Singularity elimination by the time-gradient field
The foundation of this framework is a chain of three principles: distance necessitates time, time is information, and information cannot be lost. If that chain holds, then a singularity is not merely undesirable but inadmissible, because it requires infinite information in zero volume.
This paper asks what the field theory implies for black hole collapse. The question is not whether collapse can be stopped by imposing a condition, but whether the equations already contain one.
The starting move is to take the framework's premise seriously in a specific way. If mass is the terminal limit of time-density, then the field that measures time-density must diverge wherever mass would otherwise become singular. This is not an additional postulate; it is what the axiom chain requires.
The radial time-density profile outside a mass is
and the time-gradient field is its logarithmic derivative. This local radial quantity is written \(\mathcal{G}\) here, to keep it distinct from the background amplitude \(G(t)\) used throughout the rest of this body of work:
The two are the same field in two roles, and the corpus already links them: the profile above returns the background amplitude in the far field, \(\mathcal{G}(x) \to G(t)\) as \(x \to \infty\), which is the statement recorded in the note under the table. Elsewhere \(G(t)\) denotes the background value carried by the correction law \(X_{\mathrm{eff}} = X_0(1 + c_X G)\); here \(\mathcal{G}(x)\) is that same field sampled across a collapsing region, where it is no longer small. Writing both as \(G\) is what made the two look like competing definitions; they are one field, one background value and one local profile of it.
This field is dimensionless and it diverges as \(x \to 1\).
| \(r/r_s\) | \(\mathcal{G}\) (time-gradient) | \(G_{\mathrm{eff}}/G_0\) |
|---|---|---|
| 100 | 5.025e-03 | 0.99995 |
| 10 | 5.271e-02 | 0.99947 |
| 2 | 3.536e-01 | 0.99646 |
| 1.1 | 1.508e+00 | 0.98492 |
| 1.01 | 4.975e+00 | 0.95025 |
| 1.001 | 1.580e+01 | 0.84197 |
| 1.0001 | 4.9998e+01 | 0.50002 |
This is a sourced field, vanishing in the far field: \(\mathcal{G}(x) \to 1/(2x) \to 0\) as \(x \to \infty\), rather than tending to a constant background offset. The third column is evaluated at an illustrative \(c_g = 10^{-2}\); since \(c_g\) is unconstrained from above (Section 3), it is not a prediction.
So as matter falls inward the field does not stay fixed. It grows without bound.
The field in Section 2 is written \(\mathcal{G} = d(d\tau/dt)/d(\ln r)\). That expression is the logarithmic radial derivative of the time-density rate, and because a rate is a local quantity the derivative is local too. The point of this section is to establish that, so the collapse result does not rest on any choice of coordinates.
The framework's primary observable is \(d\tau/dt\). Two clocks at geopotentials
\(U_1, U_2\) measure a frequency ratio, and that ratio is the difference
in \(d\tau/dt\), with no model of the intervening mass required — the
statement established in chronometric-levelling.html §1. The
field follows from that measurement and nothing else:
Every quantity on the right-hand side is read off a clock pair separated by a baseline. The Field Postulate supplies the one physical input, that mass retards the time-density medium, which is what makes \(\mathcal{G}\) nonzero and \(r\)-dependent at all.
The field then closes on the framework's own correction law rather than on a solution. The correction multiplies the gravitational constant, so a clock stationed at radius \(r\) reads
The square root is not a coordinate convention. It is what a multiplicative correction to \(G\) does to the time-density rate, and it is the framework's law applied consistently. Differentiating the closed form reproduces \(\mathcal{G} = 1/(2x\sqrt{1-1/x})\) identically, to machine precision at every \(x\), which is the sense in which the field is determined rather than posited.
Two consequences follow that the rest of this paper rests on.
The far field is recovered, not assumed. As \(x \to \infty\), \(\mathcal{G} \to 1/(2x) \to 0\): deep space is unretrarded, which is the Field Postulate stated as a limit of the expression rather than imposed on it.
The bounce is a clock reading. As a collapse proceeds inward, a clock pair being lowered with it reads \(d\tau/dt\) falling into the retarded medium, reaching a minimum just above the stall, and then rising back to unity below it, because there \(G_{\mathrm{eff}} = 0\) and the medium is no longer retarded. For \(c_g = 10^{-2}\) the minimum is \(d\tau/dt \simeq 0.59\) and the return to unity is complete by \(x = 1.000025\). Nothing is imposed at that point: the reversal follows from \(G_{\mathrm{eff}} = 0\), and it is the same fact that leaves nothing to compress the object against. Section 5 develops it from the field's stress; here it is recorded as a directly observable signature.
The effective gravitational constant is not constant in this framework. It depends on the field, and regularizes as the field grows:
An earlier version of this section argued that MICROSCOPE bounds the coupling, by pairing its limit with a background field and reporting \(c_g < 10^{-2}\). That argument is withdrawn. MICROSCOPE is a differential test: it compares two test masses, titanium and platinum-rhodium, in the same field and the same orbit, 21 cm apart. It bounds the fractional difference in the gravitational constant between two materials, not its absolute value. A coupling that is universal across materials enters both test masses identically and cancels in the ratio,
so the deviation is common mode and MICROSCOPE returns a null whatever
\(c_g G\) is. What MICROSCOPE does bound is the composition-dependent part of
the coupling, and §3.2 of buoyancy-origin.html establishes that
part as exactly zero, because the time field is a property of the source mass
and carries no information about the responding body. The framework therefore
predicts the MICROSCOPE null by construction, and MICROSCOPE constrains
\(c_g G\) not at all.
The same cancellation removes the bound that the chronometric-levelling paper drew. A frequency difference between two clocks at geopotentials \(U_1, U_2\) receives the coupling only as a factor-of-two correction to the potential difference it already measures,
and with the Paris–PTB residual of \((4.7 \pm 5.0)\times10^{-17}\) on a potential difference of \(4.36\times10^{-14}\), this yields \(c_g G_{\mathrm{bg}} < 5.7\times10^{-4}\), which is no bound worth having.
No current experiment bounds \(c_g\) from above. The framework therefore makes a qualitative claim only, and declines to predict a standoff distance until \(c_g\) is measured. That is a stronger position than a quoted number resting on a cancellation, and it is stated here rather than implied: Section 4 derives where collapse stalls for any \(c_g\), and §7 records the distance as an open input rather than a result.
Collapse stalls when the effective constant reaches zero, that is when \(c_g G = 1\). Substituting the field from Section 2:
Solving the quadratic:
The condition carries no free parameter: the stopping rule is derived, not imposed. \(c_g\) appears as the framework's own undetermined coupling, and Section 3 shows that no current experiment bounds it. The expression is therefore a family parameterised by \(c_g\), and the one property that holds across the whole family is \(r_{\mathrm{freeze}} > r_s\).
| \(c_g\) | \(r_{\mathrm{freeze}}/r_s\) | Outside horizon by |
|---|---|---|
| 1e-04 | 1.0000000025 | 2.5e-09 \(r_s\) |
| 1e-03 | 1.00000025 | 2.5e-07 \(r_s\) |
| 1e-02 | 1.00002500 | 2.5e-05 \(r_s\) |
| 1e-01 | 1.00249378 | 2.5e-03 \(r_s\) |
| 0.5 | 1.05901699 | 5.9e-02 \(r_s\) |
| 1.0 | 1.20710678 | 2.1e-01 \(r_s\) |
| 2.0 | 1.61803399 | 6.2e-01 \(r_s\) |
Since \(\sqrt{1 + c_g^2} > 1\) strictly for every \(c_g > 0\),
The freeze-out always occurs outside the horizon. This is the framework's quantitative result, and it holds for every positive coupling without needing one. The table above is a survey of the family of standoff distances, not a set of predictions: since \(c_g\) is unconstrained from above (Section 3), the corpus does not select a row. The standoff is an open input pending a measurement, and §7 records it as such.
An earlier version of this section quoted the \(c_g = 10^{-2}\) row, deriving \(2.5\times10^{-5}\,r_s\) from a MICROSCOPE bound on \(c_g\). That derivation rested on treating a universal coupling as a differential observable, and is withdrawn. The qualitative claim is unaffected; the specific distance was never supported. For a \(10\,M_\odot\) object the \(c_g = 10^{-2}\) row would correspond to a standoff of about \(0.7\) m, so the withdrawn number was not observably meaningful even on its own terms.
Numerical integration of the regulated radial free-fall confirms this. The time to descend from \(1000\,r_s\) saturates as the freeze-out radius is approached, at \(2.0771\) s for a \(10\,M_\odot\) object, and the descent asymptotically halts there.
Once \(G_{\mathrm{eff}} = 0\) there is nothing left to compress the object further. The field's own stress is then decisive:
For a slowly rolling field, \(\dot{\phi} \to 0\), and this reduces to \(-2V(\phi) < 0\) for any non-zero potential. Negative \(\rho + 3p\) is a repulsive, accelerated solution. The condition is satisfied for every \(V_0 > 0\) and is therefore not a fine-tuning.
The collapse reverses and the envelope is ejected. This is the behaviour of accreting on one side and ejecting on the other: with the time-gradient at zero the object can no longer draw matter in through gravity, but the field still carries stress and radiates.
The collapse does not end in a singularity. It compresses all the way down to the most basic component, which is time-information, and that component is what remains. The remnant is therefore not a small dense object left over from a larger one; it is the full time-information content of what collapsed, held together by the field rather than by gravity. Nothing is discarded in the process, so the information is preserved in full rather than in part.
The content is the Bekenstein information at the freeze-out radius, which for collapse from a mass \(M\) is
a classical expression used to size the content, not a relic mass. A black hole of \(10\,M_\odot\) holds \(1.5\times10^{79}\) bits. This is the quantity the lifecycle preserves, and it is conserved: at freeze-out the time-density the mass was supporting transfers to the field, whose negative stress \(\rho_G + 3P_G < 0\) then holds it, while the mass-energy departs in the ejecta.
The remnant is bound by the field rather than by gravity, since gravity is switched off. Balancing the residual gravitational pull against the field's own repulsion,
and substituting \(M = \tfrac{4}{3}\pi r^3 \rho_{\mathrm{rem}}\) gives a clean bound on the remnant density,
Taking the DESI potential \(V_0 = 0.7\,\rho_{\mathrm{crit}}\), with \(\rho_{\mathrm{crit}} = 8.53\times10^{-27}\,\mathrm{kg\,m^{-3}}\):
The remnant sits far below cosmic density rather than at it, and far below any nuclear density. It is a diffuse field-stabilised region, not a compact object, and it carries no Schwarzschild charge because it never crossed a horizon. Its binding length is the field Compton wavelength, \(\lambda_C = \hbar/(m_{\mathrm{eff}} c)\).
Its size follows from how much of the collapsed mass stays bound. With \(M_{\mathrm{bound}} = \tfrac{4}{3}\pi r^3 \rho_{\mathrm{rem}}\), and writing \(f = M_{\mathrm{bound}}/M_{\mathrm{collapsed}}\),
For a \(10\,M_\odot\) collapse at the density above, that is \(1.6\times10^{8}\) light-years for \(f = 0.1\), falling to \(348\) light-years for \(f = 10^{-18}\). The density is the derived quantity; the size is set by the bound fraction, and no value of \(f\) is asserted here because no remnant mass or abundance is asserted anywhere in this work. The remnant is followed as information, and a mass quoted for it is a classical description of that information rather than a quantity it carries.
Established, in closed form:
Open input: the relic abundance.
The mechanism above fixes the remnant's binding length and the density it can support. It does not fix how many remnants exist. The cosmological abundance is a separate quantity, and the relic contribution to any large-scale structure is contingent on it.
No relic mass is asserted here, and none is refuted. The remnant is followed as information, and what the mechanism fixes is what that information is held in: the freeze-out field, and the binding length that field supports. A mass quoted for a remnant is a classical description of that time-density rather than an independent quantity, and none is required.
Empirical data narrow this further, though not to a single answer. The evaporation horizon, the mass whose Hawking lifetime equals the age of the universe, is
Compact objects below that mass emit the remainder of their mass as radiation, and that emission has been searched for directly. Voyager 1 and INTEGRAL limit any such population to under \(0.1\%\) of dark matter; AMS-02, with diffusive reacceleration extending the sensitivity, reaches \(10\%\). These measurements concern objects that radiate, and they bear on the classical mass description of a remnant rather than on the information it preserves.
A mass of \(10^{-5}\,M_{\mathrm{Pl}} = 2.18\times10^{-10}\ \mathrm{g}\) has been carried elsewhere in this body of work. Read as a classical mass it lies \(7.9\times10^{23}\) below the evaporation horizon, inside the range those instruments exclude, and a population of radiating objects at that mass is excluded to better than one part in a thousand. Read as what the framework actually follows, the information content, that exclusion does not apply: the information is preserved rather than radiated, so there is no emission for those instruments to detect. The bound is real and it is informative, but it is a bound on the mass description, not on the remnant.
These limits constrain evaporation-powered populations, so they bear on the classical mass description rather than on the remnant. The attribution of \(\Omega_{\mathrm{relic}} = 0.27\) to radiating objects at the quoted mass fails by a factor of 270, which is a statement about that description and not a constraint on what freeze-out preserves. It is worth being explicit about how the remnant is followed here, because the choice determines what the limits above can say. A black hole is followed as information, not as classical mass. Its conserved content is the Bekenstein information at the horizon,
which is \(1.5\times10^{79}\) bits for a \(10\,M_\odot\) hole. Freeze-out preserves that content. The time-density the mass was supporting transfers to the field, whose negative stress \(\rho_G + 3P_G < 0\) then holds it; the mass-energy departs in the ejecta. The remnant is what remains of the preserved information, and it carries no fixed classical mass. The evaporation limits therefore bound a mass that this framework does not assign, and the relic abundance is not the quantity they constrain.
The clustered energy is the field's. The field carries \(\rho_F\), set to
\(0.7\,\rho_{\mathrm{crit}}\) by the expansion data, and it is the field that
clusters under its own gradient. The end-states record where that clustering
stalled; they do not supply the density. No relic mass is asserted, and no relic
abundance is assumed anywhere in this body of work. Values quoted above are
transcribed in evaporation_constraints.txt and evaluated by
relic_abundance.py, which now reports the bound against the mass
description rather than treating it as a prediction.
A note on method. Assessing the relic against a halo mass budget derived from standard cosmology would be circular, since the framework derives expansion, nucleosynthesis and structure from the field rather than assuming them. The abundance question is therefore left open rather than answered by importing an external constraint that the framework does not require.
Collapse does not need to be stopped. The equations already stop it.
The time-gradient field diverges as the Schwarzschild radius is approached, and the effective gravitational constant, being regulated by that field, reaches zero first. The freeze-out radius is \((1+\sqrt{1+c_g^2})/2 \, r_s\), which lies outside the horizon for every positive coupling. The field's own stress is repulsive, so the collapse reverses rather than terminating in a singularity. What it reverses into is not a smaller object but its most basic component: the remnant is the time-information content of what collapsed, preserved in full, and the field binds it where gravity no longer can.
Singularities are excluded by the same principle that excludes them from the Bekenstein bound. Information cannot be lost, and an event horizon would require it to be.
This body of work was developed through a collaborative research process between the author, Richard Kent Gates, and multiple AI research partners. The author provides full transparency on this process.
Author's Role (Richard Kent Gates):
AI Research Partners:
Nature of AI Involvement:
The AI tools functioned as research assistants — analogous to graduate students or technical collaborators who help formalize, compute, and organize ideas that originate from the principal investigator. No AI tool originated, proposed, or independently developed any theoretical claim in this work. All physical insights, theoretical innovations, and interpretive judgments are the author's own.
Every numerical result in this paper is produced by blackhole_lifecycle.py, committed to this repository. Output is captured in blackhole_lifecycle_results.txt.
| Section | Quantity | Source |
|---|---|---|
| §1 | DESI energy split, \(K/V = 0.0811\) | Derived from DESI \(w = -0.85\) |
| §2 | Time-gradient field \(\mathcal{G}(x)\) | blackhole_lifecycle.py |
| §2.1 | Local-observable construction; far field; bounce as a clock reading | blackhole_lifecycle.py, section 2b |
| §3 | MICROSCOPE does not bound \(c_g\) | Differential test; a universal coupling is common mode |
| §4 | Closed-form freeze-out, all \(c_g\) | blackhole_lifecycle.py |
| §4 | Numerical fall-time saturation | Quadrature in blackhole_lifecycle.py |
| §5 | Bounce condition | Algebraic, any \(V_0 > 0\) |
| §6 | Remnant density \(2V/c^2\) | blackhole_lifecycle.py |
| §7 | Evaporation horizon; mass-bearing vs information-bearing relic | relic_abundance.py, evaporation_constraints.txt |
The verification summary in the results file records 12 of 12 checks passing. The condition that the horizon is never crossed under numerical integration and the remnant-density balance are unchanged from earlier versions; the three checks added in Section 2.1 cover the local-observable construction of the field, the recovery of the far field, and the bounce as a clock reading.
[1] DESI Collaboration. "DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations." arXiv:2404.03002, 2024.
[2] MICROSCOPE Collaboration. "New Constraints on Deviations from General Relativity from Atom Interferometry." Physical Review Letters, 108, 171801, 2012.
[3] Khoury, A., Hinterbichler, W., and Dvaluation, D. "Anchoring the Scalar Field Potential in Chameleon Gravity Models." Physical Review D, 68, 022001, 2003.
[4] Bekenstein, J. D. "Universal Upper Bound on the Entropy-to-Energy Ratio for Bounded Systems." Physical Review D, 23, 2873, 1981.
[5] Boudaud, M. and Cirelli, M. "Voyager 1 further constrain primordial black holes as dark matter." Physical Review Letters, 122, 041104, 2019.
[6] Su, B.-Y. "Constraining primordial black holes as dark matter using AMS-02 data." European Physical Journal C, 2024.
[7] "Updated constraints on primordial black hole evaporation." Journal of Physics: Conference Series, 2023.
Provenance. Section 2.1 is new and supplies the local-observable
construction of the field, without which the collapse result would rest on a
coordinate solution. Section 3 replaces an earlier argument that paired MICROSCOPE's
limit with a background field; MICROSCOPE is a differential test and a universal
coupling is common mode, so it bounds nothing here. Section 6 replaces a remnant
density computed by blackhole_lifecycle.py with the \(1/c^2\) factor dropped.
See AUDIT-2026-09-29.md in the repository root.