Time as a Renko Chart: Time as the Accumulation of Events — An Open Framework across Proper Time, Coarse-Graining, and Relational Time.
What is the time we “feel”? This paper takes one intuition and, laying it carefully upon established physical neighbours, proposes it as an open framework (a research programme): that time is not an evenly-spaced ruler but an event-based quantity that advances one step only when enough change has accumulated. In the metaphor of financial charts, time is less like the evenly-spaced time candlestick and more like a Renko chart (a brick chart from technical trading that prints a new brick only when price moves past a fixed threshold). This intuition has two royal-road neighbours. (1) Proper time τ: in relativity the invariant accumulated along a worldline, c²dτ² = c²dt² − dx², is the time actually elapsed for that object; along a light worldline dτ=0 (light prints no bricks at all), and the faster one moves the slower proper time runs — this is standard relativity itself and the honest core of “moving clocks run slow.” (2) Relational / thermal time (Rovelli; Connes–Rovelli): time is not fundamental but emergent from change and correlation; the Renko intuition is a folk version of this. One Renko brick corresponds to Paper 77’s coarse-grained irreversible event; the reversal threshold is Paper 81’s (unreachable) recurrence, which is why time stacks one way. IMPORTANT (scope): this paper does NOT re-derive or replace special-relativistic time dilation (which follows from Lorentz invariance). “Rendering lag,” “Universe OS,” “3D game,” and “the waking of consciousness (a light sphere r=ct rising, Paper 61)” are Layer-3 NON-SCIENCE lenses (metaphors), quarantined in a dashed box — not mechanisms and not testable claims. Coordinate time, proper time, and thermodynamic (event) time are three DISTINCT times and are not fused (distinct roots, Papers 58/48); the Renko is a rhyme, not an identity. No new prediction; the origin of the arrow of time (the past hypothesis) is not solved. The contribution is only to give the intuition its correct home and correct label.
A note on the aim of this work. This is a structural interpretation and an open framework — giving a long-held intuition its home among proper time, relational/thermal time, and coarse-graining — not a solution and not a new physical law, and explicitly NOT a rewriting of special relativity. The words “Universe OS,” “rendering lag,” “3D game,” and “consciousness” are Layer-3 interpretive lenses carrying the non-science label, not testable claims. Nothing here is intended to contradict existing results.
[WHAT THE THIRD EDITION ADDS] – AN ADDITION OF TYPES ONLY; NOTHING IS RETRACTED. The open framework of the first edition (time as an accumulation of events) stands unchanged. Three things are added.
(1) THE THREE TIMES DIFFER IN LOGICAL TYPE, NOT IN VALUE. dt is a COORDINATE and has no domain – it becomes a clock only once pasted onto a congruence of worldlines (a convention, Paper 95). dtau is a FUNCTIONAL OF A CURVE (domain: worldlines). dS is a FUNCTIONAL OF A MEASURE (domain: probability measures on phase space). Hence dtau and dS are not even two numbers of differing value: they are different functionals living on different spaces, and there was never an arena in which to compare them (Paper 118). NOTE that the separation of dt from dtau is NOT a discovery of this series – it is Minkowski’s, from 1908, and within this series it is treated by Paper 108.
(2) THERE IS A FOURTH TYPE. Paper 119 reported that this series was already using one, in the Ricci flow and renormalisation-group flow of Papers 103 and 110. It is none of dt, dtau, dS: from the explicit sphere solution r^2 = r_0^2 – 2(n-1)t one reads [t] = L^2 – A SQUARED LENGTH, NOT A SECOND – and two quantities of different dimension cannot be the same quantity (Paper 116). Nor does it live inside the manifold: it advances along a path in the space of metrics Met(M), where THE VERY INSTRUMENT OF MEASUREMENT IS BEING DEFORMED. And it is not one clock but a PAIR: the parameter is a relabelable label, while what carries a direction is a monotone (Perelman’s W, Zamolodchikov’s c).
(3) THE CENTRAL UPGRADE: DECLARE THE CLOCK BEFORE COUNTING A BRICK. A brick can be counted in four ways, AND THE FOUR COUNTS DO NOT AGREE. Counted in dtau, light lays down NO BRICK AT ALL (d tau = 0 on a null worldline) – yet the same light, as radiation, is increasing dS. Counted in dS, a reversible harmonic oscillator lays down NO BRICK AT ALL (Liouville: dS = 0 without coarse-graining) – yet its proper time keeps accumulating. Counted in dt, a brick is dutifully drawn even when nothing happens – A RULER, NOT AN EVENT, the very thing this paper first doubted. Counted in the flow parameter, the unit is NOT EVEN A SECOND. Therefore, before stating how many steps of time have passed, ONE MUST DECLARE WHICH LEDGER WAS USED. This is not a correction to the first edition but a TYPE DECLARATION for the framework it opened.
RESTRAINT ADDED: the four are not reduced to one “real time” (electing any one as genuine is itself the error of type); FELT TIME IS NOT EXPLAINED – what the opening intuition points at is a further quantity that is none of the four (psychology and neuroscience), left untouched; Perelman’s W is not identified with thermodynamic entropy (Papers 88, 103, 119); the flow parameter is not called physical time (its dimension is L^2).
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Time as a Renko Chart [Paper 82]. Zenodo.
DOI: 10.5281/zenodo.21735260
ORCID: 0009-0006-8587-3450