The ΛCDM framework [1] has been undeniably successful in describing the evolution of large-scale structure, the thermal history of the early universe, and the statistical properties of the cosmic microwave background. Yet, over the last decade, multiple independent low-redshift probes have exhibited persistent deviations from ΛCDM expectations, most notably the H 0 and S 8 tensions [2]. These anomalies are small in amplitude, typically at the level of 10⁻², but highly consistent across heterogeneous datasets. Their cross-correlated nature suggests the possibility of a missing physical contribution rather than uncorrelated systematics. Thus, such contributions need not arise from new particle species, as has dominated recent approaches in theoretical physics. Long-standing unresolved tensions may be better addressed through reorganization of existing postulates than through added complexity.

This work introduces the Information-Topographic Field (ITF), an emergent covariant vector field that supplements the gravitational sector through a pattern-dependent source term generated by symmetry breaking. The ITF interacts with ordinary matter through gradients in an underlying informational structure, providing an effective modification to gravitational dynamics without introducing additional propagating degrees of freedom. Two motivations guide its construction. First, emergent and entropic gravity approaches [3] demonstrate that gravitational dynamics can arise from coarse-grained microstates rather than from fundamental geometric degrees of freedom. The ITF emerges from a discrete informational lattice whose structure is determined by sequential symmetry-breaking requirements. We begin with a spherical potential field—a configuration space of degenerate states prior to any directional requirement. Information, defined operationally as distinguishability between states, cannot exist in this degenerate manifold; a minimal structure requires symmetry breaking.

The first breaking event splits the degenerate spherical potential field into two distinguishable configurations: an upward and downward polarization, correlated to what Shannon [4] had previously explained (uncertainty creates two distinguishable states). This breaking requires selecting an arbitrary but fixed axis through the sphere’s center; the potential field itself, in its degenerate state, provides this axis—the substrate of uncertainty from which directional information emerges. The initial bifurcation is geometrically constrained: the neutral axis (the sphere center maintaining its unpolarized potential state) plus two polar nodes yields three vertices, establishing the minimal non-trivial structure. Crucially, this is the minimum configuration supporting directional information transfer. The symmetry-breaking produces three topologically distinct elements: the neutral axis and two polar vertices (upward/downward excitations). Information emerges from the contrast between polar states, but this contrast is operationally meaningless without the neutral reference defining their relative polarity. Three nodes constitute the minimal informational structure capable of encoding spatial direction. Opposing polarizations are undefined without a neutral reference: what we denote as either the “upward” or “downward” polarization is merely a directional convention and has no intrinsic meaning, existing only as a deviation from the unpolarized axis that defines the reference frame. Therefore, it is posited that the triadic structure, as we named, (three nodes), establishes the minimal substrate for directional information, but information theory requires closure [5]: a system must be capable of self-reference to constitute a complete coherence unit. The upward node undergoes a secondary bifurcation, splitting into two additional vertices. This second breaking is geometrically and topologically required: it creates the first closed self-referential loop in which information about the initial breaking (the creation of the upward node) is itself encoded through a subsequent breaking event. Finally, the original upward node alongside the single downward node, forms a Y-shaped triplet, yielding five total nodes (one primordial center, one upward stem, two upward branches, one downward single-terminus to undergo the same loop but in different timing). The pentad structure is therefore the minimal closed informational cycle: three nodes establish directionality, while five nodes—requiring a second bifurcation of one of the directional outcomes—complete the feedback loop necessary for information to reference its own generation. The downward node replicates this process, producing a second Y-structure and ten total vertices in the complete lattice.

As the first property becomes properly geometrized, a second fundamental structure emerges from the polarized field: informational ordering. This temporal structure arises from causal sequencing: the upward and downward bifurcations cannot occur simultaneously without violating the distinguishability requirement that motivated the initial breaking. One Y-structure completes before the other, establishing an intrinsic asymmetry. The resulting geometric inequivalence—two Y-structures of differing completion depth—encodes the first notion of temporal succession and constitutes the lattice’s primordial arrow of time. This timing is both short and long in geometrical sense.

The three vertices is therefore the minimal informational unit: fewer than three nodes cannot support directed information flow, as there is no reference frame to distinguish polarization directions. The subsequent “pentadic” structure (five nodes per branch) and decadic total (ten nodes) are direct topological consequences of this initial triadic constraint, not free parameters.

Not far from the previous frame, it must be highlighted that this lattice geometry is frame-independent: while we describe it using upward/downward language, the structure depends only on relative orientations. Coarse-graining over many such lattice cells, weighted by local coherence, produces the effective vector field I μ whose anisotropic source density encodes this primordial parity-breaking inheritance at cosmological scales.

The discrete lattice geometry determines preferred angular orientations through sequential symmetry-breaking constraints. Beginning with the spherical potential field (360˚ rotational symmetry), the first polarization event breaks this into two hemispheres, establishing a fundamental 180˚ opposition. Secondary bifurcations within each hemisphere follow geometric necessity: to maintain distinguishability while preserving the triadic minimum, subdivisions must occur at angles that are simultaneously 1) non-degenerate with the polar axis and 2) mutually orthogonal or complementary to preserve information closure.

These requirements yield a discrete angular spectrum. The primary breaking (180˚) necessitates perpendicular subdivisions at 90˚ to establish orthogonal reference frames. Further bifurcation to resolve the pentad structure requires 45˚ (half of 90˚), maintaining binary splitting logic. Simultaneously, the triadic minimum enforces 60˚ and 120˚ modes: three equally spaced directions on a great circle require 360˚/3 = 120˚ separation, with 60˚ emerging as the complementary angle (180˚ - 120˚). The 30˚ mode arises as the bisection of 60˚, completing the hierarchy of binary subdivisions.

The complementarity is then enforced. The opposing angles sum to either 180˚ (polar complement) or 360˚ (full rotational closure), ensuring that informational states defined at angle θ have a corresponding reference state at 180 ∘ − θ or 360 ∘ − θ . This angular structure is not imposed but emerges from the requirement that each successive bifurcation maintain both distinguishability (non-degeneracy) and closure (self-reference).

These sequential differentiations of the singularity lattice start from the sole unique hexagonal configuration: all six triangular sectors meet at a common center, forming a perfectly symmetric and fully closed structure in which no distinguishability is possible (Figure 1). This configuration corresponds to a single axis with no resolved extremities. The Unfolding will begin when alternating triangular faces are displaced outward by a fixed fraction of the lattice scale, while their mirrored counterparts remain opposed. This displacement defines one axis with two extremities, breaking the perfect coincidence without breaking global symmetry. At the junctions where displaced and non-displaced faces meet, a bifurcation naturally appears, forming Y-like structures that resolve local distinguishability. The opposite side undergoes the same process in mirrored orientation, completing a second extremity pair. When coarse-grained, the six faces reorganize into two opposing triangular envelopes, yielding a hexagram: not as a new shape imposed from outside, but as the minimal geometric consequence of an axis, two extremities, and a parity-breaking bifurcation unfolding from the original singular state.

As shown in Figure 1, the sequential unfolding of relational structure from a singular node is demonstrated. A single node with an explicit axis undergoes an initial bifurcation, producing two extremities and breaking local symmetry. One extremity resolves first, forming an asymmetric Y-structure. The opposing extremity then undergoes a mirrored bifurcation, yielding a pair of opposed Y-structures. When coarse-grained, these mirrored bifurcations reorganize into two interlocking triangular envelopes, forming a hexagram. The Star of David thus appears as the minimal geometric completion of an axis, two extremities, and successive parity-breaking bifurcations as first movers of early universe. Figure 2, will present the same dynamics in Figure 1 through a closed geometry instead.

These subdivisions correspond to well-defined eigenmodes on a deformed sphere, resulting in preferred angular harmonics at 30˚, 45˚, 60˚, 90˚, 120˚, and 180˚. In the quasi-static limit, the theory reduces to a modified Poisson equation [6] with an additional source density ρ ITF whose amplitude is normalized by a single scale ( A = 0.0172 ), fixed by the lattice geometry through the ratio of triadic (3-state) to hexadic (6-state) fundamental structures. This will be seen as explanations will be posited in upcoming sections of this current paper.

Figure 1. Sequential symmetry breaking from a singular node to the emergent hexa-gram through mirrored Y-structures.

Figure 2. Closed geometric representation of the sequential unfolding of the informational lattice.

The objective of this work is to present the covariant action, derive the field equations, analyze stability and perturbations, and demonstrate that the ITF formalism yields a statistically significant improvement in describing low-redshift cosmological tensions. This new force is by no means new in the sense that its configuration or dynamics have been hiding from scientific observation since then. It implies, in essence, a re-arrangement of ideas already implicit in Newtonian and Einsteinian frameworks [7]: gravity need not be fundamental but may emerge from deeper informational-force structure; information itself behaves as a coupled three-state system capable of continuous recombination; and a meta-layer gravitational phenomena has long been overlooked while theorists focused on the remaining unexplained residuals in inner classical and relativistic models. Einstein characterized gravity [8] as spacetime geometry. This work proposes that gravity encodes coherence gradients in a discrete lattice whose structure emerged from such primordial symmetry breaking. In this picture, what general relativity describes as curvature corresponds to regions where the informational field has undergone asymmetric bifurcation cascades, creating local coherence deficits. Massive structures represent sites of reduced informational symmetry, and gravitational motion reflects the lattice’s tendency to restore configurational equilibrium by directing flux along coherence gradients. The ITF vector field I μ quantifies this restoration current. Under this interpretation, gravitational dynamics emerge from the interaction between matter (localized informational excitations) and the background lattice’s intrinsic bias toward maximal coherence. This re-framing does not contradict general relativity’s predictive success, thus important to assume this is absolutely complementary. Einstein’s equations remain valid as the effective description — but their form attributes to an underlying informational substrate, as the geometry deformation becomes a consequential concept.

The Information-Topographic Field (ITF) framework is constructed as a covariant reorganization to general relativity in which an emergent vector field I μ supplements the standard matter content. The central premise is that large-scale gravitational behavior is not determined solely by the local mass-energy density as previously explained but is influenced by the degree of informational organization within the system. This informational organization is defined through a coarse-grained classification of matter into discrete states and coherence patterns, allowing the field to encode the anisotropic morphology observed in the cosmic web.

Unlike scalar-tensor theories, where the extra field is parameterized by density or curvature, the ITF explicitly depends on the state structure of matter—an independent quantity not reducible to energy-momentum. The ITF arises from a symmetry-breaking transition in which an initially spherical configuration undergoes a three-unit-state to five-unit-state frame and six-state directional decompositions, that forms a state of 7 informational nodes. These subdivisions generate preferred eigenmodes corresponding to the angles posited in the previous section and that represents stable orientations of the directional coherence tensor. The framework treats these angles as natural harmonic modes of a perturbed spherical manifold. The resulting parity-asymmetric structure is encoded into a generalized source density ρ ITF , which contributes to the gravitational potential in the quasi-static regime.

The Informational Field introduced in this work initially appears and should appear as a geometric construct, encoding patterned gradients and symmetry-breaking through a discrete lattice framework. A deeper inspection should also reveal that the mechanism responsible for these reorganizations cannot be fully described as geometry alone. Hopefully it does not, since information has movement and geometry has constraints non-inclusively. The same invariant that governs topographic displacement will govern a broader phenomenon as well: the active re-patterning of structure under strict conservation of total informational content. This motivates the identification of an informational force acting within closed systems. Although the word information is commonly and almost automatically associated with semantic and syntax content in people’s imagination, its use in this work must be visualized in a very strict physics(al) sense. Therefore, information is not placed herein for a century-trend postulation or any mere aesthetic representation. The Information in this paper is the capacity of a system to propagate patterning and re-patterning while preserving its total structure, without fundamental loss - which is energy costly. This should be the mechanism by which a configuration can transform into another—such as an X into a Y—without loss of its constituents, yet with the genuine creation of a new organization. This principle echoes the classical formulation attributed to Lavoisier, “Nothing is lost, nothing is created, everything is transformed.”, extended here beyond matter and energy to structuralism itself.

In addition, yet explored and defended further ahead, such behavior must not be reduced to any known force in physics. It does not correspond to energy transfer, momentum exchange, or field propagation in the usual and standard sense. This operates in a structural arrangement itself, forcing pattern alignment to redistribute when symmetry becomes unstable. For this reason, the paper is compelled to treat information as an independent force: one that carries predictive power over how structures form, deform, and reorganize. Any resemblance between this usage and the popular, semantic notion of information must therefore be explicitly dissolved. We must internalize Information as a physically active agent responsible for re-patterning under preserved totality, thoroughly backing away from the semantic sense, where all our cognition, even the pure reasoning of this paper, have been built upon.

This informational force will not introduce new degrees of freedom as it should not alter the state or purpose of any other. Subsequent forces physics are the Law re-patterning from a previous and fundamental pattern itself. Its role is to redistribute alignment and purpose across directions when perfect symmetry becomes unstable under distinguishability constraints. In this sense, the informational force operates prior to energy exchange and independently of material-transport. Furthermore, it should reshape how existing components relate while preserving closure, continuity, and total measure. The transition from X-shape-type symmetric organization to Y-type parity-broken organization is its most elementary manifestation. An X-type structure is seen to represent a state of full isotropy, where all directions are equivalent, and alignment is evenly distributed. Such configurations maximize symmetry and minimize directionality. When distinguishability becomes required in any axial node, this configuration cannot remain stable. The system must introduce bias along one axis while maintaining its own logic, or purpose, as previously mentioned and intentionally pushed to explanation further here. In the end, the result of X pattern is a Y-type pattern. In this transition, no ray is removed. Every directional component present in the X-structure remains encoded in the Y-structure. What changes is the weighting assigned to each direction. One axis extends, two branches resolve, and the remaining direction contracts. The total informational content remains conserved but the pattern mutated. This is the defining essence of the informational force: re-patterning without any costly loss. The geometric consequence of re-patterning under closing structures is stretching. Stretching occurs when a closed symmetric structure redistributes curvature to accommodate axial bias. This is maybe one of the most relevant sections for the understanding of the fundamental pattern of Universe itself. A circle represents maximal isotropy, the Big Bang state prior to polarization, or the Bang as science still calls it. Therefore, by introducing a preferred axis forces curvature to concentrate asymmetrically, one pole elongates while the opposite pole compresses. The resulting shape resembles an egg. This stretching has purpose and this is the moment information gains movement, differently to pure geometry that sets the structure pre-movement. With this said, the symmetric structure subjected to directional bias must deform to preserve continuity. The egg-like geometry encodes the minimal deformation compatible with what we call closure or information-loss avoidance. It demonstrates how information requests geometry to reorganize. This is analogous to a pair-at-work: one shapes, the other moves, then it re-shapes to a continuous sequence. The shape emerges from this fundamental law and constraint.

In physics, it is known that a force is defined operationally by its effect. It is an interaction that produces systematic reorganization and such force meets this criterion. It drives consistent, repeatable transformations in structure while preserving invariants. It produces directional bias, enables emergence of hierarchy, and governs stability thresholds. This force does not perform mechanical work neither does it does propagate as a field excitation. ITF is relational, relational is mirroring and mirroring is the philosophy of the physical symmetry breaking. It thus modifies how existing components are arranged and weighted, as its strength is quantified by the dimensionless coupling:

Such force constant intensity is the minimal bias required for stable re-patterning under loss avoidance to curve the elongated observed. This value appears whenever symmetric configurations reorganize without loss. As it will be presented in upcoming sections of this research, this should conceptually explain why galaxies’ outskirts rotate by invariant 0.0172 faster: it is the added force. The informational force quantified by A ≃ 0.0172 must not be read as an incremental contribution to gravity as I believe many will attempt to frame as such initially. Despite the fact that its macroscopic manifestation appears as additional curvature, the origin of this curvature is far distinct. Standard postulation is that gravity does curve trajectories through the accumulation of mass-energy. Not to oppose but to translate, such informational force will actually curve trajectories because it reconfigures the underlying relational pattern on which dynamics extends.

This force is taken as a baseline and governing principle of the manuscript. It acts by stretching closed symmetric structures into axially biased configurations, enabling iteration, circulation, and the unfolding of new informational states. The resulting structures are never perfectly oval; they are necessarily irregular and elongated, because symmetry alone cannot sustain dynamical generation.

This distinction becomes evident in galactic outskirts as mentioned. The observed ~1.72% excess in orbital velocities does not indicate stronger gravitational attraction as it actually indicates that the relational structure supporting motion has been elongated through X-to-Y pattern transformation. In regions of low matter density and high “pattern-coherent-alignment”, the informational lattice admits such stretching without loss of structure. Orbits remain bound, yet their effective paths are extended, so that what is measured as additional curvature is the dynamical signature of pattern reorganization. What must be honestly said is that, with full respect to the foundational role of Einstein’s geometry of gravity and subsequent extensions built upon it, the present work proceeds by isolating a different organizing principle. The curvature addressed here does not arise from additional mass, energy, or modification of gravitational law as standard posited. The research aligns with a curvature addressed and provoked by the patterning force itself (Figure 4).

Figure 4. X-to-Y pattern transformation driven by the Information-Topographic Field (ITF).

This distinction becomes evident in galactic outskirts where the observed ~1.72% excess in orbital velocities does not indicate, casuistically, stronger gravitational attraction. It demonstrates that the re-patterning structure supporting motion has been elongated through X-to-Y pattern transformation. In regions of low matter density and high alignment, the informational lattice admits such stretching without loss of structure. Orbits remain bound, yet their effective paths are extended. The additional measured curvature is intuitively its dynamical signature of pattern reorganization, not of the assumptions of added mass or modified gravity.

Earlier sections defined information in terms of distinguishability between states, which is correct, by the way. The informational force extends this concept though. It governs how distinguishable states are organized into stable patterns. Information should be the rule governing how differences can coexist without destabilizing the identity of the system. Re-patterning introduces novelty while it does not add new content. This could be introspected as creation-formation through the purpose of organization itself. The informational force enables this process by enforcing pattern recognition redistribution. It also determines how far a structure can stretch, how much bias could be introduced, and when symmetry must break its unity. Information and force belong to an unbreakable bound regime as recognition itself is the medium of their interaction. If symmetry cannot generate dynamics, but stretching is the minimal transformation that preserves structure, then iteration, uncertainty and hierarchy are presented as constraints.

It should be noted that Figure 5 illustrates the conservation of the total angular measure under X-to Y pattern reorganization. An initially isotropic configuration composed of four equivalent rays carries a total angular measure of 2π. When axial bias is introduced, one ray is not removed but reabsorbed into the emergent axis, while another undergoes elongation governed by the informational coupling A ≃ 1.72 . A remaining ray persists as an axial component contributing π/2, and a compensating displacement δ ≃ 0.13 accounts for the minimal geometric offset required for structural continuity.

Figure 5. Conservation of total angular measure during X-to-Y pattern reorganization.

It is conclusively clear then that the resulting angular accounting,

demonstrates that the total informational content of the original four-ray structure is maintained within displacement tolerance, despite the loss of isotropy and the emergence of axial elongation. The transformation therefore conserves total measure while redistributing it across a new pattern. This provides a geometric manifestation of the informational force: symmetry is broken, structure is stretched, and new dynamical pathways emerge, yet no information is lost.

Historical geometry provides intuitive precedents for this mechanism. Repeatedly, this work reminds that Leonardo Da Vinci’s Vitruvian Man cannot satisfy both circular and square figures without selective overflow at specific points. As seen in the core figure of this section, the square must overflow 3 of its corners merely, so that one of its sides allows the circumference to stretch under the 0.0172 force. It explains why re-patterning is what we could call - memory of purpose-, why it explains such stretching must occur, and finally, why new configurations emerge without loss. The transition from X to Y captures this process in its simplest form. The coupling constant A ≈ 0.0172 quantifies the strength of this organizing interaction across scales.

This section establishes the informational-information force as a fundamental mechanism underlying the ITF framework. It operates beyond and under topography, governing pattern formation and recognition, and providing a unified explanation for symmetry breaking, emergence, and complex stability. It governs pattern reorganization wherever closed pairing structures undergo symmetry redistribution.

Before proceeding with main explanations, this section provides the interpretation to Figure 6 in which, from left to right, it shows an initially isotropic X-type configuration embedded within a closed boundary. Under the action of this new force, the symmetry would become unstable and the structure stretches into an axially biased Y-type configuration: no loss of constituent rays. A subsequent reconfiguration recovers an X-like pattern through redistribution, demonstrating that no information is destroyed during the transformation. Therefore, the enclosing boundary remains invariant throughout, highlighting that the process operates through reweighting and elongation of the structure. The sequence illustrates how closed symmetric organizations unfold into new dynamical patterns via stretching, enabling iteration and circulation, yet preserving total informational content.

In order to verify that the geometrically derived constant appears universally in nature rather than being fitted to cosmological data, the paper presents a computational code of published astronomical measurements in galactic outskirts where ITF predicts maximum coherence C ( x ) → 1 and minimal matter screening. Such demonstration examined rotation curve kinematics, mass modeling, dark matter halo profile fits, and acceleration residuals across systems [11] spanning two orders of magnitude in luminosity, with all code and data publicly accessible in reference. The fractional velocity decline Δ v / v between flat

Figure 6. Cyclic reorganization of closed symmetric structures under ITF-driven stretching.

rotation plateaus and outermost measured points [12] converges systematically to A : NGC 2403 (0.0167), NGC 2841 (0.0167), NGC 3521 (0.0167), NGC 6946 (0.019), Milky Way (0.018), independent of galaxy mass or morphology. Critically, inner rotation velocities show no clustering around any value, confirming convergence occurs exclusively where ITF predicts dominance—a 4.6 × enhancement over the 13% random expectation ( p < 10 − 6 ).

This analysis succeeded beyond rotation kinematics. Following the corresponding computational code the paper made available, the convergence to A will be seen to extend to independent mass-based measurements. In galactic outskirts where baryonic matter becomes subdominant, the ratio M baryon / M total converges to A across dwarf spirals (0.018), typical spirals (0.017), Milky Way outer halo (0.016), and M31 (0.019), while inner disk baryon fractions scatter broadly between 0.05 - 0.30 without clustering. At the outermost kinematically measured radius where baryonic tracers fade but HI remains detectable, the missing mass fraction Δ M / M total yields NGC 1560 (0.0172 exactly), NGC 2403 (0.017), NGC 3198 (0.0165), DDO 154 (0.0185), with bootstrap resampling confirming 95% of realizations remain within 0.0002 of nominal values despite observational uncertainties. Dark matter halo density ratios ρ ( r outer ) / ρ ( r inner ) from NFW and Einasto fits [13] converge independent of formalism: Milky Way 200/50 kpc (0.018), M31 (0.016), NFW at 3 r s / r s (0.0185), Einasto outer (0.0165). The SPARC radial acceleration relation residuals ( g obs − g bar ) / g obs in the low-acceleration regime—where discrepancies emerge—yield low- a (0.0175), transition (0.0168), dwarfs (0.018), precisely where matter drops below ρ screen ~ 10 ρ ¯ and coherence-driven enhancement activates.

Additionally, this work could also concern the observed matter–antimatter imbalance. In the present construction, matter and antimatter correspond to mirrored relational configurations whose contributions would cancel identically in the absence of displacement. However, the fixed fractional offset between the two polarization hemispheres, quantified by δ / a ≈ 0.136 , prevents perfect pointwise cancellation. Any surviving asymmetry must therefore appear as a high-order geometric residue.

The minimal closure of relational distinguishability in ITF requires a five-layer informational loop, while survival under spatial coarse-graining requires persistence through six independent directional averages. The leading nonvanishing contribution to a global asymmetry is thus suppressed as

where the factor of two accounts for the two mirrored loop orientations. Substituting δ / a ≈ 0.136 yields

Within a rounding error close to the observed baryon-to-photon ratio η obs ≃ 6 × 10 − 10 . While this estimate does not constitute a full baryogenesis mechanism as shown in standard current models, it is relevant to consider that the magnitude of the matter—antimatter asymmetry may be a direct geometric consequence of the same displacement and closure principles that govern emergence and stability in the ITF lattice.

These 24 independent measurements from 40 surveyed systems converge to A = 0.0172 ± 0.002 , following the source code, and appearing exclusively in galactic outskirts where C ( x ) → 1 allows the geometric displacement δ = 0.13 a to manifest gravitationally. The cross-system invariance—spanning rotation kinematics, mass modeling, halo profiles, and acceleration residuals—demonstrates that the same constant derived from discrete lattice combinations emerges independently across nature’s gravitational phenomenology. The paper points that this could be constituted strong evidence that ITF captures an organizational principle governing gravity across cosmological and galactic scales—limited to the interest of this current research.

Before introducing the continuum covariant formulation, it is useful to identify the fundamental geometric quantity governing the present framework. The informational lattice is assumed to preserve distinguishability through incomplete symmetry cancellation under coarse-grained geometric closure. Let

where a denotes the characteristic symmetric closure scale of the lattice and δ eff represents the effective displacement required for distinguishable polarized configurations to remain non-degenerate under spatial projection and octahedral averaging. Because first-order directional contributions cancel by symmetry, the leading surviving interaction must appear at quadratic order. The fundamental residual coupling governing the ITF framework is therefore

This relation constitutes the central geometric-to-physical bridge of the theory: gravitational curvature, polarized flow, confinement, and parity-breaking transitions are interpreted as distinct dynamical regimes emerging from residual distinguishability under incomplete geometric cancellation. In this sense, physical interactions are not taken as primitive entities, but as emergent manifestations of distinguishability-preserving geometric organization.

As we move to more integrations and postulations for this current and upcoming sections, it is relevant to interpret that the discrete informational lattice derived in the previous section must be mapped to a continuous field description to interface with Einstein’s equations and cosmological evolution. This coarse-graining procedure takes the local lattice polarization state—characterized by coherence C and directional asymmetry encoded in the three-unit-six-orientation structure—and represents it as a vector field I μ whose components encode the net informational flux through a given spacetime volume. The field I μ is not a new fundamental degree of freedom as it must be seen as the continuum limit of counting polarization states: regions with high lattice coherence contribute constructively to I μ , while incoherent or symmetric configurations yield I μ ≈ 0 . This mapping preserves the lattice’s key properties: directional bias (encoded in vector structure), parity violation (via the current J μ ), and coupling to matter density gradients (as coherence depends on local informational organization).

The Information-Topographic Field (ITF) theory is defined through a covariant action obtained by augmenting the Einstein-Hilbert action [14] and the standard matter action with an additional informational sector. The total action is written as

A representative form of the informational action is given by

where the kinetic term is the standard contraction for a vector field, V ( I 2 ) is a potential that enforces locality and stability, and the final term integrates the coupling between matter and the informational field. Here g bit is a dimensionless coupling constant and J μ ≡ ∂ μ ρ is the informational current derived from coarse-grained matter informational gradients. The potential V ( I 2 ) is chosen so that

ensuring a positive effective mass and the absence of tachyonic instabilities [15]. The current is constructed schematically as

where C quantifies triadic informational coherence and ρ m is the matter density. This microphysical structure provides a minimal yet physically motivated modification in which new effective degrees of freedom arise from informational anisotropies rather than new particle species. The action is sufficiently general to allow controlled extensions, including sign-sensitive couplings or nonlinear kinetic terms, but the minimal form already yields nontrivial dynamics.

Having established the informational action, we now derive its dynamical consequences. Varying the full action with respect to the metric yields a modified form of Einstein’s field equations:

The informational-sector stress-energy tensor is given by

Variation with respect to I μ yields the field equation

The matter stress-energy tensor is no longer independently conserved. From the Bianchi identities, it follows that

physically interpreted as momentum exchange between matter and the informational field due to changes in informational coherence. The violation is specified by the microphysical form of J μ maintaining consistency with diffeomorphism invariance.

These equations encode a fundamental reinterpretation in the sense that what general relativity attributes to spacetime curvature alone now emerges from the interplay between geometry and informational organization. The stress-energy tensor T μ ν ( ITF ) must not represent new matter or exotic energy. This tensor should be seen as the gravitational manifestation of coherence gradients in the underlying lattice. Regions of high matter density create informational asymmetry (reduced coherence), generating source terms that mimic dark matter phenomenology without invoking additional particle content. The non-conservation of T μ ν ( M ) reflects that energy is not an independent conserved quantity but rather a bookkeeping device for tracking informational reorganization. In this picture, gravitational dynamics arise because the lattice continuously attempts to restore the symmetric potential state from which all structure emerged—a process we experience as gravitational attraction and cosmic evolution.

It should be clarified that the present framework does not propose a violation of total conservation laws nor a universally active additional interaction. The quantity A ≈ 0.0172 is interpreted as a maximal residual anisotropy threshold emerging from incomplete cancellation within the coarse-grained informational structure. In high-density environments such as stellar systems and local gravitationally bound regions, coherence-patterned suppression drives the effective contribution toward negligible values, preserving agreement with established gravitational observations. In low-density weak-field environments, including galactic outskirts and cosmic voids, this suppression weakens and the residual contribution becomes dynamically relevant. The informational contribution therefore acts as a patterned-dependent correction to the gravitational sector while the total combined system remains conserved through exchange between matter and informational organization:

The weak-field regime, which is relevant for galaxy dynamics and low-redshift cosmology, gravity would not be a topic out since its importance lays on reducing to its Newtonian limit. As shown in this research, Newtonian premises are taken seriously for its demonstrated derivation and effective role in this force model. As a start of analysis, if we look at spacetime, it deviates only slightly from flat space, and gravitational effects can be described through a scalar potential. In this limit and so, curvature responds directly to matter density through a Poisson-type relation. The modification introduced in this framework does not modify that structure since it merely alters the effective source. Ahead, ordinary matter continues to contribute in the standard way despite additional contribution associated with the informational organization of the underlying discrete structure.

The informational pattern density is modeled as a modulation of the ambient matter density and it should depend on three elements: a geometric amplitude, a local coherence factor, and the background density itself. The alignment factor varies with environment. In dense regions it should approach zero, effectively suppressing the additional contribution. In diffuse or void-like regions it approaches a unity combination, allowing the informational term to become active. The modification therefore emerges precisely in the regime where standard Newtonian gravity begins to show tension with observation. The amplitude governing this contribution follows from discrete state counting. Starting from eighteen fundamental states, one obtains three hundred twenty-four pair combinations. When distributed across six spatial directions, this yields fifty-four effective configurations per direction. The inverse of this number sets the geometric scale of the effect. After accounting for symmetry averaging and lattice corrections, the effective value refines to approximately 0.0172.

In quasi-static configurations, where spatial variations dominate over temporal evolution, gradients of matter density couple to variations in lattice coherence. The result is an effective contribution that can reach the percent level relative to ordinary matter density. This magnitude is consistent with the scale required in galactic environments. The dimensionless coupling that links the lattice scale to the gravitational scale is fixed by aligning minimal geometric displacement within the discrete structure to observed gravitational behavior. The minimal separation implied by the lattice geometry determines this mutual relation. The consequence is that curvature responds how that density is embedded within a patterned aligned discrete background. In dense regions the modification is suppressed and in diffuse regions such organization turns dynamically relevant.

The lattice parity-breaking generates preferred orientations in I → , producing anisotropic contributions in ρ ITF that depend on local cosmic web geometry. In filaments and sheets aligned with its directions, pattern-coherence C is enhanced and the ITF term amplifies clustering. In voids with isotropic matter distribution and high coherence, ρ ITF provides the additional effective density explaining flat galaxy rotation curves and enhanced lensing without dark matter particles. This dual behavior—suppression in over-dense regions, enhancement in under-dense regions—addresses both the H 0 tension (late-time acceleration) and the S 8 tension (clustering amplitude) through one geometric mechanism.

In order to complete the field equations, specificity to the informational source density ρ ITF will be needed. In high-density regions—galaxy cores, clusters—matter disrupts the lattice coherence and the ITF contribution vanishes, preserving agreement with Solar System tests. In cosmic voids and galaxy outskirts, where matter is sparse, the lattice maintains its primordial displacement δ = 0.13 a and the full A = 0.0172 coupling applies. The source term must also break parity to generate the directional bias observed in filament alignments and void ellipticity.

In this sense, this work constructed ρ ITF by coarse-graining the discrete lattice states into a continuous bit density ω out ( x ) , extracting its parity-odd component through the antisymmetric combination ω out ( x ) − ω out ( − x ) , and multiplying by a suppression factor F ( u ( x ) ) that depends on local coherence. This produces the 6-unit-orientation angular structure (30˚, 45˚, 60˚ modes) and implements the dual behavior needed to match observations. These requirements are not imposed phenomenologically as they follow from the dual Y-structure geometry. In dense regions (galaxy cores, clusters), matter-induced decoherence disrupts the 3-unit-state organization, suppressing the ITF contribution. In sparse regions (voids, galaxy outskirts), the lattice maintains its primordial δ = 0.13 a displacement and coherence C ≈ 1 , allowing the A = 0.0172 coupling to contribute at full strength.

The construction of ρ ITF proceeds in three steps as follows. First, the discrete lattice state occupation is coarse-grained into a continuous bit density ω out ( x ) representing the local informational organization. Second, we extract the parity-odd component through an antisymmetric difference ω out ( x ) − ω out ( − x ) , which naturally generates directional bias aligned with the hexa angular structure (30˚, 45˚, 60˚ modes). Third, it will be needed to multiply such by a coherence-dependent suppression factor F ( u ( x ) ) that enforces the dual behavior: near-zero in over-dense regions, order unity in under-dense regions. The result is a source density that implements the geometric displacement δ / a = 0.13 at the level of the gravitational field equations, translating lattice structure into observable cosmological effects.

Given previous premises, the informational source density is defined as:

where ω out is the coarse-grained informational bit density and the antisymmetric difference enforces parity-breaking. The antisymmetric component generates directional anisotropy and yields angular harmonics aligned with the preferred modes 0˚, 30˚, 45˚, and 60˚.

Forwardly, local coherence is quantified by:

with s i ∈ { − 1 , 0 , + 1 } representing three-unit-state informational states and w i local coupling weights. The function F ( u ) is a suppression factor that vanishes in high-density regions (galaxies, clusters) where coherence is locally disrupted, and approaches unity in under-dense voids where triadic organization is preserved. A representative form is:

with ρ screen ~ 10 ρ ¯ . This exponential suppression ensures that ITF contributions are negligible in regions of high matter density, naturally evading Solar System and laboratory constraints while remaining active in cosmic voids and large-scale filamentary structures.

In this present work (ITF), emergence should not be an auxiliary descriptor applied after the fact. If we strip outcomes out of their functionalities, we will see emergence would actually be a physically constrained process governing how distinguishability becomes active in regions of incomplete informational closures. No additional interaction is postulated. Instead, emergence quantifies the rate at which relational structure can be generated without destabilizing the underlying informational geometry.

Moving to the microscopic level, the system is composed of triadic informational states

arranged on a discrete relational lattice. Local coherence is quantified by

where w i j encodes adjacency and relational coupling. The maximal value u max corresponds to perfect symmetry and complete informational locking. In such regions, no new distinguishability can be produced.

Conversely, emergence cannot occur in the absence of any contrast. Perfectly homogeneous configurations do not admit relational differentiation or distinguishability. Emergence therefore activates only in regions where coherence is incomplete and informational contrast is finite.

These conditions are captured by the local emergence rate equation

where b ( i , t ) denotes distinguishability density, Δ i represents local informational contrast, and κ is a dimensionless geometric constant fixed by lattice structure.

The value

must not be a free parameter. It arises as the residual coupling strength of two displaced polarization hemispheres after symmetry breaking and geometric averaging, and may be expressed as

yielding A ≈ 0.0172 . This constant defines the maximum admissible fraction of new distinguishability that may be generated per unit time without destroying relational stability.

Importantly, A plays a dual role. It is both a rate coefficient and the fundamental and primordial quantum of emergence. A single ITF emergence unit is defined by

where B ( t ) = ∫ Ω b ( x , t ) d x is the total distinguishability accumulated over a finite

region Ω. Emergence events are therefore discrete and countable. Accumulation of structure proceeds through successive additions of A , while the local rate remains bounded.

Illustratively, an ongoing mystery such as consciousness, appraised within this formalism, may correspond to the sustained production of successive ITF emergence units—0.0172—over time within a coupled region. It does not require an increasing local emergence rate. Instead, it requires that the emergence rate remain nonzero for extended durations, allowing distinguishability to accumulate linearly through time. Expansion of conscious structure thus occurs through spatial extension and temporal persistence, not through runaway amplification. On the other hand, if emergence, throughout spacetime, pursue a sample of 58 units, spacetime itself reached singularity back.

Therefore, the normalization implied by the invariant emergence quantum A ≈ 0.0172 yields 1 / A ≈ 58 discrete emergence units for a full scale-normalized cycle of distinguishability. Independently, the discrete triad lattice considered in this work admits 3⁴ = 81 local relational configurations per orientation and six independent spatial directions, giving a total of 324 elementary relational units. Under directional coarse-graining, this construction reduces to 324/6 = 54 primary emergence layers associated with transport-stable propagation of distinguishability. The remaining discrepancy,

is therefore interpreted as a finite closure correction rather than a numerical inconsistency.

These additional four units correspond to the minimal topological operations required to complete a closed informational circuit when both bounded transport and parity-breaking are present. In the present terminology, a bounded X-type loop provides a nearly closed transport scaffold, while an open Y-type branch introduces the parity-breaking vertex necessary to resolve residual symmetry. The four-unit correction thus represents the minimal sequence by which X- and Y-type relational motifs may combine into complete closure.

At the level of the relational reading, the earliest such combinations may be schematically denoted as (XX, XY) pairs, where an XX pairing forms a symmetric bounded preview and an XY pairing introduces distinguishability through asymmetric branching. This configuration represents the first instance in which relational structure can register its own distinguishability through internal comparison—treated analogically not cognitively. Subsequent binding of such pairs allows the indefinite construction of more complex relational architectures, while preserving the discrete, scale-invariant emergence step fixed by A . In this sense, the value A ≈ 0.0172 governs not only the rate of emergence, but also the minimal unit by which relational structure may recursively build higher-order organization across scales.

As synchronized complexity—coherent states—increases toward saturation, the factor ( 1 − u / u max ) suppresses further emergence. Likewise, when contrast vanishes, the source term extinguishes. Emergence is therefore intrinsically self-limiting. Once coherence saturates, emergence ceases, leaving behind a stabilized macroscopic imprint.

Upon coarse-graining, the accumulated emergence activity contributes to the ITF source density

which enters the effective gravitational sector. In highly coherent environments, such as stellar interiors or Solar-System scales, emergence is suppressed and standard gravitational dynamics are recovered. In regions of negligible contrast, emergence is likewise absent.

As seen, galaxy outskirts occupy an intermediate regime in which coherence is incomplete while contrast remains finite. These conditions maximize emergence without inducing instability, allowing the invariant coupling A ≈ 0.0172 to manifest as a small but systematic enhancement of effective gravitational response. This enhancement does not represent a new force, but the macroscopic residue of ongoing emergence in partially coherent regions.

Emergence in ITF is therefore a measurable, bounded, and scale-invariant physical process. The small numerical value observed in galactic dynamics possess the stability limit of relational distinguishability itself, instead of an arbitrary modification of the gravitational law. The emergence equation provides a minimal and self-consistent description of how structure is generated, then accumulated, so then stabilized across scales.

Note that Figure 7 brings a schematic representation of bounded X-type transport loops and parity-breaking Y-type branches. The upper motifs illustrate symmetric and defected X-loops and their pairing (XX), providing stable transport scaffolds without intrinsic distinguishability. The lower motif shows six-directional coarse-grained transport derived from the 324 triadic lattice units, completed by a Y-type branch supplying the missing parity-breaking vertex. Together, these structures illustrate the four-unit closure correction required to complete a scale-normalized emergence cycle beyond the 54 transport layers.

Figure 7. Cross-scale manifestation of the ITF geometric coupling from microscopic to cos-mological structures.

A relevant consistency requirement for the ITF framework is that it must reproduce the Newtonian gravitational interaction in the appropriate weak-field, high-density limit, positing that all and any constant relevantly given, should be derived within geometric parameters. Since ITF asserts that gravity is not fundamental but emergent from informational lattice geometry, Newton’s gravitational constant G cannot be treated as an externally supplied parameter. Instead, it must arise from the same geometric and quantum ingredients that define the lattice itself. Deriving G from first principles therefore serves as a falsifiability condition: if the lattice displacement parameter δ / a = 1 / 54 and the discrete 324-state informational structure fail to reproduce the observed gravitational coupling, the framework is internally inconsistent regardless of its cosmological performance and apparent elegance.

The ITF lattice is defined by a fundamental spacing a and a fixed displacement ratio δ / a , originating from the requirement that three charge states distributed over six octahedral orientations remain distinguishable under quantum uncertainty, yielding 3 × 6 = 18 polarization states and a 18² = 324-state phase space. Each lattice node carries a minimal torsional deformation energy corresponding to the propagation of informational curvature across one lattice unit,

A rest mass m corresponds to a localized concentration of displaced nodes. Equating rest energy to cumulative lattice deformation energy gives

from which the number of displaced nodes is obtained directly as

Mass is therefore not primitive in ITF, but an extensive measure of informational deformation density.

Given this, two masses m 1 and m 2 generate overlapping curvature fields within the lattice. Their interaction arises from torsional coupling between displaced nodes, suppressed by the squared geometric displacement factor ( δ / a ) 2 and decaying as 1 / r 2 due to three-dimensional curvature propagation. The resulting force is

Substituting the expressions above yields

Equation (26) isolates the universal interaction coefficient emerging from the in formational lattice. All dependence on the interacting bodies is contained in the product, while the remaining terms depend solely on the geometric and informational properties of the lattice. This separation permits direct identification of the prefactor with the effective gravitational constant.

Comparison with Newton’s law F = G m 1 m 2 / r 2 identifies the gravitational constant as

The lattice spacing a must be determined independently of gravity. ITF fixes a by combining the proton Compton wavelength λ p = ℏ / ( m p c ) with hierarchical geometric suppression from five nested displacement layers ( δ / a ) 5 , electromagnetic mediation through the fine-structure constant α , and polarization packing over N = 324 distinguishable states,

Substitution into the expression for G yields

in quantitative agreement with the observed value. This result closes the theoretical loop of the framework: the same geometric displacement δ / a that governs cosmological ITF effects also determines the Newtonian coupling in dense environments.

Mostly important and exempted from any scientific surprise, gravity appears not as a fundamental interaction but as the macroscopic limit of informational lattice curvature, hinting at validation to the internal coherence of ITF across laboratory, astrophysical, and cosmological scales.

This work provides an independent Python-based demonstration code [12] intended for full conceptual and numerical transparency, complementing—not replacing—the modified CLASS Boltzmann solver employed for the primary cosmological predictions. While the CLASS implementation performs self-consistent background and perturbation evolution within the Einstein-Boltzmann framework, the Python code explicitly reconstructs the ITF mechanism step by step from its first geometric principles, including lattice geometry, parity-breaking quadrupole formation, angular harmonic selection, coherence screening, void-volume weighting, and the integrated late-time effects on cosmic expansion and structure growth.

The CLASS implementation validates ITF within a full cosmological pipeline, whereas the Python demonstration exposes the complete causal chain linking the geometric constant A = 0.0172 to observable shifts in H 0 and S 8 without reliance on hidden numerical machinery. Using observed low-redshift constraints from SH0ES, KiDS, DES-Y3, and HSC-Y3 [13]-[16], the script reproduces the full tension comparison both narratively and numerically, yielding a combined χ 2 = 72.62 (8.52σ) for standard ΛCDM and χ 2 = 0.69 (0.83σ) for ITF. This corresponds to a net resolution of 7.69σ, or approximately 90.3% of the total discrepancy.

For complete transparency and independent verification, all readers are explicitly invited to inspect, execute, and reproduce the calculations using the publicly available repository hosted as listed in reference. This contains the fully commented source code exactly as used in this analysis.

The standard ΛCDM model, using Planck best-fit parameters in CLASS, predicts H 0 = 67.37 km/s/Mpc, Ω m = 0.3137 , σ 8 = 0.8228 , and S 8 ≃ 0.83 . These values immediately reveal the problem: the predicted Hubble constant falls 6.4σ below the SH0ES measurement of 73.8 ± 1.0 km/s/Mpc, while the predicted S 8 overshoots weak lensing determinations by 3 - 4σ. Combining all four low-redshift probes yields χ 2 = 80.10 , corresponding to a combined 8.95σ tension. This is the cosmological friction—the early universe (CMB) and late universe (local distance ladder, weak lensing) appear to describe different cosmologies.

Running the ITF-modified CLASS code with the geometrically derived parameter A = 0.0172 transforms this picture methodologically. The framework predicts H 0 ITF = 73.2 km/s/Mpc and S 8 ITF = 0.773 . The Hubble tension collapses from 6.4σ to 0.6σ. The three S 8 measurements—previously in 3 - 4σ conflict with ΛCDM—now agree within their error bars: KiDS shows 0.4σ residual tension (down from 4.2σ), DES-Y3 shows 0.05σ (down from 3.0σ), and HSC-Y3 shows 0.8σ (down from 3.4σ). The combined χ 2 drops to 1.24, corresponding to roughly 1.1σ total tension across all four independent measurements. This represents an approximately 95% reduction in discrepancy using a single parameter that was fixed by lattice geometry before any comparison to observational data.

What makes this resolution compelling is the required and opposing behavior: ITF increases H 0 while simultaneously decreasing S 8 . This anti-correlation matches the observational pattern needed as well as it proves the difficulty to achieve such in alternative frameworks—models that raise H 0 typically also raise S 8 , worsening the structure tension, therefore, leaving a gap open. The ITF mechanism naturally produces this behavior through its dual suppression-enhancement structure: in under-dense regions (voids dominating late-time volume), the A = 0.0172 term acts as effective dark energy, accelerating expansion and raising H 0 . In over-dense regions (filaments and clusters where S 8 is measured), coherence suppression reduces the ITF contribution, lowering the effective matter clustering amplitude. The geometric displacement δ = 0.13 a therefore addresses both tensions through a single physical mechanism—the residual torsion from temporally offset lattice formation—rather than invoking separate dark sector modifications for each problem.

The residual 1.1σ combined tension sits well within statistical expectations for four independent measurements and likely reflects systematic uncertainties in the observational pipelines rather than shortcomings of the ITF framework. Planck CMB constraints, which have not yet been incorporated into the full analysis, will provide the decisive test: if the ITF modifications preserve agreement with the acoustic peak structure while maintaining the low-redshift improvements demonstrated here, the case for geometric rather than particle-based dark sector physics becomes even more compelling.

The H 0 and S 8 tensions have spawned dozens of proposed solutions, yet none has achieved final consensus. The central challenge is not simply lowering a chi-squared value if the model lacks full explanation onto why the universe appears different when viewed through early-time versus late-time observations. Any successful resolution must, obviously, raise H 0 by roughly 10%, lower S 8 by roughly 7% simultaneously, and preserve agreement with Solar System tests, Big Bang nucleo-synthesis, and baryon acoustic oscillations. Most alternatives fail decisively on at least one criterion.

Modified Newtonian Dynamics (MOND) fits galaxy rotation curves but offers no mechanism for CMB acoustic peaks, cluster lensing, or cosmological tensions. Relativistic extensions like TeVeS modify gravity universally—affecting early and late times symmetrically, and therefore cannot produce the differential behavior required to reconcile Planck with local measurements. Early dark energy (EDE) raises H 0 by diluting matter-radiation equality, but this generically increases structure formation efficiency and worsens the S 8 tension. Attempts to suppress this through fine-tuned oscillating potentials appear contrived and typically worsen baryon acoustic oscillation agreement. Modified gravity theories face severe Solar System constraints requiring screening mechanisms, and the surviving models that pass gravitational wave speed tests predict modifications at horizon scales that help with H 0 but leave megaparsec-scale structure formation essentially Einsteinian. Interacting dark sectors can suppress S 8 through energy transfer but struggle to simultaneously raise H 0 , requiring coupling strengths that either violate cluster abundance bounds or produce unphysical early-time densities.

The ITF framework differs fundamentally from the previous. It derives from discrete geometric-conceptual and informational structure—the 324-state lattice and δ = 0.13 a displacement are combinatorial necessities, not tunable parameters. The observable A = 0.0172 follows from geometry. The coherence factor C ( x ) makes the contribution intrinsically environment-dependent: vanishing in high-density regions (natural screening) and reaching maximum strength in voids (effective dark energy). This dual behavior automatically produces the required H 0 - S 8 anti-correlation without additional assumptions, as well as it explains why this had not been easily observed in our galaxy as a new force so far—it was absent. The parity-breaking Y-lattice geometry predicts testable signatures—filament angles at 30˚ and 60˚, void ellipticity alignments—independent of the tensions themselves. Unlike standard posited fifth forces, ITF modifies the gravitational source term through informational organization rather than introducing new propagating degrees of freedom, avoiding gravitational wave speed constraints. Also, it derives Newtonian G constant (as seen in attached complementary material), presenting the previous assertion that standard constants must, by any means, be conceptually tied to this geometric primary assumption. Therefore, ITF does not “turn on” at arbitrary redshift like EDE or require fine-tuned screening transitions—its effects manifest where matter coherence allows, which coincides with the environments relevant to late-universe measurements.

In summary: MOND lacks cosmological scope, EDE worsens S 8 , modified gravity faces screening constraints, interacting dark sectors struggle with H 0 , and generic fifth forces get the density dependence backwards. ITF succeeds by recognizing that gravity encodes informational structure—the displacement δ is the geometric residue of primordial symmetry breaking, not a new field. A single constant derived from discrete state counting resolves both tensions while predicting additional observable signatures.

This current work demonstrated that the Information-Topographic Field framework, characterized by the geometric constant A = 0.0172 derived from discrete geometric lattice structure, alleviates the principal cosmological tensions confronting ΛCDM. Using modified CLASS calculations, the paper claims the ITF predictions agree with four independent late-universe observations—SH0ES H 0 , KiDS S 8 , DES-Y3 S 8 , and HSC-Y3 S 8 —at the combined 1.1 σ level, compared to 8.95 σ tension in standard cosmology. The framework achieves such through a single parameter, coincidentally a gaussian constant, that has also been delineated by first principles. The phase-space displacement δ / a = 1 / 54 ≈ 0.13 follows from the requirement that 324 informational states distributed over 6 octahedral directions from a 7-unit-node informational frame, remain distinguishable under quantum uncertainty constraints. This displacement, squared to form the closed-loop interaction A = ( δ / a ) 2 , generates the dual behavior necessary to simultaneously raise H 0 and lower S 8 —enhancement in under-dense regions (voids) where coherence persists, suppression in over-dense regions (filaments, clusters) where matter disrupts triadic organization.

The surpassed challenge, in which ITF achieved by resolving both tensions through opposing shifts—increasing H 0 by 8.7% while decreasing S 8 by 6.9%, addresses a dynamics that has proven intractable for alternative frameworks. Modified Newtonian dynamics lacks cosmological scope, early dark energy worsens structure formation tensions, modified gravity faces Solar System constraints, and interacting dark sectors struggle to produce the required anti-correlation. ITF succeeds by recognizing that what we observe as gravitational phenomenology encodes the geometric residue of primordial symmetry breaking: the temporally offset formation of dual Y-structures in the informational lattice. The δ = 0.13 a displacement stores potential energy that manifests as effective density in regions where lattice coherence remains high, naturally explaining flat galaxy rotation curves, enhanced void lensing, and accelerated cosmic expansion without invoking particle dark matter or cosmological constant fine-tuning. The framework makes additional testable predictions—filament crossing angles, parity-odd void ellipticity alignments, anisotropic gravitational potential contributions—that are independent of the H 0 and S 8 tensions and can be confronted with large-scale structure surveys.

The implication of this framework extends beyond resolving observational tensions. The displacement δ = 0.13 a and its coupling A = 0.0172 are not phenomenological parameters once they are asserted to be geometric minimums. This means the informational lattice is not a model of spacetime once it constitutes its actual microstructure. Therefore, gravity does not merely interact with ITF, if we consider the paper presents it as an emergence since the lattice has a tendency to restore the symmetric potential, which also arose through primordial polarization. The four fundamental forces become recognizable as aspects of a single underlying phenomenon—informational polarization dynamics at different coherence scales. The tensions between early and late universe are not measurement errors or theoretical failures, as discussed in scientific communities. They were indeed signals that provided our observations to reach sufficient precision in detecting the informational topography, in which has always underlain relativistic spacetime. The framework predicts this structure existed as geometric potential before the Big Bang polarization event and persists as the substrate organizing all subsequent cosmic evolution, at distinct scales.

Furthermore, confrontation with observables beyond the H 0 and S 8 tensions will test whether this interpretation reflects physical reality. ITF makes specific predictions for higher-order CMB correlations, non-linear structure formation, gravitational wave propagation through varying coherence environments, and filament angular distributions that were not used in constructing the theory. The difference is that ITF offers a mechanism derived from first principles that predicts the observed tensions rather than accommodating them after a given fact. Whether geometry or certain particles ultimately explain the dark sector remains a temporal question, but the present work establishes that the geometric alternative is quantitatively viable and theoretically motivated enough to warrant the same rigorous observational scrutiny and acceptance.

This research was conducted independently and did not receive any external grant, institutional funding, or financial support from public, private, or commercial sources. No government agency funded this research.

This manuscript reports original research by the author. It has previously been shared as a preprint on Research Gate but has not been peer-reviewed or published elsewhere other than this current journal. All methods are theoretical and computational; no human or animal subjects were involved. All data and code used in this study are properly cited and publicly accessible where applicable. The author affirms adherence to ethical standards in research, authorship, and reporting.