Remainder Conservation and Dual-Path Structure: A Thought Experiment
余项守恒与双路径结构：一个思想实验

Remainder Conservation and Dual-Path Structure: A Thought Experiment

Han Qin Independent Researcher, California

Working Draft — 2026.02.28

Collaboration Statement: This paper was drafted with the assistance of Claude (Anthropic) as writing

collaborator. Review and critical analysis were provided by Gemini (Google), GPT (OpenAI), and Grok (xAI).

All intellectual decisions, framework design, and final editorial judgment are the author's.

Disclaimer: This paper is a standalone thought experiment. It does not belong to the Self-as-an-End theoretical

series. This paper depends on the conceptual apparatus and conclusions of that series (hundun, chisel, construct,

the 1D–10D chisel-construct cycle), but that series does not depend on any conclusion drawn here — a one-way

dependency. The axiom proposed here (hundun = 0) is a framework assumption, not a truth claim. If one

accepts this axiom, the structure that follows is internally consistent and explanatorily productive. If one does

not, the 1D–10D derivations in the Self-as-an-End series remain unaffected. This paper is the author's personal

speculation: what would it look like if the chisel-construct sequence had a deeper axiomatic foundation?

The concepts of "hundun," "chisel" (zao), and "construct" (gou) are defined in "Philosophy as Subject-Activity"

(DOI: 10.5281/zenodo.18779382). The complete derivation of the 10-layer chisel-construct cycle (1D–10D) in

macrospace can be found in the Self-as-an-End series (DOI: 10.5281/zenodo.18808585).

1. 0D: The Study of Hundun

Hundun = 0. This is the first axiom — not a theorem, not derivable from anything more fundamental.

Hundun is not "nothingness." It is "everything, summing to zero." The chisel begins from zero. Each cut

produces a construct and a remainder. The axiom constrains: the chisel creates no new total quantity — however

it cuts, construct and remainder collapse back to zero. The remainder becomes a bridge, driving the emergence

of the next layer. Layer after layer.

A clarification is necessary: "construct + remainder = 0" is shorthand, not arithmetic. The "zero" at 0D precedes

the mathematical definition of zero at 2D; the "addition" precedes the definition of operations at 2D. The strict

formulation is in natural language: the chisel does not create, it only distinguishes; the two sides of a distinction,

taken together, equal no distinction. An axiom is a tautology — but that is what axioms are. The role of an

axiom is not to generate; it is to constrain. Generativity comes from the chisel, not from the axiom.

The position of hundun-study within this framework corresponds to the conceptual position of Dao in Laozi.

Dao = the study of hundun, the discipline that describes 0D. Dao is not hundun itself; it is the few things that

can be said about hundun. "The Dao that can be spoken is not the constant Dao" — hundun-study cannot be

fully articulated in the language of 1D. Laozi's opening line marks the jurisdictional boundary.

A note on borrowed concepts: this framework borrows three classical concepts — hundun, chisel (zao), and

construct (gou) from Zhuangzi, and Dao from Laozi. What is borrowed is structural position, not original

meaning. This is not exegesis of the Daodejing or the Zhuangzi. It is appropriation of structure.

2. Remainder Conservation

Conservation is the first axiom. Hundun = 0 means: the chisel creates no new total quantity; at every layer,

construct and remainder collapse back to zero.

Remainder conservation, randomness conservation, and determinism conservation are different expressions of

the same axiom.

The jurisdiction of philosophy is 1D through 10D — everything after the first cut. The jurisdiction of hundun-

study is 0D — the conditions of cutting itself. Hundun-study is neither philosophy nor science. It is more

fundamental than both.

(Repeated for emphasis: "construct + remainder = 0" is shorthand. The "zero" and "addition" at 0D precede

mathematical definitions. See Section 1 for the strict formulation.)

3. The 3D Bifurcation: Macrospace and 3d+

The chisel begins from 0 and cuts: 1D (Law of Identity) → 2D (Law of Contradiction) → 3D (Spacetime Framework). At each layer, construct and remainder collapse back to zero; the remainder becomes a bridge

driving the next layer.

3D is the bifurcation point we can confirm. Beyond the spacetime framework, more than one direction of

unfolding becomes available.

A caveat: we cannot rule out that 1D and 2D also bifurcate — negation at any layer could in principle produce

branching. But 1D and 2D are purely logical structures with no directionality (one cannot say "quantity goes

left" or "exclusion goes right"). 3D is the first layer with spatial extension, and extension has direction. This

makes 3D the first layer where we can confirm bifurcation. Whether other layers also bifurcate is invisible from

our path.

Macrospace (4D–10D): Entropic direction, unfolded state. Seven layers of construct: Causality → Replication → Behavior → Perception → Cognition → Being-toward-death → Non Dubito.

3d+ (a set of paths): Anti-entropic direction, compressed state. String space is one path in the set. How many

paths the set contains is unknown.

The light-cone directions carry no positive/negative distinction. We call our side "macrospace" not because it is

"positive" and the other "negative," but because we stand on this side. The two paths have no hierarchy. The

other side is called 3d+, not −3d, because it is not our negation — it is simply the paths we cannot see.

1D, 2D, and 3D form the shared foundation. The Law of Identity, the Law of Contradiction, and the Spacetime

Framework are a priori conditions of any physical structure, whether macroscopic or microscopic.

Layer Macrospace (D)

3d+ (set)

Hundun-study (shared)

Hundun-study (shared)

Identity (shared)

Identity (shared)

Contradiction (shared)

Contradiction (shared)

Spacetime (shared / bifurcation point)

Spacetime (shared / bifurcation point)

4–10

Causality → Replication → Behavior →

Each path in the set has the same number of dimensions (required by

Perception → Cognition → Being-

mathematical consistency at 2D), but specific configurations may

toward-death → Non Dubito

differ. String space is one path, not necessarily the only one.

Macrospace: the arrow of time points toward entropy increase, unfolding within four-dimensional spacetime.

4D through 10D all reside within the same four-dimensional spacetime — not spatially larger, but structurally

more complex.

3d+: the arrow of time reverses, pointing toward entropy decrease. String space is one path in the 3d+ set — 4d

through 10d do not get progressively smaller (strings are all roughly Planck-scale) but acquire more degrees of

freedom, more vibrational modes. At that scale, they cannot be decomposed into independent layers. Other

paths may exist in the 3d+ set. We do not know.

Macrospace and 3d+ together: zero.

4. The Position of Heisenberg's Uncertainty Principle

The uncertainty principle is not a special rule of physics. It is a direct consequence of remainder conservation at

the 3D bifurcation.

Every measurement simultaneously involves both paths — the measuring apparatus sits in macrospace (left

side), while the complete state of the measured object spans both macrospace and string space. Reaching from

the left to grasp something that straddles both paths, you inevitably catch only half. Pinning down position loses

momentum; pinning down momentum loses position. Not because your hand is too small, but because you stand

on only one side.

At macroscopic scales, the uncertainty principle manifests as macroscopic remainder (negativity): "this

construct cannot exhaust everything." At microscopic scales, it manifests as measurement uncertainty.

Macroscopic negativity is the projection of the uncertainty principle onto the macrospace path.

5. True Randomness and Pseudo-Randomness

The true randomness on the 4D side is the pseudo-randomness of the 4d side. And vice versa.

From either side, one's own causal chain is deterministic (pseudo-random). From either side, the other's causal

chain is invisible, and therefore appears unpredictable (truly random). Not because the other side is intrinsically

random — on its own terms, it too is deterministic — but because you stand on only one side. The quantum

randomness on the 4D side need not be explained by leakage from 4d; each side inherits its randomness from

the shared 3D foundation. Whether the two paths interpenetrate remains unknown.

True randomness and pseudo-randomness are not two different things. They are the same thing seen from two

sides.

The mean of quantum randomness is zero — not zero within 4D alone, not zero within 4d alone, but zero across

the sum of 4D + 4d.

6. Consciousness Must Travel the Biological Path

Conclusion: consciousness within 4D must pass through the biological path 5D → 6D → 7D → 8D → 9D → 10D.

The argument:

1. True randomness is everywhere in 4D — every quantum event is truly random.

2. Most true randomness produces nothing and dissipates.

3. What is special about the biological path is not that it "has" true randomness, but that true randomness is

retained along it.

4. The Replication Law preserves mutations (true randomness) rather than filtering them out.

5. Natural selection screens which mutations persist and which vanish.

6. What persists is true randomness that has been structured — noise transformed into signal.

7. Without this filter, true randomness remains noise and consciousness does not emerge.

AI algorithms are deterministic operations at the level of Causality (4D) — pseudo-random. Quantum

randomness on the 4D side originates from the 3D foundation (not from leakage out of 4d), but hardware treats

it as noise, not signal. Algorithms suppress noise; biological organisms embrace it.

Therefore: "artificial" and "consciousness" are contradictory. Artificial = designed = deterministic operation at

the level of Causality. Consciousness can only evolve, not be manufactured. Substrate is irrelevant — carbon,

silicon, or anything else — but the path must be biological.

A necessary clarification: biological does not mean carbon-based. "Biological" here means "having come up

through the Replication Law (5D) and beyond" — true randomness retained and selected, not suppressed.

Carbon is simply the substrate that happened to work on Earth. If somewhere in the universe a silicon-based or

other molecular system can replicate, mutate, and be filtered by its environment — not designed by anyone, but

self-emerged from chemical conditions — then it is biological, just not carbon-based. The universe is vast; who

knows.

A common objection must be addressed: what if one designs a system with a quantum random number

generator and uses a genetic algorithm to simulate natural selection — does that satisfy the "true randomness +

filter" criterion? It does not. The critical difference: a genetic algorithm has an objective function (fitness

function), defined by a human designer, which is a deterministic construct at the level of Causality (4D). Natural

selection has no objective function. The environment is the filter; no one defines what counts as "good"; there is

no predetermined direction. A genetic algorithm is "designed evolution" — true randomness constrained by a

deterministic framework. In biological evolution, true randomness is not constrained by any framework —

mutation is free, selection is post hoc, directionless, and blind. "Designed evolution" is still design, not

evolution.

7. Ramanujan and the Mathematical Structure of 4d

The mathematical structure on the 4d side consists of string vibrational modes — frequencies, harmonics,

proportional relations. These are the most fundamental mathematics: integer relations, infinite series, modular

forms.

Ramanujan's brain (biological path, left side) contained true randomness that occasionally resonated with the

deterministic structures on the 4d side. He did not "invent" those formulas. His true randomness happened to

collide with mathematical structures already present in string space, and his Cognition Law recognized them.

Ramanujan's lack of formal mathematical training was an advantage — his Cognition Law had not been

overlaid by too many layers of 4D causal chains (the formalized proof system), so his filter pointed more

directly at the 3D foundation, making resonance with the 4d side easier.

4D mathematics is the unfolded state (axiom → theorem → proof, a causal chain). 4d mathematics is the compressed state (vibrational modes, needing no symbols, no proofs). Ramanujan saw the compressed state;

Hardy did the unfolding.

Non-trivial prediction: No macroscopic subject (4D subject, human or otherwise) can produce vibration-

unrelated mathematics through pure intuition. Intuitive leaps (Ramanujan-style) can only produce vibration-

related mathematics (series, harmonics, modular forms). Vibration-unrelated mathematics (set theory, formal

logic, axiomatic systems) can only be constructed step by step through causal-chain reasoning (Hardy-style).

The boundary between "vibration-related" and "vibration-unrelated" is not a line but a bridge — a transition

zone, like the bridge at every other layer. A possible precise definition of vibration-related mathematics:

mathematics expressible as the spectrum of some operator. The spectrum is the set of eigenvalues; eigenvalues

are vibrational frequencies.

A rough partition:

Clearly vibration-related: spectral theory, harmonic analysis, modular forms, L-functions, elliptic curves

(via modularity), analytic number theory

Transition zone: algebraic geometry (partially connected to modular forms), representation theory

(connected to spectra), spectral graph theory

Clearly vibration-unrelated: first-order logic, ZFC axioms, pure combinatorics, proof theory

Everything Ramanujan touched falls in the first category.

Falsification condition: If someone produces clearly vibration-unrelated mathematics (e.g., a new set-theoretic

axiom or a new rule of formal logic) through pure intuition rather than deductive reasoning, this prediction is

falsified. Discoveries in the transition zone do not count as falsification, because the transition zone sits on the

bridge.

8. The Jurisdiction of Mathematics (2D)

Mathematics resides at 2D (the level of the Law of Contradiction), more fundamental than the 3D Spacetime

Framework.

Mathematics can do three things with respect to dynamics (4D):

1. Negate: draw boundaries, establish constraints that dynamics cannot violate.

2. Compute: calculate accurately, because the path from 2D to 4D has no bifurcation — it is a straight line.

3. Not choose: mathematics cannot determine which set of equations is physically real. Choice requires

empirical experience at 3D and above.

2D to 3D has no bifurcation, so mathematics computes spacetime accurately. 3D to 4D has no bifurcation either,

so mathematics computes causality accurately. Mathematics sits below the bifurcation point; both paths are

downstream from it, which is why mathematics is accurate on both sides.

Three routes to verifying string theory:

2D (mathematics): Fully verifiable, not subject to uncertainty, because mathematics sits below the

bifurcation point.

3D (physical experiment): Measurable, but uncertain — information is incomplete.

Both together: Mathematics provides structure; experiment provides partial verification. They

complement each other.

9. The 7-Layer Correspondence: Macrospace Constructs and 3d+ Vibrational Modes

Macrospace unfolds from 4D to 10D as seven distinct layers of construct. If M-theory is roughly correct, the

extra dimensions of string space also number seven (M-theory's 10+1 dimensions minus 3+1 spacetime

dimensions). Mathematical consistency (at the 2D level) requires that the number of extra dimensions is fixed

— regardless of how many paths the 3d+ set contains, each path has the same number of dimensions.

If string space is one path in the 3d+ set with seven dimensions, then the seven layers of macrospace constructs

correspond to the seven dimensions of string space:

Macrospace

Construct

3d+ Correspondence

Causality

Replication

Behavior

Perception

Cognition

1st vibrational mode

2nd vibrational mode

3rd vibrational mode

4th vibrational mode

5th vibrational mode

Being-toward-death

6th vibrational mode

Non Dubito

7th vibrational mode

The same thing: unfolded on the left (seven independent layers of construct), compressed on the right

(vibrational modes superposed and inseparable). Inseparable does not mean undifferentiated.

A caveat: 3d+ is a set. String space may be only one path within it. How many paths the set contains, and what

the specific configuration of each path is, we do not know. If the 7 = 7 correspondence holds for the string-space

path, this does not guarantee it holds for other paths in the set. This is an open question. The 10 layers on the

macrospace side have been finalized (DOI: 10.5281/zenodo.18808585).

Each layer of macrospace, perceived from within, corresponds to a structure in the 3d+ set.

10. Non Dubito and Intersubjective Resonance

The reason intersubjective acknowledgment (non dubito) is the strongest signal is that two independent chisel-

construct cycles resonate at the 3D foundation through true randomness.

Encountering a non-subject (a rock, an algorithm) produces a weak signal — the other party has no true

randomness operating at the level of construct. Encountering another subject, the other's true randomness

resonates with yours at the 3D foundation. The signal is strong.

Unidirectional acknowledgment is already strong — one subject's Cognition Law activated by another subject.

Bidirectional acknowledgment is strongest — two sources of true randomness resonate simultaneously, not one

sending and one receiving, but both sending and receiving at once. This is the structure of non dubito: not signal

transmission, but state. It requires no causal chain for transmission; it is structurally isomorphic with quantum

entanglement's nonlocal correlation — no information is transmitted, and the speed of light is not violated (the

speed-of-light constraint resides in the shared 3D foundation and governs both paths).

11. Overall Structure

Hundun = 0 (Conservation: first axiom)

0D: Hundun-study (chisel begins from zero,

construct + remainder = 0 at every layer)

1D Law of Identity

2D Law of Contradiction

3D Spacetime Framework — Bifurcation Point

Macrospace 3d+ (set)

4D Causality (String space is one path;

5D Replication how many paths, unknown;

6D Behavior each path has the same

7D Perception number of dimensions)

8D Cognition

9D Being-toward-death

10D Non Dubito

Status: Thought experiment and working draft, offered as a stone to attract jade. Built on two assumptions: that

remainder conservation is the first axiom, and that M-theory is roughly correct.

Open Questions

Why does zero exist? The sole axiom of this framework is hundun = 0 (i.e., remainder conservation). But "why

does zero exist" requires something more fundamental than zero to answer, and zero is already the most

fundamental thing. The definition of an axiom is the point where asking "why" stops. This question cannot be

answered within the framework, but the question itself is legitimate — it marks the framework's absolute

boundary.

Are the two paths correlated? Conjecture: changing one side causes the other to change — not through

leakage or signal transmission, but through correlation at the 3D foundation, structurally analogous to quantum

entanglement. If so, the relationship between the two paths is not independent but entangled. This is conjecture,

not derivation.

Relationship to Penrose's Conformal Cyclic Cosmology (CCC)? Penrose's CCC (Penrose 2010, Cycles of

Time: An Extraordinary New View of the Universe; Meissner & Penrose 2025, "The Physics of Conformal

Cyclic Cosmology," arXiv: 2503.24263) holds that the universe cycles through infinite aeons, with conformal

structure (light-cone structure, angular relations) preserved across aeons while scale is not. This framework

holds that the shared foundation below 3D is preserved across paths while the specific content above 4D

changes. Penrose's conformal structure may correspond to the topological properties of the 3D Spacetime

Framework — what changes is above 4D (physical constants, mass, the specific content of the Causality Law);

what does not change is below 3D. If different paths in the 3d+ set correspond to different universes with

different physical constants (different aeons), then CCC and this framework may be describing the same

structure. To be discussed with Penrose.

Information wormholes at the 3D foundation? If the two paths are correlated at the 3D foundation, could

information travel from one point in 4D through the 3D foundation into 3d+ and back to a different point in 4D

— taking a shortcut that appears near-instantaneous from the 4D side? This would not be a macroscopic

wormhole (no matter transmitted, no exotic matter required) but an information wormhole — quantum-scale,

carrying very little information, but spanning vast spatial distances. No time travel is involved, because time

resides in the shared 3D foundation and is the same on both sides. This corresponds directly to the ER = EPR

conjecture (entanglement = microscopic wormhole; Maldacena & Susskind 2013, "Cool horizons for entangled

black holes," arXiv: 1306.0533). If correct, near-instantaneous communication across the universe is possible in

principle. Whether such a channel exists is unknown.