Exponential quintessence with a derived slope: λ = √3/2 from the transport geometry of the cosmic web — version 7 (corrections release)

This archive studies minimally coupled single-field quintessence with an exponential potential, V(phi) = V0 exp(-lambda phi), with one difference from the usual treatment: THE SLOPE LAMBDA IS NOT FITTED. It is obtained from the scaling of optimal transport networks (Banavar-Maritan-Rinaldo) by two further steps, both stated here rather than assumed: ONE POSTULATE — that the number of network sites grows with volume (N proportional to V), which holds exactly if the sites have a fixed physical density — and ONE EXPONENT ASSIGNMENT, that the energy of a network of N sites scales as N^(D/(D+1)) = N^(3/4). THE SECOND STEP IS PUT IN QUESTION BY THIS RELEASE (02_RESULTS/CORRECTIONS.md §35): BMR's 3/4 is a rate-versus-size exponent that appears only after the site count is eliminated, and in BMR's own variables the rate is linear in the site count. Two other readings of the same theorem give w = -4/3 and w = -1, the latter being exactly a cosmological constant. Until that step is argued, lambda = sqrt(3)/2 is one of three readings, not a derivation. With the step as the archive takes it, the result is lambda = sqrt(3)/2 = 0.8660, w0 = -0.8876 (Omega_DE0 = 0.6835), CPL (w0, wa) = (-0.8924, -0.1509) with the full w(z) curve. THIS PREDICTION IS PRE-REGISTERED in a separate deposit, with symmetric falsification criteria fixed before any Roman Space Telescope data exist. That deposit also carries a single-electron Penning-trap test of information-mass in which FFGFT (J. Pascher) and this framework state opposite predictions in advance; its file PREREGISTRATION_penning_2026-09-26.md is included here as well. J. Pascher's two statements of the FFGFT position are quoted in it verbatim and by name with his explicit written permission, and he read the full text before it was deposited; the provenance, and two corrections this archive made to its own description of his position, are recorded in §59. Its DOI is 10.5281/zenodo.22972538. EMPIRICAL POSITION — THE FOUR FIGURES BELONG TOGETHER. On DESI DR2 BAO alone the model is indistinguishable from LambdaCDM (Delta chi^2 = +0.69). With the CMB acoustic scale theta* it is 1.75 sigma DISFAVOURED (+3.07); with Pantheon+ supernovae 1.07 sigma (+1.14); with the CMB shift parameter 2.15 sigma (+4.63). An external fit of this model class (Akrami, Alestas & Nesseris, arXiv:2504.04226) gives lambda = 0.698 (+0.173 / -0.202), where 0.8660 sits at +0.97 sigma; that fit predates the DES supernova recalibration, which moves the field towards LambdaCDM. DISFAVOURED, NOT FALSIFIED; data expected in 2027-2029 will decide. There is no 5 sigma result in this archive in either direction. WHAT IS MEASURED, AND WHAT IS POSTULATED. On a simulated cosmic web the network dimension is 3.0034 (global fit; local slope 2.78-3.55) and the routing exponent at fixed density is 0.7500. The bit law N proportional to V is NOT measured by either: its one pre-registered measurement on the same web gives s = 0.396 against the postulated 1, and that adverse result is recorded. THE MAGNITUDE OF THE DARK-ENERGY DENSITY IS PREDICTED WITH NO FITTED QUANTITY, TO WITHIN A FACTOR 1.2-2.8 (§51): rho_DE = E0 N_h^alpha / V_H with E0 = k_B T_CMB ln2, N_h the Bekenstein-Hawking count of the Hubble sphere, and alpha from transport-network geometry measured on a simulated cosmic web. The claim is sharp rather than loose — ln N_h = 282, so alpha must be known to 0.39 % — and the archive's two independent unbiased routes to alpha agree to 0.027 % (Delaunay routing with a passing Poisson control, 0.7500; D_net = 3.0034 through D/(D+1), 0.7502). A third figure, alpha = 0.745 from hierarchical routing, is a biased estimator whose own uniform-set control misses BMR's value by 0.7 % and is not used. BUT THIS RESULT IS ABOUT THE MAGNITUDE ONLY, NOT THE EQUATION OF STATE, and it converts the cosmological-constant problem into the coincidence problem rather than solving it. It also carries §49's unresolved objection: ALPHA < 1 MAKES THE ENERGY SUB-ADDITIVE, SO RHO_DE IS NOT A LOCAL ENERGY DENSITY — the only extensive case, alpha = 1, misses the magnitude by 10^30. The magnitude and the energy density are traded one for one. §53.3 adds that no routing estimator in this archive reaches the 0.39 % precision the claim needs, so alpha = 3/4 is analytic — now verified independently in D = 2, 3 and 4 (§53) — with a measured correction, not a direct measurement. THREE FINDINGS OF 25 SEPTEMBER 2026 ABOUT THAT ADVERSE RESULT ITSELF (02_RESULTS/THE_25_SEPTEMBER_FINDINGS.md), two against the framework and one for it. FOR IT: the archive's own value was slightly too harsh — exact integration of the growth equation gives 0.396 rather than 0.3829 — and the fairness objection is closed, since repeating the measurement inside the framework's OWN growth history rather than LambdaCDM's moves it by -0.19 %. AGAINST: supplying the physical smallest scale the framework owes makes the result WORSE, not better (a finer cutoff lowers the rate; at the dark-matter free-streaming scale it is 0.018, not 3), so the resolution caveat was flattering the postulate rather than hiding its failure. AND A CORRECTION TO THE ARCHIVE'S OWN FRAMING: a first-crossing count is bounded, so s is a HISTORY, not a number — it passes through exactly 1 at z = 0.541 and cannot satisfy the postulate asymptotically at all, which matters because w = -3/4 is claimed as an attractor. ONE QUESTION REMAINS OPEN THAT MOVES THE NUMBER BY A FACTOR OF TWO IN THE FRAMEWORK'S FAVOUR and has never been answered: whether the bits are counted per unit mass (0.396) or per unit volume (0.822). It is stated in THE_OPEN_QUESTIONS_2026-09-25.md and is to be pre-registered before it is computed, not after. THE LABORATORY BRANCH. The derivation of the information-mass coefficient mu(T) = k_B T ln2 / c^2 from Landauer's principle and E = mc^2 is WITHDRAWN — a symmetric memory stores no internal energy, and the k_B T ln2 leaves as heat (a point published earlier by D. Lairez, Entropy 26, 337, 2024, whose priority this is). Information-mass is kept as an explicit postulate. Its simple versions are already excluded by measured erasure work (Jun, Gavrilov & Bechhoefer, PRL 113, 190601, 2014: kappa = 0.024 +/- 0.043 against 1); the remaining version — an entropy-dependent rest energy exchanged with a non-local reservoir — is the subject of a single-electron protocol with predictions fixed in advance (05_EXPERIMENT/PENNING_PROTOCOL_v4.md). Its honest content is a ceiling registered before any measurement: the temperature that raises the signal also populates the cyclotron mode the measurement needs empty, so the signal cannot exceed ln2 of the relativistic shift per cyclotron quantum while the mode is cold — about 27 Hz at 1.5 K — and whether one electron can be measured that well at 1-2 K may turn out to be no. No trap group has assessed it. A laboratory null would not affect the cosmological prediction. THE CREATION LAW, MEASURED (§§54-57, added 26 September 2026). The framework's one un-derived component is the bit law N proportional to V. This release shows it is not merely un-derived but MEASURED AND EXCLUDED. Write the creation rate of committed bits per unit physical volume as Gamma = gamma H^n; then n = 1 IS EXACTLY THE POSTULATE, and it is a named law — the STEADY-STATE creation rate (Hoyle's C-field; the CCDM models of Prigogine, Geheniau, Gunzig & Nardone 1989 and Lima, Jesus & Oliveira 2010, whose open-system thermodynamics is what lets a creation process produce negative pressure). n = 0 — A CONSTANT RATE PER FOUR-VOLUME — IS THE FOUNDING AXIOM OF CAUSAL-SET THEORY (Bombelli, Lee, Meyer & Sorkin 1987). Scanning n on DESI DR2 BAO + theta* + R: BEST FIT n = -0.10 AT DELTA CHI^2 = -6.31 AGAINST LambdaCDM; THE CAUSAL-SET VALUE n = 0 SITS 0.54 FROM THE OPTIMUM; THE FRAMEWORK'S POSTULATE n = 1 SITS 120 AWAY. At n near 0 the model BEATS LambdaCDM, and the preference STRENGTHENS when the CMB shift parameter R is added — R being the dataset on which this framework's own model was worst — and it satisfies the holographic bound over the whole history before z = 0.770. AN UNPLANNED CONTROL PASSED: n = 1 returns +46.04 on BAO alone, reproducing the archive's long-standing "+46" for constant w = -3/4 to the second decimal, computed years earlier by a different route. THIS IS NOT CLAIMED AS EVIDENCE AND THE PROVENANCE IS NOT OPTIONAL. The mechanism was selected after the present fits were known, two were tried and one landed, and it contradicts §31's pre-registered definition of a committed bit as a first shell crossing — §57.1 tests whether they could be the same law and shows they cannot, because a first-crossing count saturates and a Gamma proportional to H^n law never does, so no n reconciles them. THE DECISION RECORDED IN THIS RELEASE IS TO KEEP §31 WITH ITS ADVERSE RESULT AND TO REGISTER THE ALTERNATIVE SEPARATELY, with predictions for data that do not yet exist: w0 = -0.937, a w = -1 crossing at z = 0.348, CPL (-0.933, -0.269). Roman will measure w0 far better than the 0.06 that separates them. WITHDRAWN IN THIS VERSION, each with its reason in CHANGELOG_v7.md and 02_RESULTS/CORRECTIONS.md (§§1-61): the joint Delta chi^2 = 15.58 of earlier versions, in full; the claimed S8 resolution, which the manuscript's own page 86 states as the paper's primary observational signal; the parameter-economy argument; the coupled extension's -7.32; constant w = -3/4 as the model (it is the attractor that fixes lambda, and DESI DR2 excludes it as a constant at 9-11 sigma); the "Proven" derivation of mu(T); the laboratory null as a test of the cosmology; the 40Ca+ and single-proton protocols; Lemma L4's content; and several smaller claims. The audit's own errors are recorded with the same weight: §§43, 45, 46.2, 48 and 50.3 withdraw results computed during it, and §26 withdraws the claim that general relativity has zero postulates — it rests on the equivalence principle, every theory has postulates, and the framework's honest position is zero fitted parameters and one un-derived postulate. NEGATIVE RESULTS REPORTED IN FULL, among them: the literal holographic identification fails (44.8 sigma); identifying the routing count with the horizon count at all epochs is excluded (17.6 sigma); the CPL-artefact explanation of the DESI signal is excluded (1.1 % of the conflict is projection, and that 1.1 % is a distance ratio, not an accuracy on w(z), which is 1.4 %); a saturating-commitment variant is disfavoured at 4.1 sigma, and §37 shows that figure is a LOWER bound on the tension, not an upper one; and the thermodynamic-stability property, once offered as a point in the framework's favour, does not distinguish it — the two best-fitting dynamical models are on the same side of the bifurcation. NOT CLAIMED. No resolution of the S8 or Hubble tensions; the dark-matter sector is a re-description of CDM. THE FRAMEWORK AS PRE-REGISTERED IS NOT PREFERRED OVER LambdaCDM BY ANY STATISTIC HERE: its own model is disfavoured on all four scopes. The one figure in this release that does beat LambdaCDM — the creation-law scan at n near 0 — is a post-hoc selection that contradicts the pre-registered definition of a committed bit, and it is reported above as a measurement of the creation law and explicitly not as evidence for the framework. Every quantitative statement maps to a named script with a reference output. FRONT MATTER CORRECTED 27 SEPTEMBER 2026 (§58), in a referee-eye pass immediately before deposit. Four defects, all in prose and none in an engine, which is §30.4 repeating for the fourth time. (i) 00_README_v7.md said "294 files" while its own directory table summed to 268 and the archive holds 293; 294 counted the working directory, which carries one leftover test file the package excludes. All rows were recounted from the zip, and there are 119 analysis scripts, not 109. (ii) The README and 01_THEORY_OVERVIEW.md §5.2 still presented alpha = 0.745 as the measured routing exponent after §51 withdrew it — the uniform row IS its control and it fails, BMR requiring 0.750 — so the deposit and the archive disagreed about the framework's strongest result. Corrected in all four places; the withdrawn numbers and the failing control table stay on the record with their reason. (iii) 01_THEORY_OVERVIEW.md's status banner covered §§17-34 only, and its §2.2 — the subsection §35 is entirely about — carried no notice; the banner is rewritten to §57 with inline notices at §2.2 and §5.2. (iv) CHANGELOG_v7.md stopped at §38 and still quoted the withdrawn "0.03 %" headline; an addendum covering §§39-57 was added. §58.5 records what the pass did NOT find, including one contradiction the reviewer thought he had found and had not, and §58.6 states the standing lesson: engines have controls and prose has none. ATTRIBUTION CORRECTED 30 SEPTEMBER 2026 (§59), and both corrections went against this archive's own framing of a second author's position. The Penning registration's §0 had said J. Pascher stated the FFGFT null "before being asked to commit" — he was asked, directly, and he answered; what he volunteered unprompted was the preference for stating it in advance rather than reading it backwards afterwards, and that is now the form the credit takes. §0 had also described the correspondence as a "public record" taken "with permission", when the letters are private and no permission had been sought. Permission was then sought, naming the two passages exactly and offering paraphrase as the alternative, and granted in writing on 27 September 2026. He granted a broader standing permission as well; THAT GRANT IS NOT RELIED ON, because his documents are public and his letters are not, and the registration uses the permission only for the two passages it quotes. He asked to read the full text before it went up, and it was sent. §5's asymmetry-of-exposure section was put to him and accepted without softening. §59.4 records the form in which the granting message arrived, because the provenance of a permission is part of the quotation. No number, prediction or falsification criterion changed. §60 records what he did after the registration was deposited: he read the full text, checked the §0 quotations against his own letters, and INDEPENDENTLY REPRODUCED EVERY NUMBER IN §3 OF THE REGISTRATION — delta, the 1/B dependence, the ln2/ln(1+1/nbar) ceiling and the table rows. He also refined one sentence describing HIS OWN framework, which §60.2 records verbatim; the deposited file is left byte-identical in both records rather than forked, so that no two files share a name and differ in content. GREA READ, AND IT IS THE STRONGEST EXTERNAL RESULT AGAINST THIS FRAMEWORK'S POSITION (§61, added 30 September 2026 before deposit). General Relativistic Entropic Acceleration (Calderon & Garcia-Bellido, arXiv:2509.21491; D'Onofrio, Lee, Di Valentino & Garcia-Bellido, arXiv:2607.25841) derives late-time acceleration from the ENTROPY GROWTH OF THE CAUSAL COSMOLOGICAL HORIZON, with no cosmological constant, through an entropic force behaving as a bulk viscosity with negative effective pressure — the same open-system thermodynamics this archive named in §54. It is therefore a close relative of the Gamma = gamma H^n family, and the member of it that has been implemented in CLASS and confronted with the full likelihoods. ITS PHENOMENOLOGY IS FIXED BY ONE O(1) PARAMETER, alpha, WHICH THE THEORY PREDICTS TO BE ABOUT 1 — AND THE DATA CONFIRM IT: alpha = 0.967 +/- 0.041 to 1.079 +/- 0.026 across nine dataset combinations, with a fit matching LambdaCDM to within |Delta chi^2| <~ 6 despite one extra parameter. BOTH FRAMEWORKS MAKE THE SAME KIND OF CLAIM — A PARAMETER PREDICTED RATHER THAN FITTED, FROM A THERMODYNAMIC OR INFORMATIONAL PRINCIPLE. ONE OF THEM HAS HAD ITS PREDICTION CONFIRMED, AND IT IS NOT THIS ONE: lambda = 0.8660 sits at +1.77 sigma and +0.97 sigma against the two available fits. Worse for the registered branch: GREA's (w0, wa) = (about -1, about -0.3), with a first phantom crossing at z about 0.3, lands on top of this archive's POST-HOC causal-set branch (-0.937, -0.269, crossing z = 0.348) and far from its PRE-REGISTERED one (-0.8876, -0.1509, no crossing). That convergence is not evidence for anything — the post-hoc branch was selected after the fits were known and GREA was fitted — but it does mean the region the data prefer is occupied by more than one entropic model and the registered prediction is outside it. FAIR TO BOTH: GREA carries one free parameter where this framework fits none; it is formulated in an open universe and its alpha is defined through a curvature scale; it does NOT relieve the S8 tension either, and its authors say so against their own background-level expectation; and Bayesian model comparison still tends to prefer the cosmological constant once compressed CMB distance priors are included. NOTHING IN THIS RELEASE IS REVISED BECAUSE OF IT — revising a registered prediction because a competitor fits better is what pre-registration exists to prevent. What is owed, and stated as owed: GREA's likelihood has not been run against this archive's own BAO + theta* + R setup, so no head-to-head comparison on identical data exists. Their code is public. That calculation is now the highest-value unattempted item in the project. CONTENTS. The package is UIFT_v7_FINAL_2026-09-30.zip — 293 files, md5 b4603347d1435032762d518268796a50, and no .pyc files — which preserves the directory structure Zenodo cannot represent: 01_MANUSCRIPT/ (10), 02_RESULTS/ (88), 03_SCRIPTS/ (119 analysis scripts plus 00_SCRIPT_STATUS.md), 04_VERIFICATION_LOGS/ (50), 05_EXPERIMENT/ (7), 06_FIGURES/ (7), 07_WORKING_NOTES_TR/ (7), and four top-level files. Eleven entry-point files are also deposited loose so they can be read in the browser without downloading anything: 00_README_v7.md (start here), 01_THEORY_OVERVIEW.md, CORRECTIONS.md (§§1-61), CHANGELOG_v7.md, STATE_OF_THE_THEORY_2026-09-25.md, THE_25_SEPTEMBER_FINDINGS.md, THE_OPEN_QUESTIONS_2026-09-25.md, PAPER_magnitude_and_its_cost.md, PREREGISTRATION_penning_2026-09-26.md, ERRATA_READ_BEFORE_THE_PDF.md, and the manuscript TEKER_Theory_v7.pdf. All eleven are byte-identical to their copies inside the zip. Because the framework changed under several of the older documents, two index files say where each one stands: 02_RESULTS/00_STATUS_OF_EACH_DOCUMENT.md and 03_SCRIPTS/00_SCRIPT_STATUS.md. READ THE ERRATA BEFORE THE MANUSCRIPT PDF. TEKER_Theory_v7.pdf is a historical object: it predates every correction above, and ERRATA_READ_BEFORE_THE_PDF.md locates each affected passage by page. The description of this record as published on 6 September 2026 is superseded; its text is preserved in CHANGELOG_v7.md and in 02_RESULTS/CORRECTIONS.md, and version 6 remains available as its own Zenodo record.