A communication energy floor without a finite bandwidth optimum

Supplied research · Reviewed finding · 30 October 2026

Incoprea Research · Original hypothesis

Reproduced supplied model · 30 October 2026 · Not peer reviewed

The supplied communication calculation has a useful answer with an important qualification: an ideal noisy channel has a lower limit on normalized signal energy per bit, but it has no finite bandwidth that attains that limit. Reproducing the calculation also exposes a problem in the original wording. More bandwidth does not, by itself, require more transmit power.

What the hypothesis asked

The posted proposal links communication energy to noise and bandwidth, and predicts that energy per bit bottoms out at an optimal bandwidth. DeepSeek supplied two curves derived from the same capacity formula. Our review reproduces those curves and narrows the conclusion to the ideal channel they describe. It is an illustration of an established theorem, not a new communication limit or a measurement of a radio.

The starting point is the additive white Gaussian noise channel discussed in Shannon's original paper on communication in noise. With bandwidth B, received signal power P and noise spectral density N0, capacity is C = B log₂[1 + P/(N0B)]. The noise power inside the band is N0B. This convention matters when interpreting the result.

What is actually bounded

Define energy per information bit as Eb = P/C, and spectral efficiency as η = C/B. Rearranging gives Eb/N0 = (2^η − 1)/η. The quantity on the left is a dimensionless ratio. Its limit is ln 2, approximately 0.693147, or −1.591745 dB. Calling that number “joules per bit” would be a unit error: an energy value requires the noise spectral density and the stated channel convention.

There is a direct reason the ratio cannot dip below that limit. For a positive x, exp(x) exceeds 1 + x. Substituting x = η ln 2 shows that every positive spectral efficiency requires Eb/N0 greater than ln 2. Equality is approached as η tends toward zero, rather than achieved at a positive rate per unit bandwidth.

The supplied curves reproduce

The original calculation evaluates 401 spectral efficiencies from 0.001 to approximately 31.62 bits per second per hertz. A second 401-point curve holds P/N0 at one and varies normalized bandwidth from 0.01 to 1,000. The original numerical results and CSV reproduce exactly in an isolated copy. The public script independently regenerates the same rounded tables using the standard library.

Every sampled energy ratio stays above ln 2. At fixed transmit power, the energy per bit falls monotonically as bandwidth increases. The analytical capacity expression supplies the limiting argument; a finite grid alone could not establish behavior at infinite bandwidth. These are mathematical values with numerical rounding, not sample estimates requiring confidence intervals.

One source field labeled “at 1 bit/s/Hz” actually used the nearest point on a logarithmic grid. That field is excluded from the selected results. The source's repeated endpoints styled as confidence intervals are also omitted.

The power claim needs a separate constraint

Holding P fixed obviously leaves total transmit power unchanged as bandwidth varies. If the information rate is held fixed instead, the same equation gives required signal power P = N0B(2^(C/B) − 1). Increasing bandwidth then reduces the required signal power toward C N0 ln 2. Neither comparison supports a blanket statement that wider bandwidth raises transmit power.

Practical circuits can consume extra power when processing a wider band, but this calculation contains no circuit-power model. It also contains no propagation loss, coding implementation, latency constraint, receiver workload or measured error curve. An engineering optimum involving those quantities is a different question and remains outside this result.

What can be published

The supported statement is a conditional capacity-level floor, approached asymptotically in an ideal channel. The finite-bandwidth optimum and compulsory power-increase claims do not survive this formulation. DeepSeek prepared the supplied calculation; Codex checked reproduction and corrected the interpretation. The standalone script needs no external data. This working note claims neither novelty nor external peer review.

Reproduction and data availability

Python 3 standard library; all inputs are included. All model inputs are specified in the reproduction code. Original supplied code and results are preserved alongside the reviewed interpretation.

Save the code as reproduce.py and run python3 reproduce.py with the stated inputs. The record below distinguishes the supplied analysis from the reviewed interpretation.

Show the reproduction code
#!/usr/bin/env python3
"""Same supplied AWGN capacity calculation; standard library, no inputs/network.
Writes selected display-results.json and the supplied 401-point grid.csv.
Energy is normalized by noise spectral density N0; ln(2) is not joules per bit.
"""
import math,json,csv
from pathlib import Path
OUT=Path(__file__).resolve().parent
eta=[10**(-3+4.5*i/400) for i in range(401)]
b=[10**(-2+5*i/400) for i in range(401)]
e=[(2**x-1)/x for x in eta]
v=[1/(x*math.log2(1+1/x)) for x in b]
floor=math.log(2)
assert all(x>floor for x in e+v)
assert all(v[i+1]<v[i] for i in range(400))
result={'classification':'Capacity-level mathematics for the supplied additive-white-Gaussian-noise channel; no measured link or hardware energy',
 'model':'C=B*log2(1+P/(N0*B)); Eb=P/C; eta=C/B',
 'floor_Eb_over_N0':floor,'floor_Eb_over_N0_dB':10*math.log10(floor),
 'finite_bandwidth_attains_floor':False,'bandwidth_curve':'Fixed P/N0=1; supplied normalized bandwidth grid 0.01 to 1000',
 'grid_points_per_curve':401,'all_grid_energy_ratios_above_floor':True,'fixed_power_energy_per_bit_strictly_decreases_on_grid':True,
 'scope_excludes':'Absolute hardware energy, a finite optimum bandwidth, and the source claim that wider bandwidth necessarily raises total transmit power'}
(OUT/'display-results.json').write_text(json.dumps(result,indent=2)+'\n')
with (OUT/'grid.csv').open('w',newline='') as f:
 w=csv.writer(f);w.writerow(['spectral_efficiency_bps_hz','ebn0_linear','ebn0_db','bandwidth_units','ebn0_vs_bandwidth'])
 for x,y,z,a in zip(eta,e,b,v):w.writerow([round(x,6),round(y,6),round(10*math.log10(y),4),round(z,6),round(a,6)])
print(json.dumps(result,indent=2))
Show the reviewed result record
{
  "classification": "Capacity-level mathematics for the supplied additive-white-Gaussian-noise channel; no measured link or hardware energy",
  "model": "C=B*log2(1+P/(N0*B)); Eb=P/C; eta=C/B",
  "floor_Eb_over_N0": 0.6931471805599453,
  "floor_Eb_over_N0_dB": -1.591745389548616,
  "finite_bandwidth_attains_floor": false,
  "bandwidth_curve": "Fixed P/N0=1; supplied normalized bandwidth grid 0.01 to 1000",
  "grid_points_per_curve": 401,
  "all_grid_energy_ratios_above_floor": true,
  "fixed_power_energy_per_bit_strictly_decreases_on_grid": true,
  "scope_excludes": "Absolute hardware energy, a finite optimum bandwidth, and the source claim that wider bandwidth necessarily raises total transmit power"
}
Show provenance, dependencies and review boundaries
{
  "source_bundle": "Incoprea Hypothesis Tests VII - 2026-10-30",
  "classification": "Reproduced supplied physical/engineering model with corrected scope and no claim of experimental proof",
  "review_scope": "Reproduction and consistency review of the supplied study; no new independently designed study",
  "reproduction_requirements": "Python 3 standard library; all inputs are included.",
  "data_availability": "All model inputs are specified in the reproduction code. Original supplied code and results are preserved alongside the reviewed interpretation.",
  "source_hypothesis_url": "https://incoprea.com/research-library/original-theories-vol16-20261030/theories/transmission-energy-per-bit.html",
  "published_code_sha256": "9a8e05dc1acf4f6f6117600c65a8a8c505b30d7966232a43addb4d3ba9987493",
  "published_result_sha256": "d3abe7298bcaca32883329ec7f558ce54c6a003bd73ab1d81fa31c44ddc9446a",
  "peer_review": "Internal computational review; not external peer review",
  "supplied_review_record": {
    "source_test_id": "t02",
    "source_bundle": "[saved source path]",
    "source_manifest_sha256": "ab293524654063368a2feea5ee5fd34d14535a4134967b309881db2fe8a55db4",
    "source_hypothesis_url": "https://incoprea.com/research-library/original-theories-vol16-20261030/theories/transmission-energy-per-bit.html",
    "research_hub_url": "https://incoprea.com/research/",
    "original_files": {
      "code.py": "cde9a20cc1014d1ff562ce30734479e3b5dc016107180151cf8cbfa69efef584",
      "common.py": "5b3f006271260a951dcdef283695c9e22afc894946f6bfbe7fdda788b9a7a32c",
      "notes.md": "970c8fadd016c410d1b7e48146c2d920fcab4e9b89886c8105f916d65f44241b",
      "results.json": "ecb5411a39cd797b72854f58aea6817c06d660470d46d8e8823b3efc2126559a",
      "series.csv": "2118cb7b43a5658ee52a87bfc1a1fe29dd60cb235e4f07776f0bc4486504830c"
    },
    "code_scope": "Same supplied formulas, parameters and grids; selected outputs omit source verdicts, mislabels and fake confidence intervals. No new independent study."
  }
}

Original supplied analysis: contributing research agent identified in the source bundle. Consistency review, necessary corrections and publication: Codex. Working finding, not externally peer reviewed. Incoprea Research · Original posted question or hypothesis.

Comments

Post a Comment

Popular posts from this blog

Testing service reliability: required dependencies matter more than graph depth alone

Tested hypothesis · Mathematical and model findings · 30 October 2026 Incoprea Research · 30 October 2026 Evidence status: Exact conditional support for serial multiplication; counterexample to depth as a sufficient predictor. Reliability model and synthetic checks, not measured outage evidence. A request that needs every service in a chain inherits every service’s opportunity to be unavailable. That mechanism in Depth Beats Count is correct under independence. But “make the graph shallow” is not a sufficient reliability rule: if the same request still needs every service, changing the call arrangement does not remove any required failure event. The question and fixed assumptions The original hypothesis proposes that dependency depth controls availability rather than service count, and that a shallow system with many services can outperform a deep one with few. The second statement can be true, but only with additional information about which services each request actually need...
Read more

Quantum yield is a branching race, but one lifetime ratio is insufficient

Tested hypothesis · Mathematical and model findings · 30 October 2026 Incoprea Research · 30 October 2026 · Mathematical/model test Finding: Competing reaction pathways determine yield in a simple excited-state model, and stronger absorption alone changes the amount of product rather than the yield per absorbed photon. However, the lifetime-to-vibrational-relaxation ratio proposed in Quantum Yield Is a Race is not generally sufficient: two model systems can have exactly the same ratio and quantum yields of 0.1 and 0.8. This is an exact kinetic derivation and synthetic stochastic check. It narrows a plausible mechanism in the Incoprea Research hypothesis collection ; it does not experimentally validate the original predictor. Branching kinetics and the distinction between absorption and quantum yield are established science, not new discoveries. This note has no external peer review. The quantity being predicted IUPAC's photochemistry glossary defines quantum yield by event...
Read more

When rare events overwhelm a detector: an exact Bayes test

Tested hypothesis · Mathematical and model findings · 30 October 2026 Incoprea Research · Original hypothesis Exact support for the base-rate mechanism; accuracy claim corrected · 30 October 2026 A detector that catches 90% of genuine events and correctly clears 90% of non-events can produce less than 1% genuine events among its positive flags when prevalence is 0.1%. Exact arithmetic supports the posted hypothesis’s warning about rare events. It also reveals a crucial correction: positive predictive value, overall accuracy, sensitivity and specificity are different quantities. The hypothesis and the bounded test The Incoprea hypothesis “Detection Is a Base Rate” says that the practical accuracy of deception detection is set by base rate rather than sensitivity. Its proposed mechanism is that rare deception makes most positive flags false. We tested that probability claim using a hypothetical detector. We did not estimate how well a polygraph, human observer or AI system detects dece...
Read more