but a topological consequence of higher-dimensional rotation.

10.5 Entanglement as Shared Worldline-Plus Geometry

Entangled systems correspond to Worldline-plus structures that share a common informational origin and therefore retain synchronised normal-plane rotation.

Definition — Shared Origin

Two Worldline-plus segments are said to be entangled if they arise from a common parent Worldline-plus whose informational geometry bifurcates while preserving phase relations.

Synchronised rotation

Entangled Worldline-plus structures share the same normal-plane rotation, up to local geometric distortion.

$$ \Phi_{1}(A) = \Phi_{2}(A) $$

Their correlated behaviour follows from shared geometry, not from superluminal influence.

Locality of entanglement

Because entanglement reflects common origin and projection geometry, no nonlocal signalling is required. Correlation replaces communication.

10.6 Measurement as Projection Alignment

In BEIPE, measurement does not collapse a wavefunction. It selects a projection of higher-dimensional rotation.

Definition — Measurement Projection

Measurement corresponds to imposing a projection aligned with a chosen orientation in the observer’s three-space.

$$ P_{\theta} $$

Outcome determination

Measurement outcomes correspond to extrema of the projected rotation within the informational envelope. This yields the familiar discrete outcomes of spin measurements without invoking intrinsic randomness.

Determinism and apparent randomness

The ontology is deterministic. Apparent randomness arises from incomplete geometric knowledge: observers lack access to the full phase structure and exact projection alignment.

Ontological origin of the quantum bounds

The quantum bounds derived here are not postulates. Where a two-dimensional informational envelope is admitted, the diagonal bound follows directly from representing physical entities as Worldline-plus structures embedded in a four-dimensional non-orientable manifold.

Tsirelson’s bound is therefore a corollary of informational geometry, not an independent axiom.

10.7 Summary of Quantum Behaviour

The quantum behaviour described in this section may be summarised as follows:

1. Modulation is the projection of higher-dimensional normal-plane rotation.
2. Informational geometry constrains observable amplitudes and correlations.
3. Interference arises from projection geometry, not wave superposition.
4. Spin-one-half behaviour follows from rotation in a non-orientable two-plane.
5. Entanglement reflects shared geometric origin rather than nonlocal influence.
6. Measurement selects projections; apparent randomness reflects incomplete geometric knowledge.

11. Cosmology From the Entropic Gradient

Cosmological phenomena in BEIPE arise from the large-scale geometric encoding of informational ordering and its gradient. Unlike standard cosmology, BEIPE does not posit metric expansion, dark energy, or inflation. Instead, redshift, anisotropy, cosmic microwave background structure, and large-scale observables emerge from how informational Worldline-plus structures are ordered and projected within the block-universe Mobifold.

This section formalises these results in geometric rather than dynamical terms.

11.1 Redshift as Entropic Separation

In BEIPE, redshift measures relative entropic separation between observer and source rather than Doppler velocity or stretching of a metric background. Because informational ordering is represented by a dimensionless geometric scalar, differences in ordering are themselves dimensionless and may enter exponential relations without ambiguity.

Definition — Entropic Redshift

Entropic redshift is defined by the difference in informational ordering between source and observer.

$$ 1 + z = \exp\!\left( S(x_{s}) - S(x_{o}) \right) $$

Here the exponent is a pure number, so the expression is well-defined in purely geometric units.

$$ \Delta S = S(x_{s}) - S(x_{o}) $$

No metric expansion required

Entropic redshift arises from ordering differences rather than spacetime motion.

$$ z = \Delta S + \mathcal{O}((\Delta S)^{2}) $$

Observable frequency shifts therefore measure how far apart source and observer are ordered along informational descent, not how fast they recede in spacetime. In appropriate macroscopic limits, thermodynamic descriptions of redshift are recovered as coarse-grained approximations to this geometric relation.

11.2 Directional Dependence of Apparent Expansion

Because informational ordering is represented by a vector field on a non-orientable manifold, its projection into an observer’s three-space depends on orientation.

Directional expansion

The apparent expansion rate may be written schematically as a function of observational direction.

$$ H(\theta,\phi) = \frac{|\nabla S(\theta,\phi)|\,c}{S} $$

This dependence provides a natural explanation for observed anisotropies in local expansion measurements, dipole flows, and large-scale alignments without abandoning global homogeneity or isotropy of the Mobifold.

Cosmic anisotropy from projection

Global anisotropies arise from projection of informational ordering into the observer’s tangent three-space. The Mobifold may remain globally homogeneous and isotropic while producing direction-dependent expansion when viewed from a specific entropic ordering orientation.

This projection-induced anisotropy determines how expansion appears to a given observer, but it does not define the cosmic dipole axis itself. That axis arises from a deeper topological feature of the Mobifold.

11.3 Dipole Holonomy From Mobifold Non-Orientability

The directional variation described above operates relative to an axis fixed by the Mobifold’s global orientation-reversing holonomy. The apparent expansion dipole is therefore a projection-level manifestation of a deeper topological structure.

Half-twist holonomy

A non-orientable manifold necessarily carries global holonomy. Parallel transport around a fundamental loop induces inversion of transported vectors, defining a unique global inversion axis.

Dipole as geometric shadow

Projection of a single inversion axis onto the celestial sphere produces a dipolar harmonic.

$$ \Delta(\theta,\phi) \propto \cos\theta $$

Higher multipoles require multiple independent twisting directions. Because the Mobifold contains only one such global twist, the dipole is its unavoidable observational imprint.

Cosmological meaning

In BEIPE, the observed cosmic microwave background dipole is not kinematic and does not arise from peculiar motion. It is the geometric shadow of the Mobifold’s non-orientable holonomy registered through the observer’s ordering orientation.

11.4 Tracer-Dependent Dipoles From Projection Geometry

Different Worldline-plus types register projection geometry differently due to differences in their informational and torsional structure.

Photon dipole

Photons are torsional, modulating Worldline-plus structures. Their observed dipole reflects the Mobifold inversion axis together with accumulated projection distortion.

Neutrino dipole

Neutrinos are minimal, nearly torsion-free Worldline-plus structures. Their ordering vectors track informational ordering more faithfully, so the cosmic neutrino background dipole is predicted to align more closely with the true Mobifold inversion axis.

Radio and quasar dipoles

Intermediate Worldline-plus structures should exhibit dipoles of intermediate amplitude with small but measurable angular offsets, depending on torsional structure.

Contrast with standard cosmology

Standard cosmology predicts a single universal dipole for all tracers. Systematic angular offsets between photon, neutrino, and radio-source dipoles would falsify the kinematic interpretation and support an entropic-geometric origin.

11.5 Direction-Dependent Redshift From Observer Ordering

In BEIPE, redshift is a functional of informational ordering rather than motion.

$$ \vec{D}_{\mathrm{obs}} = -\nabla S \big|_{x_{\mathrm{obs}}} $$ $$ \vec{D}_{\mathrm{src}}(\theta,\phi) = -\nabla S \big|_{x_{\mathrm{src}}(\theta,\phi)} $$

For fixed comoving separation, the directional contribution to observed redshift is proportional to the magnitude of the difference between these ordering vectors.

$$ z(\theta,\phi) \propto \big|\vec{D}_{\mathrm{src}}(\theta,\phi) - \vec{D}_{\mathrm{obs}}\big| $$

This separation is geometric and relational rather than causal. Apparent expansion rates vary directionally even when the Mobifold remains globally homogeneous and isotropic.

11.6 CMB Structure From Curvature Eigenmodes

The curvature structure encoded by informational ordering governs multipole structure in the cosmic microwave background.

Definition — Curvature Eigenmode

A curvature eigenmode is defined by an eigenvalue relation on the informational ordering field.

$$ \nabla^{2} S_{k} = \lambda_{k} S_{k} $$

Discrete acoustic scales

Acoustic structure arises from discrete curvature eigenvalues rather than horizon-crossing dynamics.

$$ \ell_{1} \sim \sqrt{\mathrm{Re}(\lambda_{1})}\, r_{*} $$

Detailed spectral fitting requires an explicit choice of informational ordering and is left for future work.

11.7 BAO Scale as Propagated Packing Geometry

In BEIPE, the baryon acoustic oscillation scale is interpreted as a preserved geometric correlation inherited from the Universe’s initial informational packing.

BAO origin

The Main Vent corresponds to the boundary condition of maximal informational compression. Finite packing geometry admits a characteristic nearest-neighbour scale.

As informational structures separate under ordering, adjacency relations propagate outward as weak correlations.

$$ D_{\mathrm{BAO}} \propto D_{\mathrm{vent}}\, \exp\!\left(S_{\mathrm{cosmic}}\right) $$

The exponential reflects geometric separation along informational ordering, not metric expansion or causal dynamics.

11.8 Black Holes as Entropic-Stillness Wells

Regions of extreme curvature drive informational ordering toward collapse.

Definition — Entropic Stillness Region

$$ \nabla S|_{U} = 0 $$

Such regions admit Translation while preserving informational content, even though three-dimensional projection terminates at a horizon.

11.9 Cosmic Heat Death and Global Translation

If informational ordering collapses globally, all Worldline-plus structures satisfy the Translation condition and the Universe undergoes global orientation inversion.

Thermodynamic heat death corresponds to geometric entropic stillness rather than loss of information.

11.10 Summary of Cosmological Structure

The cosmological consequences of BEIPE may be summarised as follows:

1. Redshift arises from entropic separation rather than metric expansion.
2. Apparent expansion anisotropy reflects projection of informational ordering.
3. Cosmic microwave background and BAO structure arise from curvature and packing eigenmodes.
4. Black holes are entropic-stillness regions permitting Translation and preserving information.
5. Heat death corresponds to collapse of informational ordering and global orientation inversion.
6. Non-orientable holonomy produces an unavoidable dipolar anisotropy with tracer-dependent offsets.

12. Predictions and Falsifiability

A physical framework must yield empirically distinguishable predictions. BEIPE produces a range of testable consequences across cosmology, quantum behaviour, particle physics, and gravitation.

These predictions arise from informational biases acting on Worldline-plus geometry, with the Mobifold and entropic ordering encoding their geometric expression. No additional particle species or gauge fields are postulated beyond the Worldline-plus ontology itself. Masses, couplings, and correlation bounds arise as geometric invariants rather than adjustable parameters.

12.1 Directional Expansion Signatures

Directional Hubble variation

If cosmological redshift arises from informational ordering rather than metric expansion, then the apparent expansion rate must exhibit a dipolar pattern reflecting projection of the entropic ordering into the observer’s three-space.

This effect describes the appearance of anisotropic expansion and does not itself define the cosmic dipole axis, which is fixed by the Mobifold’s global orientation-reversing holonomy.

Observable signature

Large-sky surveys should detect a statistically significant anisotropy in the apparent expansion rate aligned with the observer’s entropic ordering orientation. This anisotropy should remain fixed in direction over cosmic time, even as local structures evolve.

12.2 Tracer-Dependent Dipole Offsets

Prediction

If the cosmic microwave background dipole originates from Mobifold holonomy rather than peculiar velocity, then different cosmological tracers must exhibit dipoles of differing amplitude and direction, determined by their Worldline-plus geometry and torsional structure.

Specifically:

1. The photon dipole reflects Mobifold holonomy together with curvature-induced projection distortion.
2. The cosmic neutrino dipole aligns more closely with the pure Mobifold inversion axis.
3. Radio-source and quasar dipoles lie between these two, with offsets set by torsional coupling.

Falsifiability

Standard cosmology predicts a single universal dipole direction for all tracers, fixed by peculiar velocity. Detection of systematic angular offsets between photon, neutrino, radio, and quasar dipoles would decisively falsify the kinematic interpretation and support BEIPE’s geometric origin.

12.3 Asymptotic Behaviour of Entropic Ordering at Large Entropic Age

The entropic gradient, as the geometric encoding of bias-constrained informational ordering, imposes necessary asymptotic scalings at large entropic age without invoking dynamical equations or field evolution.

These scalings follow directly from continuity and expressional biases and yield quantitative, falsifiable constraints on high-redshift cosmology.

Non-Negotiable Ontological Constraints

The relevant commitments are:

• Continuity bias: Worldline-plus structures must persist without termination or singular endpoints.
• Expressional bias: Admissible expressions favour minimal complexity, disallowing indefinite steepening of ordering gradients.
• Block-universe structure: No fundamental time or evolution; all scalings are static geometric properties.
• Entropic age as ordering parameter: Large entropic age corresponds observationally to high redshift.

These constraints forbid asymptotic behaviours that would halt admissible continuation or proliferate complexity without bound.

Derivation of Asymptotic Inequalities

At sufficiently large entropic age, the following inequalities must hold.

$$ \frac{d}{dA} |\nabla S| \le 0 \qquad \forall A > A_* $$

Rationale

Continuity requires sustained descent, while expressional bias forbids indefinite steepening that would render continuation geometrically inaccessible.

$$ |\nabla S| \to k \quad \text{or} \quad |\nabla S| \propto A^{-\alpha}, \qquad 0 < \alpha \le 1 $$

Rationale

Expressional bias favours flattening at large scales, while continuity forbids overly rapid decay that would terminate admissible continuation at finite entropic reach.

$$ |\nabla^2 S| \ll |\nabla S|^2 \qquad \forall A > A_* $$

Rationale

Late-stage curvature fluctuations must remain subordinate to global ordering. Persistent curvature dominance would introduce unnecessary structural complexity, contradicting expressional bias.

Cosmological Consequences and Falsifiability

These inequalities imply:

• flattening or decline of the apparent expansion rate at high redshift,
• asymptotic approach to a constant expansion scale without dark energy.

Precision surveys can test these predictions directly. Continued acceleration persisting to high redshift would falsify BEIPE outright.

12.4 Curvature Eigenmodes and Hadronic Spectra

Eigenmode spectrum

If hadronic resonances correspond to curvature eigenmodes of entropic ordering on braided Worldline-plus configurations, then resonance masses and widths must follow relations fixed by eigenvalue structure.

Structured mass relations

Mass ratios within hadronic families should emerge from geometric eigenvalues with a single overall scale. Failure of this mapping would falsify this element of BEIPE.

12.5 Polarisation and Informational Geometry

Diagonal bound test

If quantum correlations arise from informational geometry, then correlation functions must saturate but never exceed the universal bound associated with diagonal projection geometry.

$$ 2\sqrt{2} $$

Experimental invariance

All Bell-type experiments must converge to the same correlation envelope regardless of configuration or separation. Any robust violation of this bound would rule out BEIPE

12.6 Spin Predictions From Non-Orientability

Spin reversal symmetry

If spin-one-half behaviour arises from rotation in a non-orientable normal plane, then orientation inversion must occur after a single full rotation, with return only after a double rotation.

Mobifold-induced phase shifts

Geometric-phase experiments should detect anomalies beyond standard Berry-phase predictions, corresponding to the Mobifold’s orientation-reversing holonomy.

12.7 Black Hole Translation Effects

Translation in curvature wells

If entropic stillness regions permit Translation, then Worldline-plus structures entering deep curvature wells must undergo orientation inversion.

Information-preservation signature

Black-hole evaporation spectra should preserve informational correlations consistent with re-expression rather than destruction. Observation of genuine information loss would challenge this aspect of BEIPE.

12.8 Cosmic Heat Death and Universal Translation

Entropic limit

If global entropic ordering collapses, all Worldline-plus structures must satisfy the Translation condition, producing universal orientation inversion rather than informational loss.

Observable trend

Future surveys should detect progressive reduction in large-scale anisotropy as entropic stillness is approached. Persistent or growing anisotropy would argue against this endpoint.

12.9 Summary of Falsifiable Predictions

BEIPE makes the following testable predictions:

1. Direction-dependent expansion anchored by Mobifold holonomy.
2. Tracer-dependent dipole offsets across photon, neutrino, and radio backgrounds.
3. Flattening of the expansion rate at high redshift without dark energy.
4. Hadronic spectra governed by curvature eigenmodes rather than fitted potentials.
5. Universally bounded quantum correlations capped at the Tsirelson limit.
6. Spin-one-half phase behaviour arising from non-orientable geometry.
7. Information-preserving black-hole phenomenology via Translation.
8. Long-term drift of cosmic anisotropy toward entropic stillness.

These predictions distinguish BEIPE decisively from standard cosmology, general relativity, and quantum field theory, and provide direct empirical routes to validation or falsification.

13. Unification of Quantum Theory, Relativity, and Information

BEIPE advances a unification not merely of quantum theory and general relativity, but of quantum theory, relativistic structure, and information theory within a single ontological framework. The central claim is not that existing formalisms are mathematically incomplete, but that they are ontologically fragmented: each operates with incompatible primitives while successfully describing different aspects of the same underlying reality.

In BEIPE, information is taken as the sole physical primitive. Its admissible expressions are governed by a small set of universal biases—continuity, directionality, spatial compatibility, and expressional minimality—which determine which informational continuations are realised. The cumulative structure of these admissible continuations is realised geometrically as a fixed, four-dimensional, non-orientable block, together with a geometric entropy field, its ordering gradient, and the persistent informational structures termed Worldline-plus entities. Geometry does not act dynamically; it encodes constraint.

$$ S $$ $$ \nabla S $$

From this perspective, quantum theory, general relativity, and information theory are not competing or independent frameworks. They are descriptive regimes applied to the same bias-governed informational ontology.

Quantum theory describes how higher-dimensional informational geometry is sampled under projection into an observer’s three-space. Relativity describes the large-scale ordering of admissible informational continuation within a Lorentzian block structure. Information theory accounts for coarse-grained bookkeeping of persistence, ordering, and reconfiguration within that same structure.

The power of the framework lies in its explanatory reach. Features that appear paradoxical or axiomatic within conventional formalisms arise naturally once the underlying ontology is specified. Quantum interference and wave–particle duality reflect projection of normal-plane rotation rather than dual modes of being. Entanglement corresponds to shared informational origin and synchronised geometric phase, not nonlocal influence or collapse. Quantum correlation bounds arise as geometric saturation limits of bounded transverse structure, rather than as probabilistic axioms.

In particle physics, BEIPE replaces force mediation and confinement postulates with geometric compatibility and instability. Hadrons are stable braided configurations of Worldline-plus structures whose inseparability follows from continuity bias rather than imposed confining forces.

Electromagnetic interaction arises from long-range coupling of admissible continuation routes through the shared skein. Weak processes correspond to bias-selected reconfiguration of unstable Worldline-plus geometries, with observed parity violation reflecting the intrinsic chirality of admissible continuation in a non-orientable manifold.

Gravitation and cosmology are similarly reinterpreted. Persistent mass-carrying Worldline-plus structures encode grooves in the geometric entropy field, and nearby Worldline-plus structures realise admissible continuation ordered by the entropy gradient. Orbital dynamics and lensing follow from geometric ordering rather than from spacetime acting as a causal agent.

Cosmological redshift measures entropic separation rather than metric expansion. Large-scale anisotropies arise from projection of informational ordering into an observer’s tangent three-space. Late-time acceleration is resolved as a geometric consequence of bias-constrained ordering: indefinite steepening of the entropy gradient is forbidden, leading necessarily to flattening or sub-linear decay without invoking dark energy.

$$ -\nabla S $$

Time itself is not a fundamental dimension. It is entropic age: the accumulated ordering difference encoded by the entropy gradient. The arrow of time, gravitational redshift, and cosmological evolution all trace back to the same geometric ordering structure. Temporal experience reflects continued admissible re-expression, not motion through an independent temporal parameter.

Unification in BEIPE is therefore achieved not by embedding existing theories within a larger formalism, but by replacing incompatible primitives with a single informational substrate realised geometrically.

The framework does not, at this stage, aim to reproduce the full numerical machinery of quantum field theory or general relativity. Its aim is ontological: to clarify what physical theories are describing, why their characteristic structures arise, and which features reflect genuine geometric constraints rather than descriptive artefacts.

In this sense, BEIPE is unified, austere, deterministic, and explicitly falsifiable. Its claims stand or fall on the geometry of information itself.

14. Conclusion

The BEIPE framework advances a unified ontological account of physical phenomena across quantum, particle, gravitational, and cosmological regimes. This unification does not arise from embedding existing theories within a larger formalism, but from replacing their incompatible primitives with a single substrate: information constrained by universal biases and realised geometrically within a fixed, non-orientable four-dimensional manifold.

The central elements of the framework are deliberately minimal:

1. A four-dimensional non-orientable manifold, termed the Mobifold, which encodes the global failure of orientation arising from unavoidable orientation-inverting continuation.
2. A geometric entropy field and its associated ordering gradient, representing admissible informational continuation without invoking fundamental time or probabilistic evolution.
3. Worldline-plus structures: continuous informational threads embedded in the Mobifold, whose transverse geometry, stability, and normal-plane rotation encode quantum behaviour under projection.
4. Translation: a compulsory topological re-expression enforced when admissible continuation is exhausted, mapping Worldline-plus structures between global orientation phases while preserving informational content.

$$ \mathcal{M} $$ $$ S $$ $$ \nabla S $$

These elements are not independent assumptions. Continuity bias forbids termination of informational Worldline-plus structures. Entropic stillness exhausts admissible progression. Translation is therefore unavoidable. The impossibility of preserving a global orientation under such continuation is encoded geometrically as non-orientability.

Matter and antimatter appear as global orientation phases of the same informational structures. Spin one-half behaviour follows from rotation in a non-orientable normal plane. Quantum correlation bounds arise from finite transverse geometry rather than probabilistic axioms.

Within this ontology, familiar physical phenomena emerge as descriptive regimes:

• Quantum behaviour arises from projection of higher-dimensional informational geometry into an observer’s three-space, producing interference, entanglement, spin one-half behaviour, and bounded correlations as geometric constraints.
• Particle interactions correspond to regimes of inter-Worldline-plus geometry: braided compatibility, bias-selected reconfiguration, route alignment through the skein, and persistent deformation of entropic ordering.
• Cosmology reflects large-scale structure in informational ordering: entropic redshift replaces metric expansion; anisotropies arise from projection of ordering vectors; curvature eigenmodes yield cosmic microwave background structure; and baryon acoustic oscillations encode propagated skein-packing geometry.

A key result of the framework is its asymptotic behaviour. Under mild assumptions on bias-constrained informational unpacking, the ordering gradient cannot steepen indefinitely. Instead, it exhibits flattening or sub-linear decay at large entropic age.

This structure yields late-time cosmological behaviour without invoking dark energy, accelerated expansion, or dynamical modification of gravity. What appears as acceleration in conventional descriptions is reinterpreted as projection of a bounded geometric ordering.

BEIPE is explicitly falsifiable. Its predictions include direction-dependent expansion signatures, tracer-dependent dipole offsets, curvature-eigenmode relations in hadronic spectra, preservation of information across black-hole stillness regions, and a universal geometric bound on quantum correlations.

These predictions do not rely on free parameters tuned to existing data. They follow from the same geometric constraints that underpin the ontology itself.

Future work will focus on refining the explicit mathematical structure of informational ordering on the Mobifold, developing analytic expressions for Worldline-plus compatibility and braid eigenmodes, and extending the curvature–entropy relationship toward a fully geometric formulation.

These developments will enable quantitative application of the framework to particle masses, decay widths, early-universe structure, and cosmological energy scales.

BEIPE therefore offers a coherent, deterministic, and geometrically realised alternative to conventional field-based formulations. It restores ontology to physics not by rejecting established mathematics, but by situating it within a unified informational structure that explains both its empirical success and its conceptual limits.

Appendix A

Geometric Origin of the Correlation Ceiling in BEIPE

This appendix explains why experiments probing entangled systems consistently observe a maximum statistical contrast characterised by a specific numerical ceiling, and why this value arises necessarily within the BEIPE framework.

The aim is not to rederive quantum inequalities or appeal to Hilbert-space structure, but to identify the geometric origin of the observed ceiling itself.

In BEIPE, the bound is not a statement about probability or indeterminism. It is a statement about what geometric information can be represented and accessed through projection from a bounded transverse structure.

A.1 The Only Geometric Ingredient: A Bounded Two-Dimensional Envelope

A Worldline-plus is a one-dimensional informational object embedded in a four-dimensional manifold. Because the normal bundle of a curve in four dimensions is two-dimensional, each Worldline-plus admits a two-dimensional transverse envelope in its normal plane. This envelope encodes the full transverse informational content available for projection.

For minimal informational complexity, the envelope may be taken to be bounded by a square region of fixed side length. This choice is not probabilistic or statistical; it simply reflects the minimal orthogonal bounding region for two independent transverse degrees of freedom.

The largest possible chord of this envelope is its diagonal.

$$ d = \ell \sqrt{2} $$

Measured in the envelope’s own natural transverse unit, the maximal dimensionless reach available to any single projection is therefore fixed.

$$ \frac{d}{\ell} = \sqrt{2} $$

This value is fixed by Euclidean geometry alone.

A.2 Projection, Read-Out, and Binary Outcomes

A physical measurement does not access the full transverse envelope. It implements a projection cut: a geometric slicing of the normal-plane envelope along a chosen orientation. The detector then reports a coarse-grained binary outcome derived from the projected value, for example via a sign or threshold rule.

The underlying Worldline-plus geometry is deterministic. Apparent randomness arises because observers do not control or access the phase and orientation of the normal-plane rotation at the moment of projection. The statistical character of outcomes reflects epistemic ignorance of this geometry, not fundamental indeterminism.

A.3 Two Independent Projection Cuts

In experiments probing entanglement, a single coherent Worldline-plus configuration gives rise to two correlated read-outs. In BEIPE this corresponds to two independent projection cuts applied to the same bounded transverse envelope.

There is no temporal ordering requirement and no probabilistic branching. The two read-outs are simultaneous geometric interrogations of one object.

Each projection cut is individually limited by the same geometric ceiling. No projection can extract a larger dimensionless contrast from the envelope than that fixed by the diagonal-to-side ratio.

Because the two read-outs are independent geometric cuts of the same envelope, the maximal attainable binary contrast across the pair is therefore capped at a fixed value.

$$ 2\sqrt{2} $$

This value does not depend on how the envelope is partitioned between the two read-outs, nor on how transverse width is apportioned locally. It depends only on two facts:

1. the envelope is two-dimensional and bounded, and
2. there are two independent projection cuts.

A.4 Robustness Under Uneven Partition

Suppose the two read-outs correspond to unequal local transverse widths. Each branch still lives in a two-dimensional normal plane and each still has a maximal diagonal reach proportional to its own transverse scale.

$$ d_i = \ell_i \sqrt{2} $$

When reach is measured in each branch’s own natural transverse unit, the maximal dimensionless reach remains invariant.

$$ \frac{d_i}{\ell_i} = \sqrt{2} $$

The combined two-cut ceiling therefore remains unchanged. The value is insensitive to uneven partition and depends only on geometry and read-out count, not on microscopic details of the split.

A.5 Relation to Experimental Statistics

In experiment, the quantity observed to saturate at the correlation ceiling is a statistic constructed from relative frequencies of binary outcomes obtained under different analyser orientations. The statistic is operationally well defined; its ontological origin is not.

BEIPE rejects the inference that this ceiling reflects intrinsic probabilistic structure, operator algebra, or Hilbert-space geometry. In BEIPE, the ceiling reflects exhaustion of representable transverse geometry.

Once the diagonal reach of a bounded two-dimensional envelope is saturated across two independent projection cuts, no stronger statistical contrast can be realised without introducing additional transverse degrees of freedom or abandoning geometric continuity.

The statistical formalism of quantum theory is therefore an effective description of repeated coarse-grained sampling under epistemic ignorance of underlying geometric phase and orientation. The bound itself is geometric.

A.6 Interpretation

The numerical ceiling observed in entanglement experiments is not imported into BEIPE from quantum formalism. It is the geometric ceiling on attainable binary contrast for two independent projection cuts of a bounded two-dimensional normal-plane envelope.

Experiments observe this ceiling as a maximum statistical likelihood because they repeatedly probe the same geometric limit without access to the underlying deterministic structure.

From the BEIPE perspective, the uncertainty principle and associated probabilistic machinery do not explain the bound. They describe how ignorance of geometry manifests statistically once the geometric limit is reached.

APPENDIX B

Electromagnetism and Weak Interaction from Worldline-Plus Geometry

This section clarifies how electromagnetic and weak phenomena arise in BEIPE as consequences of inter-Worldline-plus geometry, rather than from the exchange of fundamental gauge fields.

Electromagnetism

In BEIPE, electromagnetic behaviour arises from long-range tension alignment within the skein. This interaction reflects how the admissible continuation routes of two Worldline-plus structures align or conflict geometrically.

At the geometric level, this alignment is characterised by a tension-alignment quantity defined on pairs of Worldline-plus structures.

$$ \mathcal{A} = g^{\mu\nu} T_{\mu\alpha}(\gamma_{1})\, T_{\nu}^{\ \alpha}(\gamma_{2}) $$

Negative values of the alignment quantity correspond to attractive curvature alignment, in which joint continuation reduces geometric tension. Positive values correspond to repulsive misalignment, in which continuation routes compete and separation is favoured.

To make contact with familiar electromagnetic structure, a simple worked model introduces an effective geometric charge parameter defined in terms of Worldline-plus geometry.

$$ q = \sqrt{W_1 W_2}\,\sigma_{\text{comp}} $$

Here the geometric widths encode transverse informational extent, while the compatibility factor encodes alignment of normal-plane structure. This parameter is not fundamental; it is a derived measure of geometric compatibility.

Emergent interaction form

Using this parameter, a schematic electromagnetic interaction may be written in geometric form.

$$ F^\mu_{\ \text{EM}} \propto q\, T^{\mu\nu}(\gamma_1,\gamma_2)\, N_{\nu\rho} $$

In the far-field limit around a quasi-static configuration, curvature effects dilute geometrically with distance, yielding an inverse-square dependence.

$$ F \propto \frac{q_1 q_2}{r^2} $$

The essential point is that inverse-square behaviour is not postulated as a fundamental law. It emerges naturally from geometric dilution of curvature alignment in the Mobifold, without invoking a mediating field.

Weak Interaction

Relation to electroweak resonances

In conventional particle physics, weak processes are described via exchange of massive gauge bosons with characteristic resonance signatures. Any viable framework must reproduce these experimental features.

In BEIPE, such resonances are not interpreted as fundamental point-like mediators. They are understood as unstable geometric eigenmodes of Worldline-plus configurations within the skein. These modes are short-lived curvature excitations arising during reconfiguration of informational geometry.

The weak interaction is therefore identified with geometric instability and reconfiguration at the Worldline-plus level, rather than with force mediation.

Geometric instability criterion

A configuration is geometrically unstable when curvature accumulation increases while admissible continuation diminishes. This regime corresponds to a failure of stability under entropic ordering.

$$ \frac{d}{dA}(\text{curvature}) > 0 \qquad \text{and} \qquad \alpha(A)\ \text{decreases} $$

Such configurations cannot persist indefinitely. Continuity bias therefore enforces reconfiguration into more stable Worldline-plus geometries.

To characterise the timescale associated with a weak process, one may associate a characteristic decay rate with the geometric instability eigenvalue of the configuration.

$$ \Gamma \sim \hbar c\,\sqrt{\mathrm{Im}(\lambda_{\mathrm{decay}})} $$

Here the imaginary component of the eigenvalue encodes the tendency of the configuration to unbind and reconfigure rather than persist as a stable braid. This relation is illustrative rather than predictive; it demonstrates how a purely geometric instability parameter can map onto a physical decay timescale without introducing additional force carriers.

In this picture, strong processes correspond to large instability eigenvalues and therefore prompt reconfiguration, while weak processes correspond to extraordinarily small instability components, yielding long-lived decays.

Interpretation

Within BEIPE, the defining features of weak interactions—short-lived intermediate states, parity violation, and reconfiguration into distinct final geometries—arise from instability geometry rather than from exchange of gauge bosons.

The observed chirality of weak processes reflects the intrinsic handedness of admissible continuation in the non-orientable Mobifold, not a symmetry breaking imposed by an interaction term.

Determining the detailed spectrum of instability eigenvalues for realistic Worldline-plus configurations, and thereby the quantitative hierarchy of weak decay timescales, lies beyond the scope of this work and is deferred to future analysis.

APPENDIX C

Translation Mechanics at Entropic Stillness

Translation occurs when entropic descent stalls. In the main text, admissible continuation of a Worldline-plus is governed by a descent parameter that encodes alignment with informational ordering.

Stability requires this descent parameter to remain positive and for entropic age to increase monotonically. When admissible continuation collapses, further progression in the current orientation becomes undefined.

$$ \dot{\gamma}^{\mu}(A) = -\alpha(A)\,\nabla^{\mu} S $$

Here the descent factor encodes the degree to which admissible continuation remains available. Persistent stability requires it to remain positive.

$$ \alpha(A) > 0 $$

When this factor tends toward zero, admissible progression ceases. Because Worldline-plus structures are persistent informational entities rather than objects in motion, they cannot reverse direction, stall, or terminate. Continuity bias therefore admits only a single continuation: orientation inversion.

At entropic stillness, informational ordering collapses. All directions in ordering space become effectively degenerate, and normal-plane rotation accumulates without further descent.

$$ \nabla S \rightarrow 0 $$

The geometric condition for enforced Translation may therefore be summarised as follows.

$$ \alpha(A) \to 0, \quad \nabla S \to 0 \quad \Longrightarrow \quad \text{Translation is compulsory} $$

Translation preserves informational content. Only the global orientation of expression changes.

$$ \sigma \rightarrow -\sigma $$

Here the orientation label distinguishes the two global phases of admissible continuation. Translation maps a Worldline-plus between these phases without loss or duplication of information.

Black-hole interiors and the asymptotic future state commonly described as cosmic heat death both satisfy the condition of entropic stillness. In both regimes, admissible descent collapses and Translation is therefore unavoidable.

APPENDIX D

Additional Worked Examples

This section provides explicit illustrative examples showing how familiar physical results arise within BEIPE as geometric consequences of informational ordering. These examples are schematic and explanatory rather than predictive, and are intended to clarify interpretation rather than fix numerical parameters.

Gravitational Lensing from Entropic Grooves

In BEIPE, gravitational lensing arises from entropic grooves induced by persistent mass-carrying Worldline-plus structures. Nearby Worldline-plus structures realise admissible continuation paths ordered by the local geometry of informational ordering.

$$ \dot{x}^{\mu} = -\nabla^{\mu} S $$

This relation expresses admissible continuation as descent along the local entropic ordering rather than motion through curved spacetime.

For a compact mass at a given impact parameter, the effective deflection angle in the weak-field regime recovers the familiar form observed in gravitational lensing.

$$ \theta \approx \frac{4Gm}{c^2 r} $$

For the Sun, this yields the observed deflection angle measured during solar eclipses.

$$ \theta_\odot \approx 1.75'' $$

Identification with the Newtonian potential

In the weak-field, static limit, the geometric entropy field may be identified—up to an additive constant—with a dimensionless rescaling of the Newtonian gravitational potential.

$$ S \sim -\frac{\Phi}{c^2} + \text{const} $$

Under this identification, the entropic ordering gradient is proportional to the gradient of the Newtonian potential, and the standard lensing formula follows naturally while the entropy field itself remains dimensionless.

Gravitational lensing is therefore interpreted not as curvature of spacetime, but as curvature of the entropic landscape encoded by informational ordering.

Entropic Redshift: Numerical Illustration

Using the BEIPE redshift relation, consider a simple illustrative case in which the entropic separation between source and observer is unity.

$$ 1 + z = \exp\!\bigl(S(x_s) - S(x_o)\bigr) $$

If the difference in informational ordering is one unit, the resulting redshift follows directly.

$$ z \approx e - 1 \approx 1.718 $$

Because informational ordering is dimensionless, the exponential relation is well-defined in purely geometric units. This example illustrates scaling only and does not correspond to a specific cosmological distance.

Standard redshift relations are recovered as effective descriptions in appropriate coarse-grained regimes. The example is deliberately simple and chosen solely for clarity.

Modulation Frequency from Transverse Geometry

For a Worldline-plus structure with characteristic informational width, transverse rotation in the normal plane gives rise—under projection into an observer’s three-space—to a characteristic rate of transverse re-expression.

In flat entropic conditions, this rate scales geometrically with the inverse transverse width.

$$ F_{0} \sim \frac{c}{2\pi W} $$

This expression defines a geometric sampling scale. When calibrated against a physical clock, it may be reported as a frequency, but it does not represent fundamental temporal evolution. The underlying process is geometric re-expression within a fixed block universe.

In the presence of a local gravitational environment—understood in BEIPE as an entropic groove—it is natural to write a schematic modulation of this rate.

$$ F = \frac{c}{2\pi W}\left(1 + \frac{\Phi}{c^{2}}\right) $$

Here the potential term is dimensionless. It may be understood either as the Newtonian potential in the weak-field limit or, more generally, as a potential derived from persistent entropic grooves.

$$ S \sim -\frac{\Phi}{c^{2}} + \text{const} $$

Changes in the local entropic groove therefore induce measurable spectral shifts in the modulation rate. In BEIPE, this reflects modification of projection dynamics, not redshifting of an underlying field-quantised wave.

Both the baseline rate and its modulation are geometric characteristics of projection. Any temporal interpretation arises solely from observer-dependent clock calibration.

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