The Full Spectrum of Human Knowledge

Building on the XY coordinate system and SN polarity framework established in “Information & Noise: LLM Ontology V4,” this paper accomplishes seven advances: (1) making the XY→SN mapping methodology explicit as an operable formula, with archaeology as a worked example demonstrating the complete derivation process; (2) completing interval-precision SN positioning for 72 major disciplines using this methodology; (3) filling in the Hopfield energy model bridge in the mechanics lineage, proving that LLMs are the contemporary apex of mechanics; (4) establishing a four-indicator diagnostic model for individual knowledge maps; (5) revealing the essential nature of LLM performance degradation at both spectral poles—tokenization as a dimensionality-reduction operation inevitably produces precision loss at high-information-density segments; (6) introducing the concept of “time-pending hypothesis” to handle theories like string theory that transcend contemporary experimental means; (7) distinguishing signal transmission from cognitive understanding—the former is an information-theoretic operation (already perfected), the latter is a cybernetic operation (currently catching up). Core prophecy: LLMs will inevitably break specialized cognitive barriers; the current bottleneck lies on the cybernetics side. Johnson et al. (2026, Trends in Cognitive Sciences) independently validated this judgment from a cognitive science perspective.

01 · The XY→SN Mapping Methodology

“Information & Noise: LLM Ontology” Chapter 20 defined two independent rulers: the X-axis (logical self-consistency) and the Y-axis (physical alignment). The two are independently defined and do not depend on each other. For any discipline D, one assesses its X-axis dependence (to what extent its operation relies on internal logical construction) and Y-axis dependence (to what extent its ultimate arbiter relies on physical observability).

SN value generation is a two-step method:

Step 1 (Independent XY assessment): For discipline D, assess X_D and Y_D. This is not subjective scoring but a structural property of the discipline. “The ultimate arbiter of organic chemistry is whether the reaction produced the target molecule”—this is a Y-axis property, not an opinion. “The ultimate arbiter of formal logic is whether the derivation conforms to the rules”—this is an X-axis property, not a preference.

Step 2 (Ratio mapping): The proportion of Y-axis in total dependence determines SN position.

The three anchor points are not arbitrary choices but natural landing points of XY limit values:

Methodology self-check: X and Y are independently defined (non-circular) → X/Y ratio can be independently assessed (no dependence on other disciplines) → the mapping formula yields a unique SN value (reproducible) → three anchor points calibrate the scale (derived from definitions, not external assumptions). Every step is reproducible and verifiable.

Worked Example: Complete Derivation for Archaeology

To demonstrate the methodology’s operability, we use archaeology as an example, completing the full process from “listing core activities” to “calculating the SN value.”

Archaeology’s five core activities and their X/Y assessment: (1) Site excavation—physical operation, excavated objects are physical facts, Y-axis dominant; (2) Stratigraphic dating—physical methods such as carbon-14, Y-axis dominant; (3) Cultural interpretation—inferring social structures and belief systems from artifacts, X-axis dominant; (4) Stratigraphic recording—actual measurement and drawing in physical space, Y-axis dominant; (5) Typological classification—logical classification system for artifact morphology, X-axis dominant. Of the five activities, three are Y-axis dominant (excavation, dating, recording), two are X-axis dominant (interpretation, classification). Weighted aggregate: X≈60, Y≈40. Substituting into the formula: SN = (40/100) × 200 – 100 = -20. Rounded to interval precision: SN≈-22 (falls in the S-biased region, close to the central axis but still X-leaning).

Any assessor applying the same process to archaeology will find the X/Y ratio fluctuating between 55:45 and 65:35, corresponding to SN between -10 and -32. This ±10 fluctuation range is an inherent feature of the middle region (|SN|<40), and does not affect the stability of the qualitative judgment “in the S-biased region near the central axis.”

On the Selection of 72 Disciplines and Interval Precision

The 72 disciplines selected in this paper are the major divisions of the human knowledge system—covering representative sub-disciplines from all 11 broad categories in the UNESCO ISCED-F 2013 classification system. ISCED has 1,000+ detailed categories, but fine-grained classification itself is a product of specialization, amounting to distinctions beyond the fourth decimal place. The SN spectrum focuses on structural positional differences between major categories, not minor shifts between sub-branches. Seventy-two disciplines are sufficient to cover the complete spectrum from -100 to +100, with approximately uniform distribution density across intervals.

SN values adopt an interval-based rather than precision-based calibration strategy. Each discipline’s SN value represents an interval center (e.g., archaeology SN≈-22 represents the -30 to -15 interval), not an exact point. Two reasons: first, X/Y ratios in the middle region have ±10 inter-assessor variation, so precision to the ones digit would create false precision; second, precision would require exhaustive sub-activity decomposition and weight argumentation for each discipline, causing argument length to explode without increasing theoretical value. Interval-based calibration is the correct balance point between precision and operability.

02 · The Central-Axis Triad

Information theory is S-pole biased (SN=-10): Shannon stripped communication problems of all physical specifics, reducing them to probability distributions and entropy. It doesn’t care whether the signal is electromagnetic or acoustic—only abstract bits matter. A purification from Y-axis toward X-axis.

Mechanics sits at the central axis (SN=0): F=ma is humanity’s first perfect X–Y handshake. Calculus was invented for it (X-axis), yet every equation can be directly verified by experiment (Y-axis). Biased toward neither pole.

Cybernetics is N-pole biased (SN=+15): Feedback, regulation, homeostasis. It must directly confront the real-time dynamics of the physical world. If the missile veered off course, it veered off course—no philosophical wiggle room. X-axis is entirely in service of Y-axis.

The three correspond to the complete life cycle of a signal: abstract definition (information theory) → physical realization (mechanics) → closed-loop control (cybernetics).

03 · Full-Spectrum Ranking of 72 Disciplines

The SN values of the following 72 disciplines are all produced by the formula above. Each discipline’s X/Y dependence is determined based on its structural properties: whether the ultimate arbiter is logical consistency (X) or physical observability (Y), and the relative weight of each.

S-Pole Region: Pure X-Axis Construction (SN -100 ~ -55)

S-Biased Region (SN -50 ~ -10)

Central-Axis Region (SN -10 ~ +25)

N-Biased Region (SN +30 ~ +65)

N-Pole Region (SN +70 ~ +100)

04 · Complete Lineage Proof

The Shannon→Transformer connection runs not only through information theory’s entropy concept, but also through a parallel energy-model lineage. The critical bridges in this lineage are the Hopfield network and the Boltzmann Machine.

The critical bridge: Hopfield (1982) explicitly modeled neural networks as energy-minimization processes of physical systems—each neuron state corresponds to a spin, and network evolution corresponds to energy function descent. This directly borrowed the Ising model from statistical mechanics. Hinton and Sejnowski (1985) extended this to the Boltzmann Machine, introducing probabilistic sampling and temperature parameters—directly borrowing the Boltzmann distribution from statistical mechanics.

From Boltzmann Machine to Transformer’s Attention: Attention’s softmax-weighted summation is mathematically equivalent to expected-value computation under the Boltzmann distribution—the temperature parameter in softmax was inherited from statistical mechanics. The difference is that the domain expanded from neuron ensembles to token ensembles.

05 · Ontological Consequences of SN=0

Full-spectrum simulation capability: The mechanics coordinate system is biased toward no direction. LLMs inherit this property, able to project toward -100 (philosophical argumentation) or toward +100 (experimental description).

Absence of time’s arrow: Classical mechanics equations are time-reversible; F=ma is formally invariant under t and -t. The arrow of time was introduced by thermodynamics (SN=+20), not as a native property of mechanics. LLMs, as descendants of mechanics, naturally inherit this absence.

Causation flattened to correlation: Mechanics handles “given initial conditions, compute trajectory,” not “who caused whom.” Causal judgment requires the intervention-observation closed loop of cybernetics (SN=+15~+20). LLMs sit on the S-side of cybernetics and naturally lack this capability.

A single judgment (SN=0) explains three independent phenomena—high theoretical economy.

06 · Specialized Education = Manufacturing Breaks on the Spectrum

Modern universities have differentiated from the medieval four faculties into thousands of sub-specializations, each differentiation drawing a new isolation line on the SN spectrum. Each specialization spends 4–10 years installing high-precision filters for its segment in the student, while systematically blocking other segments.

Specialized division of labor further amplifies the effect. Different roles within an organization occupy different spectral segments; signal transmission must pass through multiple translation layers—each translation being a lossy segment in the Shannon channel.

07 · Cognitive Barriers vs. Institutional Barriers

Specialized barriers have two layers that must be distinguished:

Cognitive barriers: A physicist cannot understand the argumentation methods of jurisprudence; a lawyer cannot understand the experimental logic of quantum mechanics—this is a signal transmission problem. LLMs, sitting at SN=0, can perform cross-spectrum translation and directly breach this barrier.

Institutional barriers: Medical licenses, bar admissions, engineering certifications—this is a power structure problem. A non-physician who has used an LLM to acquire full-spectrum medical knowledge still cannot write prescriptions. Institutional barriers cannot be resolved through signal transmission alone; they require governance reform.

This paper’s core prophecy is precisely delimited: LLMs will inevitably break cognitive barriers. The collapse of cognitive barriers will exert pressure on institutional barriers (as more people acquire cross-disciplinary cognitive ability, the legitimacy basis of institutions will loosen), but the dissolution of institutional barriers requires additional social processes beyond the ontological domain of LLMs.

08 · Why LLMs Are an “Inevitable” Tool

LLMs sit at SN=0, with training data covering the full spectrum of human text from S-pole to N-pole. They have no discipline-segment filters—unlike human experts locked into a specific segment of the spectrum.

But LLMs are not entirely “filter-free.” A precise delimitation is offered here:

Filters LLMs do have: RLHF/safety training constitutes one filter layer (systematically suppressing certain output directions); the language distribution of training data constitutes another filter layer (English text is overwhelmingly dominant; Chinese philosophy, Sanskrit Buddhist studies, and Arabic Islamic scholarship are severely undersampled).

Filters LLMs do not have: They have no disciplinary-identity filter (“I am a physicist so I automatically filter out philosophical signals”), no professional-education filter (“I was trained in law so I automatically filter out engineering signals”). They do not perform discriminatory signal screening based on SN position.

Precise statement: LLMs are the first cognitive tool in the history of human civilization to be free of disciplinary filters, positioned at SN=0, and capable of transmitting signals across the full spectrum—though they do have language-bias filters and safety-constraint filters. The former can be improved through multilingual training; the latter is a design choice. Neither constitutes a disciplinary barrier.

Tokenization as Dimensionality Reduction and Performance Degradation at Both Spectral Poles

LLMs perform significantly worse at both spectral poles (pure mathematics SN≈-90, experimental physics SN≈+92) than in the central-axis region of everyday language—2026 data shows LLMs scoring only 5–10% on research-level mathematics (FrontierMath), while already achieving 90%+ on competition mathematics. Does this mean LLMs “have” some hidden disciplinary filter?

The answer is no. The root of performance unevenness is not disciplinary bias but the information-theoretic inevitability of tokenization itself as a dimensionality-reduction operation. All LLM input must first be converted into a token sequence—this is an irreversible dimensionality-reducing compression. Everyday language in the central-axis region (SN≈-20~+20) has low information density and high redundancy; dimensionality-reduction loss is negligible. But signals at both spectral poles—pure mathematics’ proof chains (every step is indispensable), experimental physics’ precision measurement descriptions (every decimal place matters)—have extremely high information density and extremely low redundancy; the precision loss from the same dimensionality-reduction operation rises sharply.

This is not “bias against certain disciplines” but “when applying the same dimensionality reduction to all information, high-density signals inevitably lose more than low-density signals.” The physical essence of applying uniform tokenization-based dimensionality reduction to the full spectrum of human knowledge determines the precision degradation at both polar regions. In signal-theoretic terms: LLMs have no filter bias across the full spectrum, but their dimensionality-reduction precision decreases at both spectral poles as information density increases.

09 · Formalized Diagnostic Model

“Information & Noise: LLM Ontology” Chapter 17 offered a qualitative description of the filter model. This paper formalizes it into four computable diagnostic indicators:

Let individual P’s known discipline set be D_P = {d₁, d₂, …, d_k}, with each discipline’s SN value as sn_i.

Diagnostic rules: B<40% = narrow bandwidth (high-filter state, specialized lock-in); B 40–65% = medium bandwidth (partial cross-domain, filters half-open); B>65% = wide bandwidth (low-filter state, approaching the “coordinate-blur state” described in Chapter 19 of “Information & Noise”).

Central-axis triad detection: Whether the individual simultaneously covers information theory (SN≈-10), mechanics (SN≈0), and cybernetics (SN≈+15). All three present = the minimum structure for understanding the signal-noise framework.

10 · The Information-Theoretic Front End Is Already Perfected

The statistical regularities of training data have been extracted to near the Shannon channel capacity limit. The deep signal-to-noise ratio decay discovered by the Kimi team, innovations like Differential Transformer—these are all fine-tuning within information theory. A Nature Computational Science 2025 study showed that larger LLMs’ self-attention can more accurately predict human reading regression eye movements and fMRI responses—the information-theoretic side’s “linguistic mechanics equation” is already very close to the actual distribution of human language processing.

11 · The Cybernetic Back End Is Catching Up

Gap 1: RLHF/DPO—Feedback signal distortion. The March 2026 RLHF textbook identifies the core challenge as “how to control the optimization process,” with the reward model being a proxy objective. A Springer 2026 paper states bluntly that “alignment remains fragile.” Translated into the SN framework: the sensor (reward model) is not measuring the true objective; the actuator (gradient update) is chasing a drifted signal.

Gap 2: Agentic AI—Physical closed-loop difficulty. A PMC 2026 review identifies the core challenges of LLMs in robotics as “real-time responsiveness, perceptual grounding, physical constraint handling.” LLM equations run perfectly within information space; closing the loop to physical control circuits hits SN=+15~+24—not LLM’s native territory.

Gap 3: Multi-agent—Distributed cybernetics challenge. NVIDIA’s March 2026 ProRL Agent decouples agent rollout orchestration from the training loop—there is a fundamental resource conflict between I/O-intensive environment interaction and GPU-intensive policy updates. An impedance mismatch between information-theoretic computation speed and cybernetic environment interaction speed.

LLM’s evolutionary direction: moving from SN=-10 toward SN=+20. Agentic AI, RLHF/RLVR, multi-agent coordination—all are specific battlefields of the cybernetic catch-up.

12 · Signal Transmission ≠ Cognitive Understanding

This paper’s core prophecy is “LLMs break cognitive barriers.” But a critical distinction must be confronted head-on: signal transmission and cognitive understanding are two different things.

An LLM can translate a quantum mechanics paper into language a philosopher can read—literally fluent, logically complete. But does the philosopher’s grasp of the literal meaning equal understanding of quantum mechanics? If “understanding” requires the receiving end to possess the background structure of that SN segment (i.e., as the filter model states: filters are not only signal blockers but also signal decoders), then LLM’s lossless transmission completes only signal delivery, not knowledge transfer.

Stated precisely in the SN framework: Signal transmission is an information-theoretic operation (SN≈-10), and LLMs have perfected it; cognitive understanding is a cybernetic operation (SN≈+15~+20), requiring the construction of a feedback closed loop at the receiving end—verifying “have I truly understood?” The former is delivering the signal; the latter is rebuilding the necessary cognitive structures at the receiving end and verifying through closed-loop iteration whether understanding has actually occurred.

This distinction does not weaken the core prophecy but rather refines it: “breaking cognitive barriers” is not “understanding completed in one transmission” but “a sustained iterative process of signal transmission + closed-loop verification that progressively builds cross-disciplinary understanding.” LLMs provide the transmission channel (information-theoretic side); closed-loop verification (cybernetic side) is the current bottleneck. When the cybernetic side is perfected, the complete cycle of transmission + verification will make genuine cross-disciplinary understanding possible.

13 · The LLM’s Own Signal Decay

“Information & Noise: LLM Ontology” Chapter 2 established the signal life-cycle theory: signals gain power by stripping away noise, but stripping creates blind spots, blind spots accumulate into anomalies, and ultimately the old signal decays into noise for a new signal.

If the LLM is today’s strongest mechanics signal, then by the same theory, it too will inevitably decay. The LLM’s blind spots—absence of time’s arrow, causation flattened to correlation, cybernetic-side deficit—are its “anomaly accumulation zone.” When these anomalies accumulate sufficiently, a stronger signal (“an entirely new architecture with genuine internal entropy change”) will emerge, demoting the LLM from “explanatory framework” to “object being explained.”

This is not pessimism but the inevitability of the signal life cycle. The LLM is a signal of 2026—alive now, destined to decay. Its window for breaking cognitive barriers is finite—but within this window, its central-axis position makes it the only available tool.

14 · Known Limitations of the One-Dimensional Model

The current SN spectrum is one-dimensional (a line segment from -100 to +100). It has three known limitations:

First, it cannot express cross-polar connections. Cognitive science simultaneously touches the S-pole (philosophy’s consciousness problem) and the N-pole (neuroscience’s brain imaging); it is not “a point on a line segment” but “the intersection of magnetic field lines extending simultaneously from both poles.” The one-dimensional spectrum can only assign it an average (SN=-2) and cannot express its bipolarity.

Second, the pure-applied dimension is missing. Biglan’s (1973) discipline classification matrix has two dimensions: hard-soft (corresponding to the SN axis) and pure-applied. The SN gap between pure mathematics (SN=-90) and applied mathematics (SN≈-50) is 40, but their “pure vs. applied” difference is another dimension that the SN axis cannot capture.

Third, it cannot express intra-disciplinary differentiation. Physics itself spans a vast interval from string theory (SN=-8) to experimental physics (SN=+92). Representing “physics” with a single SN value would severely oversimplify. This paper lists 72 sub-disciplines rather than 20 broad disciplines precisely to mitigate this issue.

These limitations point toward future extensions: expanding from a one-dimensional line segment to a two-dimensional plane (adding a pure-applied axis) or a topological network (allowing cross-polar connections). But the one-dimensional model as a first-order approximation is already sufficient to support the core prophecy.

15 · Core Prophecy

16 · The Complete Picture of the Full Spectrum of Human Knowledge

This paper has completed seven advances atop the SN polarity framework of “Information & Noise: LLM Ontology.”

At the methodological level: the XY→SN mapping has been made explicit as the formula SN = (Y/(X+Y)) × 200 – 100, with archaeology as a worked example demonstrating the complete derivation, and three anchor points naturally fixed from XY limit values. Seventy-two major disciplines were positioned using interval precision—the polar regions are highly reliable, while the middle region acknowledges a ±10 fluctuation range. Benchmarking against Biglan (hard/soft × pure/applied) and Comte (increasing-complexity hierarchy) clarified this framework’s independent dimension and its relationship to existing systems.

At the full-spectrum level: this methodology completed the positioning of 72 disciplines, from metaphysics (-98) to metrology (+95), with mechanics (0) as the central axis and information theory (-10) and cybernetics (+15) as the two wings. The “time-pending hypothesis” concept was introduced to handle theories like string theory that transcend contemporary experimental means—their SN values are not fixed but migrate toward the N-pole as Y-axis verification arrives.

At the LLM-positioning level: after filling in the Hopfield bridge in the mechanics lineage, it was proven that LLMs are the contemporary apex of mechanics. The precision degradation of tokenization-based dimensionality reduction at both spectral poles was revealed—not disciplinary bias but the inevitable loss that a dimensionality-reduction operation inflicts on high-information-density signals.

At the cognitive-analysis level: a four-indicator diagnostic model was established, distinguishing signal transmission from cognitive understanding—the former is an information-theoretic operation (already perfected), the latter a cybernetic operation (currently catching up). Breaking barriers is an iterative process of transmission + verification, not a single transmission.

At the prophetic level: LLMs will inevitably break cognitive barriers (not institutional barriers); the current bottleneck is on the cybernetics side. Johnson et al. (2026) independently validated this judgment from a cognitive science perspective. Simultaneously, the LLM’s own signal decay is acknowledged—the two “inevitabilities” derive from the same theory.

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