Hodge: source and evidence

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1 \chapter{What the exact replay verifies}
2 The purpose of the evidence bundle is to make each finite claim independently inspectable. The final result is not a single undifferentiated pass flag. It is a collection of named inputs, arithmetic domains, frozen certificates, execution records and logical boundaries.
3 
4 The proof-critical September verifiers use integers, rational numbers and exact symbolic polynomial rings. Discovery used bounded enumeration and optimization where helpful. The discovered certificates were accepted only after exact replay. A floating-point optimizer's successful status is not the final proof object.
5 
6 \section{The current finite contract}
7 \begin{longtable}{p{0.25\linewidth}p{0.22\linewidth}p{0.43\linewidth}}
8 \toprule
9 Evidence layer & Fresh result & Mathematical boundary\\
10 \midrule
11 Degree-70 carrier & 24 grouped checks passed & Matrix, character, weighted degree, coordinate strata and radical ideal identities. The geometry uses classical inputs.\\
12 Degree-210 transfer & 46 grouped checks passed & All target/source conjugates, signed identity, standard blocks, stabilizing pairs and dual premise.\\
13 Capacity and cycles & 137 grouped assertions passed & Seven exact capacities, symbolic determinant controls and the specified cycle support range.\\
14 Five-phase supports & 241,920 monomials & All $b$, all first-five permutations, and $2\le q\le64$. Distinct from all Fermat packets.\\
15 Five-phase balance & 218,836 instances & The retained finite parameter and unit-conjugacy range. Uniformity uses a separate proof.\\
16 K23 reconstruction & 364 certificates & Generated independently from every valid phase pair and ordered unit conjugate of seven retained inputs.\\
17 K23 ages & 8,336 comparisons & Exact Hodge-age and opposite-CM-sign checks on those regenerated certificates.\\
18 Dyadic census & Moduli $8,16,32,64,128$ & The complete finite packet census and implemented grammar at those moduli.\\
19 Dyadic weighted identity & 87,376 coefficients & Fresh range $3\le s\le9$. The historical larger range was not fully replayed.\\
20 Weil exterior algebra & 924 basis vectors & Rational projector, intersection pairing and Poincaré-dual identities. No cycle-space census.\\
21 Projector toy models & 18 tolerance checks & Floating-point illustrations with declared tolerances, and not exact variety-level certificates.\\
22 \bottomrule
23 \end{longtable}
24 
25 These counts are grouped assertions or enumerated instances. They are not counts of independent geometric theorems. Several assertions check different consequences of one frozen certificate, which is appropriate for detecting implementation errors and should not be inflated into independent proof multiplicity.
26 
27 \section{What changed in the K23 replay}
28 The archived K23 verifier read the retained certificate JSON and checked its fields. The October audit independently rebuilt the phase grammar from the original rational packet addresses. For each unit conjugate it enumerated valid complementary mod-5 phase pairs, checked the nonzero residual entries, formed the curve, surface and lower-curve blocks, and compared Hodge ages.
29 
30 The resulting counts by retained packet were
31 \[
32 40,\quad60,\quad48,\quad48,\quad48,\quad48,\quad72,
33 \]
34 which total 364. This stronger bounded replay guards against a certificate file whose generator omitted or misclassified a valid case. It still covers only the seven declared inputs. It neither proves the universal mixed-conductor grammar nor constructs a Chow correspondence.
35 
36 \section{Why the degree-210 dual is global}
37 The dual uses numerator vector $(31,87,57,101,84,60)$ and denominator 420. For invariant nonnegative columns of degrees 1 through 13, the exact enumerator checks all 27,131 compositions. Exactly 143 satisfy the congruence, and each has dual numerator at least 420.
38 
39 For degree at least 14, every coordinate cost is at least 31, so the numerator is at least $14\cdot31=434>420$. Thus the finite premise and the uniform tail cover every nonzero invariant column. The global impossibility statement is proved by this argument, rather than extrapolated from the degree-18 search.
40 
41 A separate residue-graph calculation finds the same minimum nonempty return cost 420. The independent implementation helps detect an enumeration mistake. The direct finite inequalities and the tail bound remain the stated certificate.
42 
43 \section{Why the capacity certificates are global}
44 The seven capacities are proved by matching a feasible primal representation with an exact rational dual. An integer potential on the finite residue graph verifies every edge inequality and every last-edge lower bound. Any nonempty zero-sum word decomposes into a prefix and a last edge. Those inequalities therefore bound every invariant column, at arbitrary degree.
45 
46 The potential proof does not assume the correctness of the algorithm used to discover the potential. Once the finite inequalities are verified, its argument is a finite sum. The primal representation proves equality at the claimed value. A relaxed capacity optimizer may use fewer than six columns, so equality of capacity does not supply a nonsingular carrier matrix or normal projective geometry.
47 
48 \section{Uniform formula versus finite benchmark}
49 The five-phase determinant formula is proved by wedge products and polynomial identities for every positive integer parameter in its stated range. Finite determinants at $q=2,3,4,5,7,11,17$ independently test the implementation and row sign. Socle pairing is separately checked for every $b$ at $q=2,7,11$.
50 
51 The dyadic identity similarly has a displayed coefficient derivation. Its fresh finite replay checks a bounded range. The later all-degree tree classification is a distinct manuscript proof. Selected independent reconstruction in this edition found no concrete contradiction in its basis/tree/coarea argument once the odd-character nonvanishing leaf was authenticated. That selected review does not silently reverse the August 15 publication-status downgrade. A full uniform write-up and correspondence-level account remain explicit obligations.
52 
53 \section{Reproduction commands and arithmetic}
54 The evidence directory includes a portable driver for the bounded replay. Its manifests declare inputs, expected mathematical values, code hashes and arithmetic domains. The driver was tested. The latest exact verifiers need the Python standard library for the transfer and exact symbolic dependencies for the cycle/gradient calculations.
55 
56 The preserved discovery scripts can take different times, produce a different valid support, or fail to terminate within the same time allowance on another solver version. None of these outcomes invalidates replay of the retained frozen identity. A newly discovered identity requires its own exact residue equality, balance checks and geometric-factor proof.
57 
58 Replaying a report changes its execution timestamp and output hash. It does not change the frozen character input, matrix, polynomial identity or expected rational value. File checksums identify the exact artifacts used in each recorded run, and the bundle keeps historical sources separate from current replay outputs.
59 
60 \chapter{Corrections, failures and retractions}
61 Research progress here includes removing false implications. The negative records identify a failed method or inference, and they do not become counterexamples to the Hodge conjecture.
62 
63 \section{A positive compressed Gram matrix is insufficient}
64 The original August cycle-frame theorem asserted that positivity of a compressed frame on a target subspace implied that the cycles spanned that subspace. Without containment or a suitable invariant restriction, this is false.
65 
66 Take $V=\Span_{\R}\{e_1\}$ in $\R^2$ and let the only cycle vector be $c=e_1+e_2$. Then
67 \[
68 S=cc^t=\begin{pmatrix}1&1\\1&1\end{pmatrix}.
69 \]
70 The compressed Gram matrix on $V$ is the positive $1\times1$ matrix $(1)$, while $e_1$ does not belong to $\Span(c)$. Positivity proves that the target direction is detected, not that synthesis returns it inside the original cycle span.
71 
72 The corrected theorem requires $c_j\in V$, obtainable through an explicitly algebraic rational packet projector when available. Without that condition, the exact containment test is
73 \[
74 \rank_{\Q}[C\ H_{\Q}]=\rank_{\Q}C.
75 \]
76 A rational target basis $H_{\Q}$ is distinct from an orthonormal real basis, since orthonormalization can introduce irrational entries. The old source is preserved unchanged and this correction governs the current reader.
77 
78 \section{The universal direct matrix constructor is false}
79 The invariant-column positive-anticanonical matrix criterion was a useful search target. It is not a universal theorem for balanced Fermat packets. The infinite five-phase family has capacity exactly one for the stated coprime parameters, so no matrix of that type can have normalized weight sum greater than one.
80 
81 The degree-210 target has its own exact capacity-one dual. Yet the signed transfer makes it algebraic under the degree-70 carrier proof and classical operations. Thus positive transfer and negative matrix obstruction coexist on the same packet. Universal applicability of the direct constructor should be removed from a plausible-gap list and replaced by the proved method limitation.
82 
83 \section{A parameterized weight sum did not identify a new packet family}
84 An exploratory modification replaced the exponent 7 in the old carrier matrix by a parameter. The formal normalized weights remained positive and summed to $1+7/(10q)$. That calculation alone did not verify a desired full-support character on the projective quotient. It established no uniform new primitive-packet family.
85 
86 The missing step was character covariance and deck invariance for the claimed target, together with normality and boundary control for the resulting family. The positivity calculation is retained as exploration and is not promoted to a Hodge theorem.
87 
88 \section{An invalid exploratory input}
89 The exploratory modulus-45 tuple
90 \[
91 (1,10,19,28,36,37)
92 \]
93 has sum 131, whereas a middle-balanced six-tuple at that modulus must sum 135. It is not even an admissible starting Hodge character. Its reported candidate capacity $34/45$ is therefore not a Hodge result.
94 
95 The invalid record remains in the historical discovery evidence and is excluded from the seven authoritative capacity inputs. The exact verifier checks every unit-conjugate sum before accepting a capacity claim. The general lower bound $C(a)\ge1$ for a balanced character also flags the erroneous subunit value.
96 
97 \section{Bounded search failures are inconclusive}
98 The atomic chain/loop family failed to find the degree-70 carrier, and an early Hermite-normal-form membership experiment exceeded its 20-second allowance without a conclusion. The later bounded mixed-integer discovery succeeded and was followed by exact replay.
99 
100 Bounded degree-18 matrix scans failed on selected targets. Those failures alone do not prove global infeasibility. Only the targets covered by the exact dual certificate or the symbolic infinite-family proof have the stated global obstruction.
101 
102 \section{The five-phase cycles were already known}
103 The explicit $Z_\kappa$ equations recover the Aoki standard family. The October source audit obtained the original article and checked that its elementary-symmetric construction becomes these equations through Newton identities. A direct determinant and period recomputation is valuable reproducible work, and is not a newly solved open Hodge family.
104 
105 The earlier retrieval failure is preserved in the archive. It no longer justifies leaving the equations' historical overlap unspecified in this edition. No priority is claimed for the construction, algebraicity or determinant.
106 
107 \section{The Weil graph failure was defeated}
108 A first-order obstruction to deforming the original graph does not bound every representative of the same class. The exact audit found replacement divisor factors along the selected mixed direction. Those factors defeat the tested counterexample path.
109 
110 No nonalgebraicity was proved. The abstract nilpotent was not realized as an algebraic semistable degeneration, and a pointwise tangent rank for one graph was not extended to all cycle-space components. These are distinct missing premises, and none is hidden behind the pass status of the exterior algebra.
111 
112 \section{Formal source inspection is not a compiler pass}
113 The current Lean attempt was blocked before elaboration. The restored compiler reported that it failed to locate its application, and the Mathlib files were unavailable. No fresh Lean pass is claimed.
114 
115 Even a successful compilation of the preserved source would verify only its definitions and hypotheses. The final theorem assumes the algebraic seed and closure rules through an abstract predicate. It does not prove those geometric hypotheses. The current report separates finite arithmetic scripts, assumed closure operations and absent Chow realizations.
116 
117 \chapter{The live research obligations}
118 The general Hodge conjecture remains open in this reader. The different missing implications are stated separately because solving one would not automatically solve the others.
119 
120 \section{Arbitrary smooth projective varieties}
121 The real obstruction projector vanishes exactly when the algebraic and rational Hodge spaces agree. It provides no algorithm for computing the complete algebraic space. A finite cycle family, actual cohomology coordinates and an exact spanning certificate would close a specified $(X,p)$ case.
122 
123 An alternative is an explicitly algebraic divisor-lowering transfer with proved packet preservation and invertibility. Its polarized adjoint is an additional algebraicity hypothesis. The cohomological transpose under a chosen metric cannot simply be declared a Chow correspondence.
124 
125 The analytic branch requires vanishing of the complete finite obstruction $\Phi$, after the proposed stabilization. The range contraction solves only the range equation. Hodge-type Chern characters or trace moments do not force an endomorphism-valued defect to vanish.
126 
127 \section{Uniform Fermat grammar coverage}
128 The exact balanced residue equation is available at every degree. The present constructive grammar admits many packets, and finite audits describe the retained degree range. A proof that every full-conductor balanced packet is admitted by one of its channels is still required.
129 
130 Mixed-conductor Kummer patterns observed in a finite table cannot supply that universal premise. A correctly typed nonzero CM map for every currently admitted K23 case would solve those cases. It would not prove that arbitrary residual packets have that K23 form.
131 
132 \section{The typed Chow bridge}
133 Before stating a rank-one isomorphism, fix the coefficient field, the diagonal character projector, the rational Galois orbit and any multiplicities caused by conductor reduction. Construct an actual algebraic kernel and prove a nonzero period or intersection coefficient.
134 
135 Matching CM types is an identification at Hodge level. The rational tensor can contain off-diagonal embeddings and need not have the dimension of the lower curve factor. These typing requirements belong to the target, rather than editorial details after a claimed proof.
136 
137 \section{The special constructions}
138 The degree-70 carrier argument and degree-33 torus/cubic argument remain manuscript geometric constructions. Their finite identities and classical leaves have been checked at the stated scope. A submission-ready treatment should give the resolved graphs, full-support boundary action and rational Hodge lifting with complete notation.
139 
140 The degree-210 signed transfer has an exact certificate and uses checked classical operations. Its algebraicity conclusion inherits the degree-70 geometric proposition. The actual period proof for the five-phase surfaces is more explicit, and reconstructs a known standard family. Neither construction exhausts the Fermat packet space.
141 
142 \section{The Weil family}
143 The remaining sixfold question asks for one algebraic threefold with a nonzero pair of Weil periods at each fully mixed member under consideration. The algebraic projector and isogeny action then generate both rational directions.
144 
145 The seed graph, product locus and replacement factorization demonstrate several sources. Their failure to cover every member is an open coverage issue. A counterexample would require proving that all algebraic threefolds have zero Weil periods at a specified member. No such universal exclusion is obtained here.

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